discrete mathematics i - computer science departmenttiskin/teach/dm1/o.pdf · discrete maths in...
TRANSCRIPT
![Page 1: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1.jpg)
Discrete Mathematics ICS127
Lecturer: Dr. Alex Tiskinhttp://www.dcs.warwick.ac.uk/˜tiskin
Department of Computer Science
University of Warwick
Discrete Mathematics I – p. 1/292
![Page 2: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/2.jpg)
Discrete Mathematics IMathematics relevant to computer scienceUsed in other CS courses
29 lectures in Autumn TermWeekly problem sheets, seminars from week 2 —please sign up!
Website: http://www.dcs.warwick.ac.uk/~tiskin/teach/dm1.html
Seminar signup open now
Discrete Mathematics I – p. 2/292
![Page 3: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/3.jpg)
Discrete Mathematics IClass test in week 7 (new from 2002/03!)
Exam in week 1 of Summer TermResults at the end of Summer Term
Discrete Mathematics I – p. 3/292
![Page 4: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/4.jpg)
Discrete Mathematics ILecture notes available at lectures
Website: http://www.dcs.warwick.ac.uk/~tiskin/teach/dm1.html
Forum: http://forums.warwick.ac.uk, thenclick on Departments > Computer Science> UGyear1 > CS127
Please participate!
Discrete Mathematics I – p. 4/292
![Page 5: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/5.jpg)
Discrete Mathematics IDiscrete Maths II — a Summer Term optionLecturer: Dr. Mike Joy
Discrete maths in depth, highly recommended!
Discrete Mathematics I – p. 5/292
![Page 6: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/6.jpg)
Discrete Mathematics IRecommended books:
Discrete MathematicsRoss and Wright (Prentice Hall, 2003)
Discrete Mathematics and its ApplicationsRosen (McGraw-Hill, 2003)
Discrete Mathematics for Computer ScientistsTruss (Addison-Wesley, 1999)
Discrete Mathematics I – p. 6/292
![Page 7: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/7.jpg)
Discrete Mathematics IWhich is the best?
• Ross and Wright: the most helpful. . .
• Rosen: the most interesting. . .
• Truss: the most advanced. . .
Hundreds more, the choice is yours!
Discrete Mathematics I – p. 7/292
![Page 8: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/8.jpg)
Discrete Mathematics IWhich is the best?
• Ross and Wright: the most helpful. . .
• Rosen: the most interesting. . .
• Truss: the most advanced. . .
Hundreds more, the choice is yours!
Discrete Mathematics I – p. 7/292
![Page 9: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/9.jpg)
Discrete Mathematics IWhich is the best?
• Ross and Wright: the most helpful. . .
• Rosen: the most interesting. . .
• Truss: the most advanced. . .
Hundreds more, the choice is yours!
Discrete Mathematics I – p. 7/292
![Page 10: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/10.jpg)
Discrete Mathematics IWhich is the best?
• Ross and Wright: the most helpful. . .
• Rosen: the most interesting. . .
• Truss: the most advanced. . .
Hundreds more, the choice is yours!
Discrete Mathematics I – p. 7/292
![Page 11: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/11.jpg)
Discrete Mathematics IWhich is the best?
• Ross and Wright: the most helpful. . .
• Rosen: the most interesting. . .
• Truss: the most advanced. . .
Hundreds more, the choice is yours!
Discrete Mathematics I – p. 7/292
![Page 12: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/12.jpg)
Discrete Mathematics IAlso:
Proofs and Fundamentals: a First Course in AbstractMathematicsBloch (Birkhäuser, 2002)
Does not cover whole course, but helps with proofs
Discrete Mathematics I – p. 8/292
![Page 13: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/13.jpg)
Discrete Mathematics IAlso:
Proofs and Fundamentals: a First Course in AbstractMathematicsBloch (Birkhäuser, 2002)
Does not cover whole course, but helps with proofs
Discrete Mathematics I – p. 8/292
![Page 14: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/14.jpg)
Introduction
Discrete Mathematics I – p. 9/292
![Page 15: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/15.jpg)
IntroductionMathematics:
the science of abstraction
Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, . . .What does “5” mean?
Five apples, five hats, five lottery numbers. . .Abstraction of all five-element sets
What set does 0 represent? The empty set
Discrete Mathematics I – p. 10/292
![Page 16: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/16.jpg)
IntroductionMathematics: the science of abstraction
Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, . . .What does “5” mean?
Five apples, five hats, five lottery numbers. . .Abstraction of all five-element sets
What set does 0 represent? The empty set
Discrete Mathematics I – p. 10/292
![Page 17: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/17.jpg)
IntroductionMathematics: the science of abstraction
Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, . . .What does “5” mean?
Five apples, five hats, five lottery numbers. . .Abstraction of all five-element sets
What set does 0 represent? The empty set
Discrete Mathematics I – p. 10/292
![Page 18: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/18.jpg)
IntroductionMathematics: the science of abstraction
Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, . . .What does “5” mean?
Five apples, five hats, five lottery numbers. . .Abstraction of all five-element sets
What set does 0 represent? The empty set
Discrete Mathematics I – p. 10/292
![Page 19: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/19.jpg)
IntroductionMathematics: the science of abstraction
Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, . . .What does “5” mean?
Five apples, five hats, five lottery numbers. . .Abstraction of all five-element sets
What set does 0 represent?
The empty set
Discrete Mathematics I – p. 10/292
![Page 20: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/20.jpg)
IntroductionMathematics: the science of abstraction
Natural numbers: 0, 1, 2, 3, 4, 5, 6, 7, . . .What does “5” mean?
Five apples, five hats, five lottery numbers. . .Abstraction of all five-element sets
What set does 0 represent? The empty set
Discrete Mathematics I – p. 10/292
![Page 21: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/21.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 22: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/22.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 23: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/23.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}
{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 24: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/24.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 25: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/25.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 26: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/26.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 27: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/27.jpg)
IntroductionA set is any collection of elements
{Peter Pan, Gingerbread Man, Wrestling Fan}
{♠,♥,♣,♦}{0, 1, 2, 3, 4, 5, 6, 7, . . . } = N natural numbers
{0, 2, 4, 6, 8, 10, 12, 14, . . . } = Neven even naturals
{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
First two sets finite, last three infinite
Discrete Mathematics I – p. 11/292
![Page 28: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/28.jpg)
IntroductionThe empty set: {} = ∅Plays a role for sets similar to 0 for numbers
Discrete Mathematics I – p. 12/292
![Page 29: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/29.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 30: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/30.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?
Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 31: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/31.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 32: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/32.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?
Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 33: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/33.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 34: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/34.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?
No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 35: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/35.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 36: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/36.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why?
It can be proved!
Discrete Mathematics I – p. 13/292
![Page 37: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/37.jpg)
IntroductionSome “tough” questions:
Do infinite sets have “sizes”?Yes, but these are beyond natural numbers
Can infinite sets have different sizes?Yes, a huge variety of possible sizes
Is there a set of all sets?No! Not even a set of all (infinite) set sizes
Why? It can be proved!
Discrete Mathematics I – p. 13/292
![Page 38: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/38.jpg)
IntroductionNatural sciences are based on evidence
Mathematics is based on proof(but evidence helps to understand proofs)
Discrete Mathematics I – p. 14/292
![Page 39: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/39.jpg)
IntroductionNatural sciences are based on evidence
Mathematics is based on proof(but evidence helps to understand proofs)
Discrete Mathematics I – p. 14/292
![Page 40: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/40.jpg)
IntroductionTo write proofs, we need a special language:
• precise (unambiguous)
• concise (clear and relatively brief)
The “grammar” of this language is logic
Discrete Mathematics I – p. 15/292
![Page 41: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/41.jpg)
IntroductionAll eagles can flySome pigs cannot fly
Therefore, some pigs are not eagles
Proof.
Consider all creatures. If it is an eagle, it can fly.Hence, if it cannot fly, it is not an eagle.
There is a creature that is a pig and cannot fly.By the above statement, it is not an eagle.
Discrete Mathematics I – p. 16/292
![Page 42: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/42.jpg)
IntroductionAll eagles can flySome pigs cannot fly
Therefore, some pigs are not eagles
Proof.
Consider all creatures. If it is an eagle, it can fly.Hence, if it cannot fly, it is not an eagle.
There is a creature that is a pig and cannot fly.By the above statement, it is not an eagle.
Discrete Mathematics I – p. 16/292
![Page 43: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/43.jpg)
IntroductionAll eagles can flySome pigs cannot fly
Therefore, some pigs are not eagles
Proof.
Consider all creatures. If it is an eagle, it can fly.Hence, if it cannot fly, it is not an eagle.
There is a creature that is a pig and cannot fly.By the above statement, it is not an eagle.
Discrete Mathematics I – p. 16/292
![Page 44: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/44.jpg)
IntroductionThe same in mathematical notation:
If for all x, eagle(x) ⇒ canfly(x),and for some y, pig(y) ∧ ¬canfly(y),
then for some z, pig(z) ∧ ¬eagle(z)
Proof. Consider all x ∈ Creatures .
By first condition, ¬canfly(x) ⇒ ¬eagle(x).
Take any y such that pig(y) ∧ ¬canfly(y).By the above, we have pig(y) ∧ ¬eagle(y).Take z = y.
Discrete Mathematics I – p. 17/292
![Page 45: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/45.jpg)
IntroductionSome concepts are basic, i.e. need no definitionExamples: set, natural number
All other concepts must be definedExamples: finite set, even number
Some facts are axioms, i.e. need no proofExample: equal sets have the same elements
All other facts must be provedExample: the set of even numbers is infinite
This is the axiomatic method
Discrete Mathematics I – p. 18/292
![Page 46: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/46.jpg)
IntroductionSome concepts are basic, i.e. need no definitionExamples: set, natural number
All other concepts must be definedExamples: finite set, even number
Some facts are axioms, i.e. need no proofExample: equal sets have the same elements
All other facts must be provedExample: the set of even numbers is infinite
This is the axiomatic method
Discrete Mathematics I – p. 18/292
![Page 47: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/47.jpg)
IntroductionSome concepts are basic, i.e. need no definitionExamples: set, natural number
All other concepts must be definedExamples: finite set, even number
Some facts are axioms, i.e. need no proofExample: equal sets have the same elements
All other facts must be provedExample: the set of even numbers is infinite
This is the axiomatic method
Discrete Mathematics I – p. 18/292
![Page 48: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/48.jpg)
IntroductionSome concepts are basic, i.e. need no definitionExamples: set, natural number
All other concepts must be definedExamples: finite set, even number
Some facts are axioms, i.e. need no proofExample: equal sets have the same elements
All other facts must be provedExample: the set of even numbers is infinite
This is the axiomatic method
Discrete Mathematics I – p. 18/292
![Page 49: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/49.jpg)
IntroductionSome concepts are basic, i.e. need no definitionExamples: set, natural number
All other concepts must be definedExamples: finite set, even number
Some facts are axioms, i.e. need no proofExample: equal sets have the same elements
All other facts must be provedExample: the set of even numbers is infinite
This is the axiomatic method
Discrete Mathematics I – p. 18/292
![Page 50: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/50.jpg)
IntroductionCourse structure:
• Logic
• Sets
• More fun: relations, functions, graphs
Any questions?
Discrete Mathematics I – p. 19/292
![Page 51: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/51.jpg)
IntroductionCourse structure:
• Logic
• Sets
• More fun: relations, functions, graphs
Any questions?
Discrete Mathematics I – p. 19/292
![Page 52: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/52.jpg)
Logic
Discrete Mathematics I – p. 20/292
![Page 53: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/53.jpg)
Logic
In everyday life, we use all sorts of sentences:
Five is less than ten. Welcome to Tweedy’s farm!Pigs can fly. What’s in the pies?There is life on Mars. It’s not as bad as it looks. . .
A statement is a sentence that is either true or false(but not both!).
Discrete Mathematics I – p. 21/292
![Page 54: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/54.jpg)
Logic
In everyday life, we use all sorts of sentences:
Five is less than ten.
Welcome to Tweedy’s farm!
Pigs can fly.
What’s in the pies?
There is life on Mars.
It’s not as bad as it looks. . .
A statement is a sentence that is either true or false(but not both!).
Discrete Mathematics I – p. 21/292
![Page 55: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/55.jpg)
Logic
In everyday life, we use all sorts of sentences:
Five is less than ten. Welcome to Tweedy’s farm!Pigs can fly. What’s in the pies?There is life on Mars. It’s not as bad as it looks. . .
A statement is a sentence that is either true or false(but not both!).
Discrete Mathematics I – p. 21/292
![Page 56: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/56.jpg)
Logic
In everyday life, we use all sorts of sentences:
Five is less than ten. Welcome to Tweedy’s farm!Pigs can fly. What’s in the pies?There is life on Mars. It’s not as bad as it looks. . .
A statement is a sentence that is either true or false(but not both!).
Discrete Mathematics I – p. 21/292
![Page 57: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/57.jpg)
Logic
False and true are Boolean values: B = {F, T}
(After G. Boole, 1815–1864)
value(5 < 10) = T
value(“Pigs can fly”) = F
value(“It’s not as bad as it looks”) — ?
value(“The pie is not as bad as it looks”) = F
Discrete Mathematics I – p. 22/292
![Page 58: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/58.jpg)
Logic
False and true are Boolean values: B = {F, T}(After G. Boole, 1815–1864)
value(5 < 10) = T
value(“Pigs can fly”) = F
value(“It’s not as bad as it looks”) — ?
value(“The pie is not as bad as it looks”) = F
Discrete Mathematics I – p. 22/292
![Page 59: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/59.jpg)
Logic
False and true are Boolean values: B = {F, T}(After G. Boole, 1815–1864)
value(5 < 10) = T
value(“Pigs can fly”) = F
value(“It’s not as bad as it looks”) — ?
value(“The pie is not as bad as it looks”) = F
Discrete Mathematics I – p. 22/292
![Page 60: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/60.jpg)
Logic
False and true are Boolean values: B = {F, T}(After G. Boole, 1815–1864)
value(5 < 10) = T
value(“Pigs can fly”) = F
value(“It’s not as bad as it looks”) — ?
value(“The pie is not as bad as it looks”) = F
Discrete Mathematics I – p. 22/292
![Page 61: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/61.jpg)
Logic
False and true are Boolean values: B = {F, T}(After G. Boole, 1815–1864)
value(5 < 10) = T
value(“Pigs can fly”) = F
value(“It’s not as bad as it looks”) — ?
value(“The pie is not as bad as it looks”) = F
Discrete Mathematics I – p. 22/292
![Page 62: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/62.jpg)
Logic
Often need compound statements:
(5 < 10) AND (Pigs can fly)
. . . i.e. T AND F = F — similar e.g. to 3 + 4 = 7
Discrete Mathematics I – p. 23/292
![Page 63: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/63.jpg)
Logic
Often need compound statements:
(5 < 10) AND (Pigs can fly)
. . . i.e. T AND F = F — similar e.g. to 3 + 4 = 7
Discrete Mathematics I – p. 23/292
![Page 64: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/64.jpg)
Logic
Boolean operators on B:
¬A NOT A negationA ∧B A AND B conjunctionA ∨B A OR B disjunctionA ⇒ B IF A THEN B implicationA ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 65: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/65.jpg)
Logic
Boolean operators on B:
¬A NOT A negation
A ∧B A AND B conjunctionA ∨B A OR B disjunctionA ⇒ B IF A THEN B implicationA ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 66: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/66.jpg)
Logic
Boolean operators on B:
¬A NOT A negationA ∧B A AND B conjunction
A ∨B A OR B disjunctionA ⇒ B IF A THEN B implicationA ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 67: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/67.jpg)
Logic
Boolean operators on B:
¬A NOT A negationA ∧B A AND B conjunctionA ∨B A OR B disjunction
A ⇒ B IF A THEN B implicationA ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 68: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/68.jpg)
Logic
Boolean operators on B:
¬A NOT A negationA ∧B A AND B conjunctionA ∨B A OR B disjunctionA ⇒ B IF A THEN B implication
A ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 69: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/69.jpg)
Logic
Boolean operators on B:
¬A NOT A negationA ∧B A AND B conjunctionA ∨B A OR B disjunctionA ⇒ B IF A THEN B implicationA ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 70: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/70.jpg)
Logic
Boolean operators on B:
¬A NOT A negationA ∧B A AND B conjunctionA ∨B A OR B disjunctionA ⇒ B IF A THEN B implicationA ⇔ B A ⇒ B AND B ⇒ A equivalence
Definition: by truth tables
Discrete Mathematics I – p. 24/292
![Page 71: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/71.jpg)
Logic
Negation (NOT A): ¬A
True if A false, false if A true
A ¬A
F T
T F
Discrete Mathematics I – p. 25/292
![Page 72: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/72.jpg)
Logic
Negation (NOT A): ¬A
True if A false, false if A true
A ¬A
F T
T F
Discrete Mathematics I – p. 25/292
![Page 73: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/73.jpg)
Logic
Examples:
¬[5 < 10]
⇐⇒ ¬T ⇐⇒ F
¬[Pigs can fly] ⇐⇒ ¬F ⇐⇒ T
Discrete Mathematics I – p. 26/292
![Page 74: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/74.jpg)
Logic
Examples:
¬[5 < 10] ⇐⇒ ¬T
⇐⇒ F
¬[Pigs can fly] ⇐⇒ ¬F ⇐⇒ T
Discrete Mathematics I – p. 26/292
![Page 75: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/75.jpg)
Logic
Examples:
¬[5 < 10] ⇐⇒ ¬T ⇐⇒ F
¬[Pigs can fly] ⇐⇒ ¬F ⇐⇒ T
Discrete Mathematics I – p. 26/292
![Page 76: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/76.jpg)
Logic
Examples:
¬[5 < 10] ⇐⇒ ¬T ⇐⇒ F
¬[Pigs can fly]
⇐⇒ ¬F ⇐⇒ T
Discrete Mathematics I – p. 26/292
![Page 77: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/77.jpg)
Logic
Examples:
¬[5 < 10] ⇐⇒ ¬T ⇐⇒ F
¬[Pigs can fly] ⇐⇒ ¬F
⇐⇒ T
Discrete Mathematics I – p. 26/292
![Page 78: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/78.jpg)
Logic
Examples:
¬[5 < 10] ⇐⇒ ¬T ⇐⇒ F
¬[Pigs can fly] ⇐⇒ ¬F ⇐⇒ T
Discrete Mathematics I – p. 26/292
![Page 79: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/79.jpg)
Logic
Conjunction (A AND B): A ∧B
True if both A and B true
A B A ∧B
F F F
F T F
T F F
T T T
Discrete Mathematics I – p. 27/292
![Page 80: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/80.jpg)
Logic
Conjunction (A AND B): A ∧B
True if both A and B true
A B A ∧B
F F F
F T F
T F F
T T T
Discrete Mathematics I – p. 27/292
![Page 81: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/81.jpg)
Logic
Example:
[5 < 10] ∧ [Pigs can fly]
⇐⇒ T ∧ F ⇐⇒ F
Discrete Mathematics I – p. 28/292
![Page 82: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/82.jpg)
Logic
Example:
[5 < 10] ∧ [Pigs can fly] ⇐⇒ T ∧ F
⇐⇒ F
Discrete Mathematics I – p. 28/292
![Page 83: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/83.jpg)
Logic
Example:
[5 < 10] ∧ [Pigs can fly] ⇐⇒ T ∧ F ⇐⇒ F
Discrete Mathematics I – p. 28/292
![Page 84: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/84.jpg)
Logic
Disjunction (A OR B): A ∨B
True if either A or B true (or both)
A B A ∨B
F F F
F T T
T F T
T T T
Discrete Mathematics I – p. 29/292
![Page 85: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/85.jpg)
Logic
Disjunction (A OR B): A ∨B
True if either A or B true (or both)
A B A ∨B
F F F
F T T
T F T
T T T
Discrete Mathematics I – p. 29/292
![Page 86: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/86.jpg)
Logic
Example:
[5 < 10] ∨ [Pigs can fly]
⇐⇒ T ∨ F ⇐⇒ T
Discrete Mathematics I – p. 30/292
![Page 87: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/87.jpg)
Logic
Example:
[5 < 10] ∨ [Pigs can fly] ⇐⇒ T ∨ F
⇐⇒ T
Discrete Mathematics I – p. 30/292
![Page 88: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/88.jpg)
Logic
Example:
[5 < 10] ∨ [Pigs can fly] ⇐⇒ T ∨ F ⇐⇒ T
Discrete Mathematics I – p. 30/292
![Page 89: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/89.jpg)
Logic
Implication (IF A THEN B)
In everyday life, often ambiguous:
If the bird is happy, then it sings loud
Happy — definitely singsUnhappy — may or may not sing
Discrete Mathematics I – p. 31/292
![Page 90: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/90.jpg)
Logic
Implication (IF A THEN B)
In everyday life, often ambiguous:
If the bird is happy, then it sings loud
Happy — definitely singsUnhappy — may or may not sing
Discrete Mathematics I – p. 31/292
![Page 91: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/91.jpg)
Logic
Implication (IF A THEN B)
In everyday life, often ambiguous:
If the bird is happy, then it sings loud
Happy — definitely singsUnhappy — ?
may or may not sing
Discrete Mathematics I – p. 31/292
![Page 92: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/92.jpg)
Logic
Implication (IF A THEN B)
In everyday life, often ambiguous:
If the bird is happy, then it sings loud
Happy — definitely singsUnhappy — may or may not sing
Discrete Mathematics I – p. 31/292
![Page 93: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/93.jpg)
Logic
Implication (IF A THEN B): A ⇒ B
True if A false; true if B true; false otherwise
A B A ⇒ B
F F T
F T T
T F F
T T T
Everything implies truth; false implies anything
Discrete Mathematics I – p. 32/292
![Page 94: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/94.jpg)
Logic
Implication (IF A THEN B): A ⇒ B
True if A false; true if B true; false otherwise
A B A ⇒ B
F F T
F T T
T F F
T T T
Everything implies truth; false implies anything
Discrete Mathematics I – p. 32/292
![Page 95: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/95.jpg)
Logic
Implication (IF A THEN B): A ⇒ B
True if A false; true if B true; false otherwise
A B A ⇒ B
F F T
F T T
T F F
T T T
Everything implies truth; false implies anything
Discrete Mathematics I – p. 32/292
![Page 96: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/96.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 97: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/97.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ]
⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 98: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/98.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F
[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 99: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/99.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 100: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/100.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ]
⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 101: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/101.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T
[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 102: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/102.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 103: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/103.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ]
⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 104: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/104.jpg)
Logic
Examples:[
[5 < 10] ⇒ [Pigs fly]]
⇐⇒ [T ⇒ F ] ⇐⇒ F[
[Pigs fly] ⇒ [5 < 10]]
⇐⇒ [F ⇒ T ] ⇐⇒ T[
[Pigs fly] ⇒ [5 > 10]]
⇐⇒ [F ⇒ F ] ⇐⇒ T
Discrete Mathematics I – p. 33/292
![Page 105: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/105.jpg)
Logic
Example (by G. Hardy):2 + 2 = 5 =⇒ I am Count Dracula
“Proof”:2 + 2 = 5 =⇒ 4 = 5 =⇒ 5 = 4 =⇒ 2 = 1
Dracula and I are two =⇒Dracula and I are one
Discrete Mathematics I – p. 34/292
![Page 106: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/106.jpg)
Logic
Example (by G. Hardy):2 + 2 = 5 =⇒ I am Count Dracula
“Proof”:2 + 2 = 5 =⇒ 4 = 5 =⇒ 5 = 4 =⇒ 2 = 1
Dracula and I are two =⇒Dracula and I are one
Discrete Mathematics I – p. 34/292
![Page 107: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/107.jpg)
Logic
Example (by G. Hardy):2 + 2 = 5 =⇒ I am Count Dracula
“Proof”:2 + 2 = 5 =⇒ 4 = 5 =⇒ 5 = 4 =⇒ 2 = 1
Dracula and I are two =⇒Dracula and I are one
Discrete Mathematics I – p. 34/292
![Page 108: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/108.jpg)
Logic
Example: 2 + 2 = 5 =⇒ Grass is green
Proof:2 + 2 = 5 =⇒ 4 = 5 =⇒ 5 = 4 =⇒
4 + 5 = 5 + 4 =⇒ T
T =⇒ Grass is green
Discrete Mathematics I – p. 35/292
![Page 109: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/109.jpg)
Logic
Example: 2 + 2 = 5 =⇒ Grass is green
Proof:2 + 2 = 5 =⇒ 4 = 5 =⇒ 5 = 4 =⇒
4 + 5 = 5 + 4 =⇒ T
T =⇒ Grass is green
Discrete Mathematics I – p. 35/292
![Page 110: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/110.jpg)
Logic
Example: 2 + 2 = 5 =⇒ Grass is green
Proof:2 + 2 = 5 =⇒ 4 = 5 =⇒ 5 = 4 =⇒
4 + 5 = 5 + 4 =⇒ T
T =⇒ Grass is green
Discrete Mathematics I – p. 35/292
![Page 111: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/111.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 112: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/112.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B
B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 113: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/113.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 114: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/114.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B
B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 115: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/115.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 116: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/116.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B
B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 117: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/117.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 118: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/118.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B
B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 119: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/119.jpg)
Logic
Implication A ⇒ B can have many disguises:
A implies B B is implied by A
A leads to B B follows from A
A is stronger than B B is weaker than A
A is sufficient for B B is necessary for A
Discrete Mathematics I – p. 36/292
![Page 120: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/120.jpg)
Logic
Examples:
For a number to be divisible by 4, it is
necessary
thatit is even
For a triangle to be isosceles, it is sufficient that it isequilateral
Discrete Mathematics I – p. 37/292
![Page 121: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/121.jpg)
Logic
Examples:
For a number to be divisible by 4, it is necessary thatit is even
For a triangle to be isosceles, it is sufficient that it isequilateral
Discrete Mathematics I – p. 37/292
![Page 122: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/122.jpg)
Logic
Examples:
For a number to be divisible by 4, it is necessary thatit is even
For a triangle to be isosceles, it is
sufficient
that it isequilateral
Discrete Mathematics I – p. 37/292
![Page 123: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/123.jpg)
Logic
Examples:
For a number to be divisible by 4, it is necessary thatit is even
For a triangle to be isosceles, it is sufficient that it isequilateral
Discrete Mathematics I – p. 37/292
![Page 124: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/124.jpg)
Logic
Equivalence (A IF AND ONLY IF B): A ⇔ B
True if A and B agree; false otherwise
A B A ⇔ B
F F T
F T F
T F F
T T T
IF AND ONLY IF often contracted to IFF
Discrete Mathematics I – p. 38/292
![Page 125: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/125.jpg)
Logic
Equivalence (A IF AND ONLY IF B): A ⇔ B
True if A and B agree; false otherwise
A B A ⇔ B
F F T
F T F
T F F
T T T
IF AND ONLY IF often contracted to IFF
Discrete Mathematics I – p. 38/292
![Page 126: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/126.jpg)
Logic
Equivalence (A IF AND ONLY IF B): A ⇔ B
True if A and B agree; false otherwise
A B A ⇔ B
F F T
F T F
T F F
T T T
IF AND ONLY IF often contracted to IFF
Discrete Mathematics I – p. 38/292
![Page 127: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/127.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 128: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/128.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ]
⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 129: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/129.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F
[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 130: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/130.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 131: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/131.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ]
⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 132: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/132.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F
[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 133: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/133.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 134: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/134.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ]
⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 135: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/135.jpg)
Logic
Examples:[
[5 < 10] ⇔ [Pigs fly]]
⇐⇒ [T ⇔ F ] ⇐⇒ F[
[Pigs fly] ⇔ [5 < 10]]
⇐⇒ [F ⇔ T ] ⇐⇒ F[
[Pigs fly] ⇔ [5 > 10]]
⇐⇒ [F ⇔ F ] ⇐⇒ T
Discrete Mathematics I – p. 39/292
![Page 136: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/136.jpg)
Logic
Implication and equivalence are often used to statetheorems
Examples:
Axiom. If n is in N, then n + 1 is in N.That is, for all n, [n in N] =⇒ [n + 1 in N]
Theorem. Number n is even iff n + 1 is odd.That is, for all n in N, [n even] ⇐⇒ [n + 1 odd]
Discrete Mathematics I – p. 40/292
![Page 137: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/137.jpg)
Logic
Implication and equivalence are often used to statetheorems
Examples:
Axiom. If n is in N, then n + 1 is in N.
That is, for all n, [n in N] =⇒ [n + 1 in N]
Theorem. Number n is even iff n + 1 is odd.That is, for all n in N, [n even] ⇐⇒ [n + 1 odd]
Discrete Mathematics I – p. 40/292
![Page 138: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/138.jpg)
Logic
Implication and equivalence are often used to statetheorems
Examples:
Axiom. If n is in N, then n + 1 is in N.That is, for all n, [n in N] =⇒ [n + 1 in N]
Theorem. Number n is even iff n + 1 is odd.That is, for all n in N, [n even] ⇐⇒ [n + 1 odd]
Discrete Mathematics I – p. 40/292
![Page 139: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/139.jpg)
Logic
Implication and equivalence are often used to statetheorems
Examples:
Axiom. If n is in N, then n + 1 is in N.That is, for all n, [n in N] =⇒ [n + 1 in N]
Theorem. Number n is even iff n + 1 is odd.
That is, for all n in N, [n even] ⇐⇒ [n + 1 odd]
Discrete Mathematics I – p. 40/292
![Page 140: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/140.jpg)
Logic
Implication and equivalence are often used to statetheorems
Examples:
Axiom. If n is in N, then n + 1 is in N.That is, for all n, [n in N] =⇒ [n + 1 in N]
Theorem. Number n is even iff n + 1 is odd.That is, for all n in N, [n even] ⇐⇒ [n + 1 odd]
Discrete Mathematics I – p. 40/292
![Page 141: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/141.jpg)
Logic
More examples (from geometry):
Axiom. If two points are distinct, then there is exactlyone line connecting them.
Theorem. A triangle has two equal sides, if and onlyif it has two equal angles.
Discrete Mathematics I – p. 41/292
![Page 142: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/142.jpg)
Logic
More examples (from geometry):
Axiom. If two points are distinct, then there is exactlyone line connecting them.
Theorem. A triangle has two equal sides, if and onlyif it has two equal angles.
Discrete Mathematics I – p. 41/292
![Page 143: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/143.jpg)
Logic
Implication and equivalence are used in proofs
Example:
Theorem. If number n is even, then n + 2 is even.
Proof.[n even] ⇒ [n + 1 odd] ⇒ [(n + 1) + 1 even]
Discrete Mathematics I – p. 42/292
![Page 144: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/144.jpg)
Logic
Implication and equivalence are used in proofs
Example:
Theorem. If number n is even, then n + 2 is even.
Proof.[n even] ⇒ [n + 1 odd] ⇒ [(n + 1) + 1 even]
Discrete Mathematics I – p. 42/292
![Page 145: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/145.jpg)
Logic
Implication and equivalence are used in proofs
Example:
Theorem. If number n is even, then n + 2 is even.
Proof.[n even] ⇒ [n + 1 odd] ⇒ [(n + 1) + 1 even]
Discrete Mathematics I – p. 42/292
![Page 146: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/146.jpg)
Logic
When proving “⇔”, must prove both “⇒” and “⇐”!
Example (a stronger theorem):
Theorem. Number n is even, iff n + 2 is even.
Proof.
“⇒” as before
“⇐”: [n + 2 = (n + 1) + 1 even] ⇒ [n + 1 odd] ⇒[n even]
Discrete Mathematics I – p. 43/292
![Page 147: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/147.jpg)
Logic
When proving “⇔”, must prove both “⇒” and “⇐”!
Example (a stronger theorem):
Theorem. Number n is even, iff n + 2 is even.
Proof.
“⇒” as before
“⇐”: [n + 2 = (n + 1) + 1 even] ⇒ [n + 1 odd] ⇒[n even]
Discrete Mathematics I – p. 43/292
![Page 148: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/148.jpg)
Logic
When proving “⇔”, must prove both “⇒” and “⇐”!
Example (a stronger theorem):
Theorem. Number n is even, iff n + 2 is even.
Proof.
“⇒” as before
“⇐”: [n + 2 = (n + 1) + 1 even] ⇒ [n + 1 odd] ⇒[n even]
Discrete Mathematics I – p. 43/292
![Page 149: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/149.jpg)
Logic
To cut down on brackets, we use priorities
Highest priority: ¬, then ∧, ∨, then ⇒, ⇔
Example:
¬(A ∧B) ⇔ ¬A ∨ ¬B means¬(A ∧B) ⇔ ((¬A) ∨ (¬B))
Discrete Mathematics I – p. 44/292
![Page 150: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/150.jpg)
Logic
To cut down on brackets, we use priorities
Highest priority: ¬, then ∧, ∨, then ⇒, ⇔Example:
¬(A ∧B) ⇔ ¬A ∨ ¬B means¬(A ∧B) ⇔ ((¬A) ∨ (¬B))
Discrete Mathematics I – p. 44/292
![Page 151: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/151.jpg)
Logic
Truth table completely define logical operators.
Not always convenient:
(¬(T ∧ F ) ∨ ¬(F ⇒ T )) ⇔ (¬¬(F ∨ T ) ∨ F )
— true or false?
((A ∨B) ∧ C) ∨ ((¬A ∧ ¬B) ∨ ¬C)
— true for all A, B, C?
To simplify expressions, will use laws of logic
Discrete Mathematics I – p. 45/292
![Page 152: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/152.jpg)
Logic
Truth table completely define logical operators.
Not always convenient:
(¬(T ∧ F ) ∨ ¬(F ⇒ T )) ⇔ (¬¬(F ∨ T ) ∨ F )
— true or false?
((A ∨B) ∧ C) ∨ ((¬A ∧ ¬B) ∨ ¬C)
— true for all A, B, C?
To simplify expressions, will use laws of logic
Discrete Mathematics I – p. 45/292
![Page 153: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/153.jpg)
Logic
Truth table completely define logical operators.
Not always convenient:
(¬(T ∧ F ) ∨ ¬(F ⇒ T )) ⇔ (¬¬(F ∨ T ) ∨ F )
— true or false?
((A ∨B) ∧ C) ∨ ((¬A ∧ ¬B) ∨ ¬C)
— true for all A, B, C?
To simplify expressions, will use laws of logic
Discrete Mathematics I – p. 45/292
![Page 154: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/154.jpg)
Logic
Truth table completely define logical operators.
Not always convenient:
(¬(T ∧ F ) ∨ ¬(F ⇒ T )) ⇔ (¬¬(F ∨ T ) ∨ F )
— true or false?
((A ∨B) ∧ C) ∨ ((¬A ∧ ¬B) ∨ ¬C)
— true for all A, B, C?
To simplify expressions, will use laws of logic
Discrete Mathematics I – p. 45/292
![Page 155: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/155.jpg)
Logic
Laws of Boolean logic (hold for any A, B, C):
¬¬A ⇐⇒ A double negation
A ∧ A ⇐⇒ A ∧ idempotentA ∨ A ⇐⇒ A ∨ idempotent
A ∧B ⇐⇒ B ∧ A ∧ commutativeA ∨B ⇐⇒ B ∨ A ∨ commutative
Discrete Mathematics I – p. 46/292
![Page 156: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/156.jpg)
Logic
Laws of Boolean logic (hold for any A, B, C):
¬¬A ⇐⇒ A double negation
A ∧ A ⇐⇒ A ∧ idempotentA ∨ A ⇐⇒ A ∨ idempotent
A ∧B ⇐⇒ B ∧ A ∧ commutativeA ∨B ⇐⇒ B ∨ A ∨ commutative
Discrete Mathematics I – p. 46/292
![Page 157: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/157.jpg)
Logic
Laws of Boolean logic (hold for any A, B, C):
¬¬A ⇐⇒ A double negation
A ∧ A ⇐⇒ A ∧ idempotentA ∨ A ⇐⇒ A ∨ idempotent
A ∧B ⇐⇒ B ∧ A ∧ commutativeA ∨B ⇐⇒ B ∨ A ∨ commutative
Discrete Mathematics I – p. 46/292
![Page 158: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/158.jpg)
Logic
More laws of Boolean logic:
(A ∧B) ∧ C ⇐⇒ A ∧ (B ∧ C) ∧ associative(A ∨B) ∨ C ⇐⇒ A ∨ (B ∨ C) ∨ associative
A ∧ (B ∨ C) ⇐⇒ (A ∧B) ∨ (A ∧ C)∧ distributes over ∨
A ∨ (B ∧ C) ⇐⇒ (A ∨B) ∧ (A ∨ C)∨ distributes over ∧
Compare a · (b + c) = a · b + a · c,but a + b · c 6= (a + b) · (a + c)
Discrete Mathematics I – p. 47/292
![Page 159: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/159.jpg)
Logic
More laws of Boolean logic:
(A ∧B) ∧ C ⇐⇒ A ∧ (B ∧ C) ∧ associative(A ∨B) ∨ C ⇐⇒ A ∨ (B ∨ C) ∨ associative
A ∧ (B ∨ C) ⇐⇒ (A ∧B) ∨ (A ∧ C)∧ distributes over ∨
A ∨ (B ∧ C) ⇐⇒ (A ∨B) ∧ (A ∨ C)∨ distributes over ∧
Compare a · (b + c) = a · b + a · c,but a + b · c 6= (a + b) · (a + c)
Discrete Mathematics I – p. 47/292
![Page 160: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/160.jpg)
Logic
More laws of Boolean logic:
(A ∧B) ∧ C ⇐⇒ A ∧ (B ∧ C) ∧ associative(A ∨B) ∨ C ⇐⇒ A ∨ (B ∨ C) ∨ associative
A ∧ (B ∨ C) ⇐⇒ (A ∧B) ∨ (A ∧ C)∧ distributes over ∨
A ∨ (B ∧ C) ⇐⇒ (A ∨B) ∧ (A ∨ C)∨ distributes over ∧
Compare a · (b + c) = a · b + a · c,but a + b · c 6= (a + b) · (a + c)
Discrete Mathematics I – p. 47/292
![Page 161: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/161.jpg)
Logic
More laws of Boolean logic:
(A ∧B) ∧ C ⇐⇒ A ∧ (B ∧ C) ∧ associative(A ∨B) ∨ C ⇐⇒ A ∨ (B ∨ C) ∨ associative
A ∧ (B ∨ C) ⇐⇒ (A ∧B) ∨ (A ∧ C)∧ distributes over ∨
A ∨ (B ∧ C) ⇐⇒ (A ∨B) ∧ (A ∨ C)∨ distributes over ∧
Compare a · (b + c) = a · b + a · c,but a + b · c 6= (a + b) · (a + c)
Discrete Mathematics I – p. 47/292
![Page 162: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/162.jpg)
Logic
De Morgan’s laws:
¬(A ∧B) ⇐⇒ ¬A ∨ ¬B¬(A ∨B) ⇐⇒ ¬A ∧ ¬B
Thus, A ∧B ⇐⇒ ¬(¬A ∨ ¬B),so ∧ can be expressed via ¬, ∨Alternatively, A ∨B ⇐⇒ ¬(¬A ∧ ¬B),so ∨ can be expressed via ¬, ∧(Cannot remove both ∧, ∨ at the same time!)
Discrete Mathematics I – p. 48/292
![Page 163: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/163.jpg)
Logic
De Morgan’s laws:
¬(A ∧B) ⇐⇒ ¬A ∨ ¬B¬(A ∨B) ⇐⇒ ¬A ∧ ¬B
Thus, A ∧B ⇐⇒ ¬(¬A ∨ ¬B),so ∧ can be expressed via ¬, ∨
Alternatively, A ∨B ⇐⇒ ¬(¬A ∧ ¬B),so ∨ can be expressed via ¬, ∧(Cannot remove both ∧, ∨ at the same time!)
Discrete Mathematics I – p. 48/292
![Page 164: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/164.jpg)
Logic
De Morgan’s laws:
¬(A ∧B) ⇐⇒ ¬A ∨ ¬B¬(A ∨B) ⇐⇒ ¬A ∧ ¬B
Thus, A ∧B ⇐⇒ ¬(¬A ∨ ¬B),so ∧ can be expressed via ¬, ∨Alternatively, A ∨B ⇐⇒ ¬(¬A ∧ ¬B),so ∨ can be expressed via ¬, ∧
(Cannot remove both ∧, ∨ at the same time!)
Discrete Mathematics I – p. 48/292
![Page 165: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/165.jpg)
Logic
De Morgan’s laws:
¬(A ∧B) ⇐⇒ ¬A ∨ ¬B¬(A ∨B) ⇐⇒ ¬A ∧ ¬B
Thus, A ∧B ⇐⇒ ¬(¬A ∨ ¬B),so ∧ can be expressed via ¬, ∨Alternatively, A ∨B ⇐⇒ ¬(¬A ∧ ¬B),so ∨ can be expressed via ¬, ∧(Cannot remove both ∧, ∨ at the same time!)
Discrete Mathematics I – p. 48/292
![Page 166: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/166.jpg)
Logic
Still more laws of Boolean logic:
A ∧ T ⇐⇒ A A ∨ F ⇐⇒ A identity laws
A ∧ F ⇐⇒ F A ∨ T ⇐⇒ Tannihilation laws
A ∧ ¬A ⇐⇒ F A ∨ ¬A ⇐⇒ Tlaws of excluded middle
A ∧ (A ∨B) ⇐⇒ A ⇐⇒ A ∨ (A ∧B)absorption laws
Discrete Mathematics I – p. 49/292
![Page 167: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/167.jpg)
Logic
Still more laws of Boolean logic:
A ∧ T ⇐⇒ A A ∨ F ⇐⇒ A identity laws
A ∧ F ⇐⇒ F A ∨ T ⇐⇒ Tannihilation laws
A ∧ ¬A ⇐⇒ F A ∨ ¬A ⇐⇒ Tlaws of excluded middle
A ∧ (A ∨B) ⇐⇒ A ⇐⇒ A ∨ (A ∧B)absorption laws
Discrete Mathematics I – p. 49/292
![Page 168: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/168.jpg)
Logic
Still more laws of Boolean logic:
A ∧ T ⇐⇒ A A ∨ F ⇐⇒ A identity laws
A ∧ F ⇐⇒ F A ∨ T ⇐⇒ Tannihilation laws
A ∧ ¬A ⇐⇒ F A ∨ ¬A ⇐⇒ Tlaws of excluded middle
A ∧ (A ∨B) ⇐⇒ A ⇐⇒ A ∨ (A ∧B)absorption laws
Discrete Mathematics I – p. 49/292
![Page 169: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/169.jpg)
Logic
Still more laws of Boolean logic:
A ∧ T ⇐⇒ A A ∨ F ⇐⇒ A identity laws
A ∧ F ⇐⇒ F A ∨ T ⇐⇒ Tannihilation laws
A ∧ ¬A ⇐⇒ F A ∨ ¬A ⇐⇒ Tlaws of excluded middle
A ∧ (A ∨B) ⇐⇒ A ⇐⇒ A ∨ (A ∧B)absorption laws
Discrete Mathematics I – p. 49/292
![Page 170: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/170.jpg)
Logic
Finally,
(A ⇒ B) ⇐⇒ (¬A ∨B) ⇐⇒ ¬(A ∧ ¬B)
(A ⇔ B) ⇐⇒ (A ⇒ B) ∧ (B ⇒ A) ⇐⇒(A ∧B) ∨ (¬A ∧ ¬B)
So, ⇒ and ⇔ are redundant (but convenient)
Discrete Mathematics I – p. 50/292
![Page 171: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/171.jpg)
Logic
Finally,
(A ⇒ B) ⇐⇒ (¬A ∨B) ⇐⇒ ¬(A ∧ ¬B)
(A ⇔ B) ⇐⇒ (A ⇒ B) ∧ (B ⇒ A) ⇐⇒(A ∧B) ∨ (¬A ∧ ¬B)
So, ⇒ and ⇔ are redundant (but convenient)
Discrete Mathematics I – p. 50/292
![Page 172: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/172.jpg)
Logic
Finally,
(A ⇒ B) ⇐⇒ (¬A ∨B) ⇐⇒ ¬(A ∧ ¬B)
(A ⇔ B) ⇐⇒ (A ⇒ B) ∧ (B ⇒ A) ⇐⇒(A ∧B) ∨ (¬A ∧ ¬B)
So, ⇒ and ⇔ are redundant (but convenient)
Discrete Mathematics I – p. 50/292
![Page 173: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/173.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 174: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/174.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 175: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/175.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T
T F F F F
T F
F T F T T
F T
F T T F T
F F
F T T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 176: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/176.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T
F F F F
T F F
T F T T
F T F
T T F T
F F F
T T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 177: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/177.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T F
F F F
T F F T
F T T
F T F T
T F T
F F F T
T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 178: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/178.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T F F F
F
T F F T F T
T
F T F T T F
T
F F F T T T
T
? ?
Discrete Mathematics I – p. 51/292
![Page 179: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/179.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 180: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/180.jpg)
Logic
All these laws can be verified by truth tables
Example: ¬(A ∧B) ⇐⇒ (¬A ∨ ¬B)
A B A ∧B ¬(A ∧B) ¬A ¬B (¬A ∨ ¬B)
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
? ?
Discrete Mathematics I – p. 51/292
![Page 181: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/181.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 182: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/182.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A)
⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 183: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/183.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A)
⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 184: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/184.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A)
⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 185: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/185.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B)
⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 186: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/186.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 187: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/187.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 188: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/188.jpg)
Logic
Using laws of logic
Prove: (A ⇒ B) ⇐⇒ (¬B ⇒ ¬A).
Proof.
(¬B ⇒ ¬A) ⇐⇒ (¬¬B ∨ ¬A) ⇐⇒(B ∨ ¬A) ⇐⇒ (¬A ∨B) ⇐⇒ (A ⇒ B)
¬B ⇒ ¬A is the contapositive of A ⇒ B
B ⇒ A is the converse of A ⇒ B
Statement A ⇒ B is equivalent to its contrapositive,but not to its converse
Equivalence with contrapositive allows proof bycontradiction
Discrete Mathematics I – p. 52/292
![Page 189: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/189.jpg)
Logic
Holmes: I see our visitor was absent-minded. . .
Watson: But why!??
Holmes: Elementary, my dear Watson! Alert peoplenever leave things behind. But he left his walkingstick. Therefore, he must be absent-minded.
A = “person is alert”B = “person does not leave things behind”
Holmes’ argument: (A ⇒ B) =⇒ (¬B ⇒ ¬A)
Discrete Mathematics I – p. 53/292
![Page 190: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/190.jpg)
Logic
Holmes: I see our visitor was absent-minded. . .
Watson: But why!??
Holmes: Elementary, my dear Watson! Alert peoplenever leave things behind. But he left his walkingstick. Therefore, he must be absent-minded.
A = “person is alert”B = “person does not leave things behind”
Holmes’ argument: (A ⇒ B) =⇒ (¬B ⇒ ¬A)
Discrete Mathematics I – p. 53/292
![Page 191: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/191.jpg)
Logic
Holmes: I see our visitor was absent-minded. . .
Watson: But why!??
Holmes: Elementary, my dear Watson! Alert peoplenever leave things behind. But he left his walkingstick. Therefore, he must be absent-minded.
A = “person is alert”B = “person does not leave things behind”
Holmes’ argument: (A ⇒ B) =⇒ (¬B ⇒ ¬A)
Discrete Mathematics I – p. 53/292
![Page 192: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/192.jpg)
Logic
Holmes: I see our visitor was absent-minded. . .
Watson: But why!??
Holmes: Elementary, my dear Watson! Alert peoplenever leave things behind. But he left his walkingstick. Therefore, he must be absent-minded.
A = “person is alert”B = “person does not leave things behind”
Holmes’ argument: (A ⇒ B) =⇒ (¬B ⇒ ¬A)
Discrete Mathematics I – p. 53/292
![Page 193: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/193.jpg)
Logic
Sign in a restaurant:
Good food is not cheap.
Cheap food is not good.Is it repeating the same thing twice?
Yes! The statements are contrapositive to each other.
Discrete Mathematics I – p. 54/292
![Page 194: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/194.jpg)
Logic
Sign in a restaurant:
Good food is not cheap.
Cheap food is not good.Is it repeating the same thing twice?
Yes! The statements are contrapositive to each other.
Discrete Mathematics I – p. 54/292
![Page 195: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/195.jpg)
Logic
Somebody walks into a pub and says:
If I drink, everybody drinks!
Can this statement be true?
Discrete Mathematics I – p. 55/292
![Page 196: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/196.jpg)
Logic
Answer: yes, it can!
Proof. We know the world is nonempty.
Case 1: Suppose everybody in the world drinks. Thenevery person can say that.
Case 2: Suppose Joe does not drink. Then Joe can saythat.
Discrete Mathematics I – p. 56/292
![Page 197: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/197.jpg)
Logic
Answer: yes, it can!
Proof. We know the world is nonempty.
Case 1: Suppose everybody in the world drinks. Thenevery person can say that.
Case 2: Suppose Joe does not drink. Then Joe can saythat.
Discrete Mathematics I – p. 56/292
![Page 198: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/198.jpg)
Logic
Answer: yes, it can!
Proof. We know the world is nonempty.
Case 1: Suppose everybody in the world drinks. Thenevery person can say that.
Case 2: Suppose Joe does not drink. Then Joe can saythat.
Discrete Mathematics I – p. 56/292
![Page 199: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/199.jpg)
Logic
Somebody walks into a pub and says:
If anybody drinks, I drink!
Can this statement be true?
Discrete Mathematics I – p. 57/292
![Page 200: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/200.jpg)
Logic
Answer: yes, it can!
Proof. We know the world is nonempty.
Case 1: Suppose nobody in the world drinks. Thenevery person can say that.
Case 2: Suppose Jack drinks. Then Jack can saythat.
Discrete Mathematics I – p. 58/292
![Page 201: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/201.jpg)
Logic
Answer: yes, it can!
Proof. We know the world is nonempty.
Case 1: Suppose nobody in the world drinks. Thenevery person can say that.
Case 2: Suppose Jack drinks. Then Jack can saythat.
Discrete Mathematics I – p. 58/292
![Page 202: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/202.jpg)
Logic
Answer: yes, it can!
Proof. We know the world is nonempty.
Case 1: Suppose nobody in the world drinks. Thenevery person can say that.
Case 2: Suppose Jack drinks. Then Jack can saythat.
Discrete Mathematics I – p. 58/292
![Page 203: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/203.jpg)
Logic
So far — statements about individual objects:
Five is less than tenThe pie is not as bad as it looks
Often need to say more:
Some natural numbers are less than tenAll pies are not as bad as they look
Discrete Mathematics I – p. 59/292
![Page 204: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/204.jpg)
Logic
So far — statements about individual objects:
Five is less than tenThe pie is not as bad as it looks
Often need to say more:
Some natural numbers are less than tenAll pies are not as bad as they look
Discrete Mathematics I – p. 59/292
![Page 205: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/205.jpg)
Logic
Some natural numbers are less than ten
Could try to specify an instance:
Five is less than ten
What if we do not have an instance?
Discrete Mathematics I – p. 60/292
![Page 206: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/206.jpg)
Logic
Some natural numbers are less than ten
Could try to specify an instance:
Five is less than ten
What if we do not have an instance?
Discrete Mathematics I – p. 60/292
![Page 207: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/207.jpg)
Logic
Some natural numbers are less than ten
Could try to specify an instance:
Five is less than ten
What if we do not have an instance?
Discrete Mathematics I – p. 60/292
![Page 208: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/208.jpg)
Logic
Could try to use Boolean operators:
Some natural numbers are less than ten
(0 < 10) ∨ (1 < 10) ∨ (2 < 10) ∨ (3 < 10) ∨ · · ·All pies are not as bad as they look
(Chicken pie. . . ) ∧ (Mushroom pie. . . ) ∧(Cabbage pie. . . ) ∧ · · ·Cannot have infinite conjunction/disjunction!
Discrete Mathematics I – p. 61/292
![Page 209: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/209.jpg)
Logic
Could try to use Boolean operators:
Some natural numbers are less than ten
(0 < 10) ∨ (1 < 10) ∨ (2 < 10) ∨ (3 < 10) ∨ · · ·All pies are not as bad as they look
(Chicken pie. . . ) ∧ (Mushroom pie. . . ) ∧(Cabbage pie. . . ) ∧ · · ·Cannot have infinite conjunction/disjunction!
Discrete Mathematics I – p. 61/292
![Page 210: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/210.jpg)
Logic
Could try to use Boolean operators:
Some natural numbers are less than ten
(0 < 10) ∨ (1 < 10) ∨ (2 < 10) ∨ (3 < 10) ∨ · · ·
All pies are not as bad as they look
(Chicken pie. . . ) ∧ (Mushroom pie. . . ) ∧(Cabbage pie. . . ) ∧ · · ·Cannot have infinite conjunction/disjunction!
Discrete Mathematics I – p. 61/292
![Page 211: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/211.jpg)
Logic
Could try to use Boolean operators:
Some natural numbers are less than ten
(0 < 10) ∨ (1 < 10) ∨ (2 < 10) ∨ (3 < 10) ∨ · · ·All pies are not as bad as they look
(Chicken pie. . . ) ∧ (Mushroom pie. . . ) ∧(Cabbage pie. . . ) ∧ · · ·Cannot have infinite conjunction/disjunction!
Discrete Mathematics I – p. 61/292
![Page 212: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/212.jpg)
Logic
Could try to use Boolean operators:
Some natural numbers are less than ten
(0 < 10) ∨ (1 < 10) ∨ (2 < 10) ∨ (3 < 10) ∨ · · ·All pies are not as bad as they look
(Chicken pie. . . ) ∧ (Mushroom pie. . . ) ∧(Cabbage pie. . . ) ∧ · · ·
Cannot have infinite conjunction/disjunction!
Discrete Mathematics I – p. 61/292
![Page 213: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/213.jpg)
Logic
Could try to use Boolean operators:
Some natural numbers are less than ten
(0 < 10) ∨ (1 < 10) ∨ (2 < 10) ∨ (3 < 10) ∨ · · ·All pies are not as bad as they look
(Chicken pie. . . ) ∧ (Mushroom pie. . . ) ∧(Cabbage pie. . . ) ∧ · · ·Cannot have infinite conjunction/disjunction!
Discrete Mathematics I – p. 61/292
![Page 214: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/214.jpg)
Logic
A predicate is a sentence with variables
Becomes true or false when values are substituted forvariables
Values are taken from a particular set (the range)
Always assume range is nonempty
Discrete Mathematics I – p. 62/292
![Page 215: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/215.jpg)
Logic
A predicate is a sentence with variables
Becomes true or false when values are substituted forvariables
Values are taken from a particular set (the range)
Always assume range is nonempty
Discrete Mathematics I – p. 62/292
![Page 216: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/216.jpg)
Logic
Examples:
x < 10 (x in N)
“Pie p is not as bad as it looks” (p in Pies)
Can be true or false, depending on x, p
Discrete Mathematics I – p. 63/292
![Page 217: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/217.jpg)
Logic
Examples:
x < 10 (x in N)
“Pie p is not as bad as it looks” (p in Pies)
Can be true or false, depending on x, p
Discrete Mathematics I – p. 63/292
![Page 218: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/218.jpg)
Logic
Examples:
x < 10 (x in N)
“Pie p is not as bad as it looks” (p in Pies)
Can be true or false, depending on x, p
Discrete Mathematics I – p. 63/292
![Page 219: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/219.jpg)
Logic
A predicate can have more than one variable
Examples:
x < y (x, y in N)
“Pie p is better than pie q” (p, q in Pies)
Discrete Mathematics I – p. 64/292
![Page 220: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/220.jpg)
Logic
A predicate can have more than one variable
Examples:
x < y (x, y in N)
“Pie p is better than pie q” (p, q in Pies)
Discrete Mathematics I – p. 64/292
![Page 221: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/221.jpg)
Logic
A predicate can have more than one variable
Examples:
x < y (x, y in N)
“Pie p is better than pie q” (p, q in Pies)
Discrete Mathematics I – p. 64/292
![Page 222: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/222.jpg)
Logic
Predicate with no variables: ordinary statement
Examples:
5 < 10
“My pie is better than your pie”
Discrete Mathematics I – p. 65/292
![Page 223: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/223.jpg)
Logic
Predicate with no variables: ordinary statement
Examples:
5 < 10
“My pie is better than your pie”
Discrete Mathematics I – p. 65/292
![Page 224: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/224.jpg)
Logic
Predicate with no variables: ordinary statement
Examples:
5 < 10
“My pie is better than your pie”
Discrete Mathematics I – p. 65/292
![Page 225: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/225.jpg)
Logic
Let P (x) be a predicate with variable x
Can make statements by quantifiers
Existential (FOR SOME x, P (x)): ∃x : P (x)
Universal (FOR ALL x, P (x)): ∀x : P (x)
A particular range of x always assumed
Discrete Mathematics I – p. 66/292
![Page 226: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/226.jpg)
Logic
Let P (x) be a predicate with variable x
Can make statements by quantifiers
Existential (FOR SOME x, P (x)): ∃x : P (x)
Universal (FOR ALL x, P (x)): ∀x : P (x)
A particular range of x always assumed
Discrete Mathematics I – p. 66/292
![Page 227: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/227.jpg)
Logic
Let P (x) be a predicate with variable x
Can make statements by quantifiers
Existential (FOR SOME x, P (x)): ∃x : P (x)
Universal (FOR ALL x, P (x)): ∀x : P (x)
A particular range of x always assumed
Discrete Mathematics I – p. 66/292
![Page 228: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/228.jpg)
Logic
Let P (x) be a predicate with variable x
Can make statements by quantifiers
Existential (FOR SOME x, P (x)): ∃x : P (x)
Universal (FOR ALL x, P (x)): ∀x : P (x)
A particular range of x always assumed
Discrete Mathematics I – p. 66/292
![Page 229: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/229.jpg)
Logic
Range often made explicit
Examples:
∃x ∈ N : x < 10
∀p ∈ Pies : “p is not as bad as it looks”
∀x ∈ N : ∃y ∈ N : x < y
∃y ∈ N : ∀x ∈ N : x < y — note the difference!
Discrete Mathematics I – p. 67/292
![Page 230: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/230.jpg)
Logic
Range often made explicit
Examples:
∃x ∈ N : x < 10
∀p ∈ Pies : “p is not as bad as it looks”
∀x ∈ N : ∃y ∈ N : x < y
∃y ∈ N : ∀x ∈ N : x < y — note the difference!
Discrete Mathematics I – p. 67/292
![Page 231: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/231.jpg)
Logic
Range often made explicit
Examples:
∃x ∈ N : x < 10
∀p ∈ Pies : “p is not as bad as it looks”
∀x ∈ N : ∃y ∈ N : x < y
∃y ∈ N : ∀x ∈ N : x < y — note the difference!
Discrete Mathematics I – p. 67/292
![Page 232: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/232.jpg)
Logic
Range often made explicit
Examples:
∃x ∈ N : x < 10
∀p ∈ Pies : “p is not as bad as it looks”
∀x ∈ N : ∃y ∈ N : x < y
∃y ∈ N : ∀x ∈ N : x < y — note the difference!
Discrete Mathematics I – p. 67/292
![Page 233: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/233.jpg)
Logic
Range often made explicit
Examples:
∃x ∈ N : x < 10
∀p ∈ Pies : “p is not as bad as it looks”
∀x ∈ N : ∃y ∈ N : x < y
∃y ∈ N : ∀x ∈ N : x < y — note the difference!
Discrete Mathematics I – p. 67/292
![Page 234: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/234.jpg)
Logic
Quantifier variable can be changed
∀p ∈ Pies : “p is not as bad as it looks” ⇐⇒∀π ∈ Pies : “π is not as bad as it looks”
Variable under a quantifier bound, otherwise free
Example:
∃y ∈ N : x > y x free, y bound
Truth value depends on x, but not on y
Discrete Mathematics I – p. 68/292
![Page 235: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/235.jpg)
Logic
Quantifier variable can be changed
∀p ∈ Pies : “p is not as bad as it looks” ⇐⇒
∀π ∈ Pies : “π is not as bad as it looks”
Variable under a quantifier bound, otherwise free
Example:
∃y ∈ N : x > y x free, y bound
Truth value depends on x, but not on y
Discrete Mathematics I – p. 68/292
![Page 236: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/236.jpg)
Logic
Quantifier variable can be changed
∀p ∈ Pies : “p is not as bad as it looks” ⇐⇒∀π ∈ Pies : “π is not as bad as it looks”
Variable under a quantifier bound, otherwise free
Example:
∃y ∈ N : x > y x free, y bound
Truth value depends on x, but not on y
Discrete Mathematics I – p. 68/292
![Page 237: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/237.jpg)
Logic
Quantifier variable can be changed
∀p ∈ Pies : “p is not as bad as it looks” ⇐⇒∀π ∈ Pies : “π is not as bad as it looks”
Variable under a quantifier bound, otherwise free
Example:
∃y ∈ N : x > y x free, y bound
Truth value depends on x, but not on y
Discrete Mathematics I – p. 68/292
![Page 238: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/238.jpg)
Logic
Quantifier variable can be changed
∀p ∈ Pies : “p is not as bad as it looks” ⇐⇒∀π ∈ Pies : “π is not as bad as it looks”
Variable under a quantifier bound, otherwise free
Example:
∃y ∈ N : x > y x free, y bound
Truth value depends on x, but not on y
Discrete Mathematics I – p. 68/292
![Page 239: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/239.jpg)
Logic
Quantifier variable can be changed
∀p ∈ Pies : “p is not as bad as it looks” ⇐⇒∀π ∈ Pies : “π is not as bad as it looks”
Variable under a quantifier bound, otherwise free
Example:
∃y ∈ N : x > y x free, y bound
Truth value depends on x, but not on y
Discrete Mathematics I – p. 68/292
![Page 240: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/240.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
P (u, v) ⇐⇒ u > v u, v free
Q1(x) ⇐⇒ P (x, 5) ⇐⇒ x > 5 x free
Q2(z) ⇐⇒ P (3, z) ⇐⇒ 3 > z z free
Q3(y) ⇐⇒ ∀x : P (x, y) ⇐⇒ ∀u : P (u, y)⇐⇒ ∀u : u > y y free x, u bound
Q4(v) ⇐⇒ ∃y : P (v, y) ⇐⇒ ∃w : P (v, w)⇐⇒ ∃w : v > w v free y, w bound
Discrete Mathematics I – p. 69/292
![Page 241: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/241.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
P (u, v) ⇐⇒ u > v u, v free
Q1(x) ⇐⇒ P (x, 5) ⇐⇒ x > 5 x free
Q2(z) ⇐⇒ P (3, z) ⇐⇒ 3 > z z free
Q3(y) ⇐⇒ ∀x : P (x, y) ⇐⇒ ∀u : P (u, y)⇐⇒ ∀u : u > y y free x, u bound
Q4(v) ⇐⇒ ∃y : P (v, y) ⇐⇒ ∃w : P (v, w)⇐⇒ ∃w : v > w v free y, w bound
Discrete Mathematics I – p. 69/292
![Page 242: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/242.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
P (u, v) ⇐⇒ u > v u, v free
Q1(x) ⇐⇒ P (x, 5) ⇐⇒ x > 5 x free
Q2(z) ⇐⇒ P (3, z) ⇐⇒ 3 > z z free
Q3(y) ⇐⇒ ∀x : P (x, y) ⇐⇒ ∀u : P (u, y)⇐⇒ ∀u : u > y y free x, u bound
Q4(v) ⇐⇒ ∃y : P (v, y) ⇐⇒ ∃w : P (v, w)⇐⇒ ∃w : v > w v free y, w bound
Discrete Mathematics I – p. 69/292
![Page 243: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/243.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
P (u, v) ⇐⇒ u > v u, v free
Q1(x) ⇐⇒ P (x, 5) ⇐⇒ x > 5 x free
Q2(z) ⇐⇒ P (3, z) ⇐⇒ 3 > z z free
Q3(y) ⇐⇒ ∀x : P (x, y) ⇐⇒ ∀u : P (u, y)⇐⇒ ∀u : u > y y free x, u bound
Q4(v) ⇐⇒ ∃y : P (v, y) ⇐⇒ ∃w : P (v, w)⇐⇒ ∃w : v > w v free y, w bound
Discrete Mathematics I – p. 69/292
![Page 244: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/244.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
P (u, v) ⇐⇒ u > v u, v free
Q1(x) ⇐⇒ P (x, 5) ⇐⇒ x > 5 x free
Q2(z) ⇐⇒ P (3, z) ⇐⇒ 3 > z z free
Q3(y) ⇐⇒ ∀x : P (x, y) ⇐⇒ ∀u : P (u, y)⇐⇒ ∀u : u > y y free x, u bound
Q4(v) ⇐⇒ ∃y : P (v, y) ⇐⇒ ∃w : P (v, w)⇐⇒ ∃w : v > w v free y, w bound
Discrete Mathematics I – p. 69/292
![Page 245: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/245.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
P (u, v) ⇐⇒ u > v u, v free
Q1(x) ⇐⇒ P (x, 5) ⇐⇒ x > 5 x free
Q2(z) ⇐⇒ P (3, z) ⇐⇒ 3 > z z free
Q3(y) ⇐⇒ ∀x : P (x, y) ⇐⇒ ∀u : P (u, y)⇐⇒ ∀u : u > y y free x, u bound
Q4(v) ⇐⇒ ∃y : P (v, y) ⇐⇒ ∃w : P (v, w)⇐⇒ ∃w : v > w v free y, w bound
Discrete Mathematics I – p. 69/292
![Page 246: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/246.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
Q5 ⇐⇒ ∃z : P (0, z)⇐⇒ ∃z : 0 > z ⇐⇒ F z bound
Q6 ⇐⇒ ∀y : ∃x : P (x, y)⇐⇒ ∀y : ∃x : x > y ⇐⇒ T x, y bound
Q7 ⇐⇒ P (3, 5) ⇐⇒ 3 > 5 ⇐⇒ F
Discrete Mathematics I – p. 70/292
![Page 247: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/247.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
Q5 ⇐⇒ ∃z : P (0, z)⇐⇒ ∃z : 0 > z ⇐⇒ F z bound
Q6 ⇐⇒ ∀y : ∃x : P (x, y)⇐⇒ ∀y : ∃x : x > y ⇐⇒ T x, y bound
Q7 ⇐⇒ P (3, 5) ⇐⇒ 3 > 5 ⇐⇒ F
Discrete Mathematics I – p. 70/292
![Page 248: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/248.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
Q5 ⇐⇒ ∃z : P (0, z)⇐⇒ ∃z : 0 > z ⇐⇒ F z bound
Q6 ⇐⇒ ∀y : ∃x : P (x, y)⇐⇒ ∀y : ∃x : x > y ⇐⇒ T x, y bound
Q7 ⇐⇒ P (3, 5) ⇐⇒ 3 > 5 ⇐⇒ F
Discrete Mathematics I – p. 70/292
![Page 249: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/249.jpg)
Logic
P (x, y) ⇐⇒ x > y x, y free
Q5 ⇐⇒ ∃z : P (0, z)⇐⇒ ∃z : 0 > z ⇐⇒ F z bound
Q6 ⇐⇒ ∀y : ∃x : P (x, y)⇐⇒ ∀y : ∃x : x > y ⇐⇒ T x, y bound
Q7 ⇐⇒ P (3, 5) ⇐⇒ 3 > 5 ⇐⇒ F
Discrete Mathematics I – p. 70/292
![Page 250: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/250.jpg)
Logic
Suppose set S finite: S = {a1, . . . , an}
∀x ∈ S : P (x) ⇐⇒ P (a1) ∧ · · · ∧ P (an)
∃x ∈ S : P (x) ⇐⇒ P (a1) ∨ · · · ∨ P (an)
On a finite range, quantifiers can be expressed byBoolean operators
Not so on an infinite range
Discrete Mathematics I – p. 71/292
![Page 251: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/251.jpg)
Logic
Suppose set S finite: S = {a1, . . . , an}∀x ∈ S : P (x) ⇐⇒ P (a1) ∧ · · · ∧ P (an)
∃x ∈ S : P (x) ⇐⇒ P (a1) ∨ · · · ∨ P (an)
On a finite range, quantifiers can be expressed byBoolean operators
Not so on an infinite range
Discrete Mathematics I – p. 71/292
![Page 252: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/252.jpg)
Logic
Suppose set S finite: S = {a1, . . . , an}∀x ∈ S : P (x) ⇐⇒ P (a1) ∧ · · · ∧ P (an)
∃x ∈ S : P (x) ⇐⇒ P (a1) ∨ · · · ∨ P (an)
On a finite range, quantifiers can be expressed byBoolean operators
Not so on an infinite range
Discrete Mathematics I – p. 71/292
![Page 253: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/253.jpg)
Logic
Suppose set S finite: S = {a1, . . . , an}∀x ∈ S : P (x) ⇐⇒ P (a1) ∧ · · · ∧ P (an)
∃x ∈ S : P (x) ⇐⇒ P (a1) ∨ · · · ∨ P (an)
On a finite range, quantifiers can be expressed byBoolean operators
Not so on an infinite range
Discrete Mathematics I – p. 71/292
![Page 254: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/254.jpg)
Logic
Laws of predicate logic:
(∀x : T ) ⇐⇒ T (∀x : F ) ⇐⇒ F
(∃x : T ) ⇐⇒ T (∃x : F ) ⇐⇒ F
∀x : (P (x) ∧Q) ⇐⇒ (∀x : P (x)) ∧Q
(if Q does not contain free x)
Holds for ∀, ∃, and for each of ¬, ∧, ∨, ⇒, ⇔
Discrete Mathematics I – p. 72/292
![Page 255: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/255.jpg)
Logic
Laws of predicate logic:
(∀x : T ) ⇐⇒ T (∀x : F ) ⇐⇒ F
(∃x : T ) ⇐⇒ T (∃x : F ) ⇐⇒ F
∀x : (P (x) ∧Q) ⇐⇒ (∀x : P (x)) ∧Q
(if Q does not contain free x)
Holds for ∀, ∃, and for each of ¬, ∧, ∨, ⇒, ⇔
Discrete Mathematics I – p. 72/292
![Page 256: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/256.jpg)
Logic
Laws of predicate logic:
(∀x : T ) ⇐⇒ T (∀x : F ) ⇐⇒ F
(∃x : T ) ⇐⇒ T (∃x : F ) ⇐⇒ F
∀x : (P (x) ∧Q) ⇐⇒ (∀x : P (x)) ∧Q
(if Q does not contain free x)
Holds for ∀, ∃, and for each of ¬, ∧, ∨, ⇒, ⇔
Discrete Mathematics I – p. 72/292
![Page 257: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/257.jpg)
Logic
Laws of predicate logic:
(∀x : T ) ⇐⇒ T (∀x : F ) ⇐⇒ F
(∃x : T ) ⇐⇒ T (∃x : F ) ⇐⇒ F
∀x : (P (x) ∧Q) ⇐⇒ (∀x : P (x)) ∧Q
(if Q does not contain free x)
Holds for ∀, ∃, and for each of ¬, ∧, ∨, ⇒, ⇔
Discrete Mathematics I – p. 72/292
![Page 258: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/258.jpg)
Logic
Laws of predicate logic:
(∀x : T ) ⇐⇒ T (∀x : F ) ⇐⇒ F
(∃x : T ) ⇐⇒ T (∃x : F ) ⇐⇒ F
∀x : (P (x) ∧Q) ⇐⇒ (∀x : P (x)) ∧Q
(if Q does not contain free x)
Holds for ∀, ∃, and for each of ¬, ∧, ∨, ⇒, ⇔
Discrete Mathematics I – p. 72/292
![Page 259: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/259.jpg)
Logic
Laws of predicate logic:
(∀x : T ) ⇐⇒ T (∀x : F ) ⇐⇒ F
(∃x : T ) ⇐⇒ T (∃x : F ) ⇐⇒ F
∀x : (P (x) ∧Q) ⇐⇒ (∀x : P (x)) ∧Q
(if Q does not contain free x)
Holds for ∀, ∃, and for each of ¬, ∧, ∨, ⇒, ⇔
Discrete Mathematics I – p. 72/292
![Page 260: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/260.jpg)
Logic
De Morgan’s laws for predicates:
¬∀x : P (x) ⇐⇒ ∃x : ¬P (x)
¬∃x : P (x) ⇐⇒ ∀x : ¬P (x)
Discrete Mathematics I – p. 73/292
![Page 261: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/261.jpg)
Logic
Quantifiers — handle with care!
∀x : (P (x) ∧Q(x)) ⇐⇒ (∀x : P (x)) ∧ (∀x : Q(x))
∃x : (P (x) ∨Q(x)) ⇐⇒ (∃x : P (x)) ∨ (∃x : Q(x))
But:
∀x : (P (x) ∨Q(x)) 6⇒ (∀x : P (x)) ∨ (∀x : Q(x))(“⇐” still holds)
∃x : (P (x) ∧Q(x)) 6⇐ (∃x : P (x)) ∧ (∃x : Q(x))(“⇒” still holds)
Discrete Mathematics I – p. 74/292
![Page 262: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/262.jpg)
Logic
Quantifiers — handle with care!
∀x : (P (x) ∧Q(x)) ⇐⇒ (∀x : P (x)) ∧ (∀x : Q(x))
∃x : (P (x) ∨Q(x)) ⇐⇒ (∃x : P (x)) ∨ (∃x : Q(x))
But:
∀x : (P (x) ∨Q(x)) 6⇒ (∀x : P (x)) ∨ (∀x : Q(x))
(“⇐” still holds)
∃x : (P (x) ∧Q(x)) 6⇐ (∃x : P (x)) ∧ (∃x : Q(x))(“⇒” still holds)
Discrete Mathematics I – p. 74/292
![Page 263: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/263.jpg)
Logic
Quantifiers — handle with care!
∀x : (P (x) ∧Q(x)) ⇐⇒ (∀x : P (x)) ∧ (∀x : Q(x))
∃x : (P (x) ∨Q(x)) ⇐⇒ (∃x : P (x)) ∨ (∃x : Q(x))
But:
∀x : (P (x) ∨Q(x)) 6⇒ (∀x : P (x)) ∨ (∀x : Q(x))(“⇐” still holds)
∃x : (P (x) ∧Q(x)) 6⇐ (∃x : P (x)) ∧ (∃x : Q(x))(“⇒” still holds)
Discrete Mathematics I – p. 74/292
![Page 264: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/264.jpg)
Logic
Quantifiers — handle with care!
∀x : (P (x) ∧Q(x)) ⇐⇒ (∀x : P (x)) ∧ (∀x : Q(x))
∃x : (P (x) ∨Q(x)) ⇐⇒ (∃x : P (x)) ∨ (∃x : Q(x))
But:
∀x : (P (x) ∨Q(x)) 6⇒ (∀x : P (x)) ∨ (∀x : Q(x))(“⇐” still holds)
∃x : (P (x) ∧Q(x)) 6⇐ (∃x : P (x)) ∧ (∃x : Q(x))
(“⇒” still holds)
Discrete Mathematics I – p. 74/292
![Page 265: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/265.jpg)
Logic
Quantifiers — handle with care!
∀x : (P (x) ∧Q(x)) ⇐⇒ (∀x : P (x)) ∧ (∀x : Q(x))
∃x : (P (x) ∨Q(x)) ⇐⇒ (∃x : P (x)) ∨ (∃x : Q(x))
But:
∀x : (P (x) ∨Q(x)) 6⇒ (∀x : P (x)) ∨ (∀x : Q(x))(“⇐” still holds)
∃x : (P (x) ∧Q(x)) 6⇐ (∃x : P (x)) ∧ (∃x : Q(x))(“⇒” still holds)
Discrete Mathematics I – p. 74/292
![Page 266: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/266.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 267: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/267.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 268: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/268.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧
∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 269: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/269.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y))
⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 270: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/270.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y))
⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 271: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/271.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y))
⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 272: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/272.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 273: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/273.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 274: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/274.jpg)
Logic
Using quantifiers — an example
P (x) true for at least one x in S: ∃x ∈ S : P (x)
P (x) true for exactly one x in S:
∃x ∈ S : (P (x) ∧ ∀y ∈ S : P (y) ⇒ (x = y)) ⇐⇒∃x ∈ S : (P (x)∧∀y ∈ S : (x 6= y) ⇒ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ∀y ∈ S : (x = y) ∨ ¬P (y)) ⇐⇒∃x ∈ S : (P (x) ∧ ¬∃y ∈ S : (x 6= y) ∧ P (y))
Notation: ∃!x ∈ S : P (x)
Exercise: P (x) true for all but one x in S
Discrete Mathematics I – p. 75/292
![Page 275: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/275.jpg)
Sets
Discrete Mathematics I – p. 76/292
![Page 276: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/276.jpg)
SetsSet: a basic (undefined) concept
By a set we shall understand any collectioninto a whole M of definite, distinct objects ofour intuition or of our thought. These objectsare called the elements of M .
G. Cantor (1845–1918)
Discrete Mathematics I – p. 77/292
![Page 277: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/277.jpg)
SetsSet: a basic (undefined) concept
By a set we shall understand any collectioninto a whole M of definite, distinct objects ofour intuition or of our thought. These objectsare called the elements of M .
G. Cantor (1845–1918)
Discrete Mathematics I – p. 77/292
![Page 278: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/278.jpg)
SetsAnything can be an element of a set
Planets = {Mercury, Venus, . . . , Pluto}Neven = {0, 2, 4, 6, 8, 10, . . .}Junk = {239, banana, ace of spades}A set can be an element of another set
SuperJunk = {239, Junk , ∅} ={239, {banana, ace of spades, 239}, ∅}
Discrete Mathematics I – p. 78/292
![Page 279: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/279.jpg)
SetsAnything can be an element of a set
Planets = {Mercury, Venus, . . . , Pluto}
Neven = {0, 2, 4, 6, 8, 10, . . .}Junk = {239, banana, ace of spades}A set can be an element of another set
SuperJunk = {239, Junk , ∅} ={239, {banana, ace of spades, 239}, ∅}
Discrete Mathematics I – p. 78/292
![Page 280: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/280.jpg)
SetsAnything can be an element of a set
Planets = {Mercury, Venus, . . . , Pluto}Neven = {0, 2, 4, 6, 8, 10, . . .}
Junk = {239, banana, ace of spades}A set can be an element of another set
SuperJunk = {239, Junk , ∅} ={239, {banana, ace of spades, 239}, ∅}
Discrete Mathematics I – p. 78/292
![Page 281: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/281.jpg)
SetsAnything can be an element of a set
Planets = {Mercury, Venus, . . . , Pluto}Neven = {0, 2, 4, 6, 8, 10, . . .}Junk = {239, banana, ace of spades}
A set can be an element of another set
SuperJunk = {239, Junk , ∅} ={239, {banana, ace of spades, 239}, ∅}
Discrete Mathematics I – p. 78/292
![Page 282: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/282.jpg)
SetsAnything can be an element of a set
Planets = {Mercury, Venus, . . . , Pluto}Neven = {0, 2, 4, 6, 8, 10, . . .}Junk = {239, banana, ace of spades}A set can be an element of another set
SuperJunk = {239, Junk , ∅} ={239, {banana, ace of spades, 239}, ∅}
Discrete Mathematics I – p. 78/292
![Page 283: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/283.jpg)
SetsAnything can be an element of a set
Planets = {Mercury, Venus, . . . , Pluto}Neven = {0, 2, 4, 6, 8, 10, . . .}Junk = {239, banana, ace of spades}A set can be an element of another set
SuperJunk = {239, Junk , ∅} ={239, {banana, ace of spades, 239}, ∅}
Discrete Mathematics I – p. 78/292
![Page 284: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/284.jpg)
SetsOrder of elements does not matter
Junk = {banana, ace of spades, 239}
Repetition of elements does not matter
Junk ={banana, banana, ace of spades, 239, 239, 239}
Discrete Mathematics I – p. 79/292
![Page 285: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/285.jpg)
SetsOrder of elements does not matter
Junk = {banana, ace of spades, 239}Repetition of elements does not matter
Junk ={banana, banana, ace of spades, 239, 239, 239}
Discrete Mathematics I – p. 79/292
![Page 286: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/286.jpg)
SetsThe empty set: ∅ = {}
A singleton: any one-element set
MorningStars = {Venus}NonpositiveNaturals = {0}EmptySets = {∅} 6= ∅
Discrete Mathematics I – p. 80/292
![Page 287: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/287.jpg)
SetsThe empty set: ∅ = {}A singleton: any one-element set
MorningStars = {Venus}NonpositiveNaturals = {0}EmptySets = {∅} 6= ∅
Discrete Mathematics I – p. 80/292
![Page 288: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/288.jpg)
SetsThe empty set: ∅ = {}A singleton: any one-element set
MorningStars = {Venus}
NonpositiveNaturals = {0}EmptySets = {∅} 6= ∅
Discrete Mathematics I – p. 80/292
![Page 289: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/289.jpg)
SetsThe empty set: ∅ = {}A singleton: any one-element set
MorningStars = {Venus}NonpositiveNaturals = {0}
EmptySets = {∅} 6= ∅
Discrete Mathematics I – p. 80/292
![Page 290: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/290.jpg)
SetsThe empty set: ∅ = {}A singleton: any one-element set
MorningStars = {Venus}NonpositiveNaturals = {0}EmptySets = {∅} 6= ∅
Discrete Mathematics I – p. 80/292
![Page 291: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/291.jpg)
SetsElement x is in set A: x ∈ A
Jupiter ∈ Planets , orange 6∈ Junk
Set A is a subset of set B, if all elements of A are alsoelements of B (but not necessarily the other wayround)
A ⊆ B ⇐⇒ ∀x : x ∈ A ⇒ x ∈ B
Discrete Mathematics I – p. 81/292
![Page 292: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/292.jpg)
SetsElement x is in set A: x ∈ A
Jupiter ∈ Planets , orange 6∈ Junk
Set A is a subset of set B, if all elements of A are alsoelements of B (but not necessarily the other wayround)
A ⊆ B ⇐⇒ ∀x : x ∈ A ⇒ x ∈ B
Discrete Mathematics I – p. 81/292
![Page 293: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/293.jpg)
SetsElement x is in set A: x ∈ A
Jupiter ∈ Planets , orange 6∈ Junk
Set A is a subset of set B, if all elements of A are alsoelements of B (but not necessarily the other wayround)
A ⊆ B ⇐⇒ ∀x : x ∈ A ⇒ x ∈ B
Discrete Mathematics I – p. 81/292
![Page 294: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/294.jpg)
SetsElement x is in set A: x ∈ A
Jupiter ∈ Planets , orange 6∈ Junk
Set A is a subset of set B, if all elements of A are alsoelements of B (but not necessarily the other wayround)
A ⊆ B ⇐⇒ ∀x : x ∈ A ⇒ x ∈ B
Discrete Mathematics I – p. 81/292
![Page 295: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/295.jpg)
SetsIn particular, for any set A: A ⊆ A ∅ ⊆ A
∅ ⊆ ∅Neven ⊆ N
∅ ⊆ {banana} ⊆ {banana, 239} ⊆ Junk
Discrete Mathematics I – p. 82/292
![Page 296: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/296.jpg)
SetsIn particular, for any set A: A ⊆ A ∅ ⊆ A
∅ ⊆ ∅
Neven ⊆ N
∅ ⊆ {banana} ⊆ {banana, 239} ⊆ Junk
Discrete Mathematics I – p. 82/292
![Page 297: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/297.jpg)
SetsIn particular, for any set A: A ⊆ A ∅ ⊆ A
∅ ⊆ ∅Neven ⊆ N
∅ ⊆ {banana} ⊆ {banana, 239} ⊆ Junk
Discrete Mathematics I – p. 82/292
![Page 298: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/298.jpg)
SetsIn particular, for any set A: A ⊆ A ∅ ⊆ A
∅ ⊆ ∅Neven ⊆ N
∅ ⊆ {banana} ⊆ {banana, 239} ⊆ Junk
Discrete Mathematics I – p. 82/292
![Page 299: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/299.jpg)
SetsWhat are the axioms of set theory?
The Law of Extensionality:
Two sets with all the same elements are equal
For any sets A, B: (A ⊆ B) ∧ (B ⊆ A) ⇒ A = B
In particular, there is only one empty set
Discrete Mathematics I – p. 83/292
![Page 300: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/300.jpg)
SetsWhat are the axioms of set theory?
The Law of Extensionality:
Two sets with all the same elements are equal
For any sets A, B: (A ⊆ B) ∧ (B ⊆ A) ⇒ A = B
In particular, there is only one empty set
Discrete Mathematics I – p. 83/292
![Page 301: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/301.jpg)
SetsWhat are the axioms of set theory?
The Law of Extensionality:
Two sets with all the same elements are equal
For any sets A, B: (A ⊆ B) ∧ (B ⊆ A) ⇒ A = B
In particular, there is only one empty set
Discrete Mathematics I – p. 83/292
![Page 302: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/302.jpg)
SetsWhat are the axioms of set theory?
The Law of Extensionality:
Two sets with all the same elements are equal
For any sets A, B: (A ⊆ B) ∧ (B ⊆ A) ⇒ A = B
In particular, there is only one empty set
Discrete Mathematics I – p. 83/292
![Page 303: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/303.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}In particular: {x ∈ S | T} = S {x ∈ S | F} = ∅
Discrete Mathematics I – p. 84/292
![Page 304: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/304.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}In particular: {x ∈ S | T} = S {x ∈ S | F} = ∅
Discrete Mathematics I – p. 84/292
![Page 305: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/305.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}
In particular: {x ∈ S | T} = S {x ∈ S | F} = ∅
Discrete Mathematics I – p. 84/292
![Page 306: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/306.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}In particular: {x ∈ S | T} =
S {x ∈ S | F} = ∅
Discrete Mathematics I – p. 84/292
![Page 307: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/307.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}In particular: {x ∈ S | T} = S
{x ∈ S | F} = ∅
Discrete Mathematics I – p. 84/292
![Page 308: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/308.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}In particular: {x ∈ S | T} = S {x ∈ S | F} =
∅
Discrete Mathematics I – p. 84/292
![Page 309: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/309.jpg)
SetsLet P (x) be a predicate
{x | P (x)}: the set of all x, such that P (x) is true
Range often made explicit: {x ∈ S | P (x)}In particular: {x ∈ S | T} = S {x ∈ S | F} = ∅
Discrete Mathematics I – p. 84/292
![Page 310: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/310.jpg)
SetsExamples:
{x ∈ N | x > 0} = {1, 2, 3, 4, 5, 6, . . .}
{x ∈ Planets | x is red} = {Mars}{x ∈ N | x ≥ 0} = N
{x ∈ Planets | x is a banana} = ∅
Discrete Mathematics I – p. 85/292
![Page 311: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/311.jpg)
SetsExamples:
{x ∈ N | x > 0} = {1, 2, 3, 4, 5, 6, . . .}{x ∈ Planets | x is red} = {Mars}
{x ∈ N | x ≥ 0} = N
{x ∈ Planets | x is a banana} = ∅
Discrete Mathematics I – p. 85/292
![Page 312: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/312.jpg)
SetsExamples:
{x ∈ N | x > 0} = {1, 2, 3, 4, 5, 6, . . .}{x ∈ Planets | x is red} = {Mars}{x ∈ N | x ≥ 0} = N
{x ∈ Planets | x is a banana} = ∅
Discrete Mathematics I – p. 85/292
![Page 313: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/313.jpg)
SetsExamples:
{x ∈ N | x > 0} = {1, 2, 3, 4, 5, 6, . . .}{x ∈ Planets | x is red} = {Mars}{x ∈ N | x ≥ 0} = N
{x ∈ Planets | x is a banana} = ∅
Discrete Mathematics I – p. 85/292
![Page 314: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/314.jpg)
SetsAnother axiom of set theory
The Law of Abstraction:
For any predicate P (x), there is a set {x | P (x)}
Discrete Mathematics I – p. 86/292
![Page 315: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/315.jpg)
SetsAnother axiom of set theory
The Law of Abstraction:
For any predicate P (x), there is a set {x | P (x)}
Discrete Mathematics I – p. 86/292
![Page 316: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/316.jpg)
SetsAnother axiom of set theory
The Law of Abstraction:
For any predicate P (x), there is a set {x | P (x)}Extensionality + Abstraction = Set Theory
Discrete Mathematics I – p. 86/292
![Page 317: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/317.jpg)
SetsAnother axiom of set theory
The Law of Abstraction:
For any predicate P (x), there is a set {x | P (x)}Extensionality + Abstraction = CONTRADICTION
Discrete Mathematics I – p. 86/292
![Page 318: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/318.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 319: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/319.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 320: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/320.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 321: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/321.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}
B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 322: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/322.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 323: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/323.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B
B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 324: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/324.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 325: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/325.jpg)
SetsRussell’s paradox
A barber shaves everyone who does not shavehimself. Who shaves the barber?
Let P (x) = x 6∈ x
Let B = {x | P (x)} = {x | x 6∈ x}B ∈ B — true or false?
B ∈ B =⇒ B 6∈ B B 6∈ B =⇒ B ∈ B
Contradiction!
Discrete Mathematics I – p. 87/292
![Page 326: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/326.jpg)
SetsRussell’s paradox can only be explained byinconsistency of axioms
Set theory can be fixed — no details here
Extensionality + Abstraction =Naive set theory
Discrete Mathematics I – p. 88/292
![Page 327: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/327.jpg)
SetsRussell’s paradox can only be explained byinconsistency of axioms
Set theory can be fixed — no details here
Extensionality + Abstraction =Naive set theory
Discrete Mathematics I – p. 88/292
![Page 328: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/328.jpg)
SetsRussell’s paradox can only be explained byinconsistency of axioms
Set theory can be fixed — no details here
Extensionality + Abstraction =
Naive set theory
Discrete Mathematics I – p. 88/292
![Page 329: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/329.jpg)
SetsRussell’s paradox can only be explained byinconsistency of axioms
Set theory can be fixed — no details here
Extensionality + Abstraction = Naive set theory
Discrete Mathematics I – p. 88/292
![Page 330: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/330.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}Venn diagrams (illustration only!):
A B A ∩B A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 331: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/331.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}
Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}Venn diagrams (illustration only!):
A B A ∩B A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 332: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/332.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}
Venn diagrams (illustration only!):
A B A ∩B A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 333: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/333.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}Venn diagrams (illustration only!):
A B
A ∩B A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 334: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/334.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}Venn diagrams (illustration only!):
A B A ∩B
A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 335: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/335.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}Venn diagrams (illustration only!):
A B A ∩B A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 336: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/336.jpg)
SetsOperations on sets:
Intersection: A ∩B = {x | (x ∈ A) ∧ (x ∈ B)}Union: A ∪B = {x | (x ∈ A) ∨ (x ∈ B)}Venn diagrams (illustration only!):
A B A ∩B A ∪B
Sets A, B are called disjoint, if A ∩B = ∅
Discrete Mathematics I – p. 89/292
![Page 337: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/337.jpg)
SetsMore operations on sets:
Difference: A \B = {x | (x ∈ A) ∧ (x 6∈ B)}
A B A \B B \ A
If A, B disjoint, then A \B = A, B \ A = B
Discrete Mathematics I – p. 90/292
![Page 338: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/338.jpg)
SetsMore operations on sets:
Difference: A \B = {x | (x ∈ A) ∧ (x 6∈ B)}
A B
A \B B \ A
If A, B disjoint, then A \B = A, B \ A = B
Discrete Mathematics I – p. 90/292
![Page 339: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/339.jpg)
SetsMore operations on sets:
Difference: A \B = {x | (x ∈ A) ∧ (x 6∈ B)}
A B A \B
B \ A
If A, B disjoint, then A \B = A, B \ A = B
Discrete Mathematics I – p. 90/292
![Page 340: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/340.jpg)
SetsMore operations on sets:
Difference: A \B = {x | (x ∈ A) ∧ (x 6∈ B)}
A B A \B B \ A
If A, B disjoint, then A \B = A, B \ A = B
Discrete Mathematics I – p. 90/292
![Page 341: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/341.jpg)
SetsMore operations on sets:
Difference: A \B = {x | (x ∈ A) ∧ (x 6∈ B)}
A B A \B B \ A
If A, B disjoint, then A \B = A, B \ A = B
Discrete Mathematics I – p. 90/292
![Page 342: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/342.jpg)
SetsLet S be a fixed (universal) set, A ⊆ S
Complement of A (with respect to S): A = S \ A
S
A
A
Discrete Mathematics I – p. 91/292
![Page 343: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/343.jpg)
SetsLet S be a fixed (universal) set, A ⊆ S
Complement of A (with respect to S): A = S \ A
S
A
A
Discrete Mathematics I – p. 91/292
![Page 344: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/344.jpg)
SetsLet S be a fixed (universal) set, A ⊆ S
Complement of A (with respect to S): A = S \ A
S
A
A
Discrete Mathematics I – p. 91/292
![Page 345: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/345.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}
A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 346: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/346.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B =
{a, b, c, f, g} A ∩B = {a, c}A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 347: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/347.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g}
A ∩B = {a, c}A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 348: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/348.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B =
{a, c}A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 349: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/349.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}
A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 350: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/350.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B =
{b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 351: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/351.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B = {b}
B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 352: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/352.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B = {b} B \ A =
{f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 353: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/353.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B = {b} B \ A = {f, g}
Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 354: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/354.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 355: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/355.jpg)
SetsExamples:
A = {a, b, c}, B = {c, a, f, g}A ∪B = {a, b, c, f, g} A ∩B = {a, c}A \B = {b} B \ A = {f, g}Note: A ∪B = (A ∩B) ∪ (A \B) ∪ (B \ A)
Also: (A ∪B) \ (A ∩B) = (A \B) ∪ (B \ A)
Discrete Mathematics I – p. 92/292
![Page 356: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/356.jpg)
SetsLaws of set operations (hold for any A, B, C):
¯A = A double complement
A ∩ A = A ∩ idempotentA ∪ A = A ∪ idempotent
A ∩B = B ∩ A ∩ commutativeA ∪B = B ∪ A ∪ commutative
Discrete Mathematics I – p. 93/292
![Page 357: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/357.jpg)
SetsLaws of set operations (hold for any A, B, C):
¯A = A double complement
A ∩ A = A ∩ idempotentA ∪ A = A ∪ idempotent
A ∩B = B ∩ A ∩ commutativeA ∪B = B ∪ A ∪ commutative
Discrete Mathematics I – p. 93/292
![Page 358: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/358.jpg)
SetsLaws of set operations (hold for any A, B, C):
¯A = A double complement
A ∩ A = A ∩ idempotentA ∪ A = A ∪ idempotent
A ∩B = B ∩ A ∩ commutativeA ∪B = B ∪ A ∪ commutative
Discrete Mathematics I – p. 93/292
![Page 359: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/359.jpg)
SetsMore laws of set operations:
(A ∩B) ∩ C = A ∩ (B ∩ C) ∩ associative(A ∪B) ∪ C = A ∪ (B ∪ C) ∪ associative
A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)∩ distributes over ∪
A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)∪ distributes over ∩
Compare with arithmetic and Boolean logic
Discrete Mathematics I – p. 94/292
![Page 360: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/360.jpg)
SetsMore laws of set operations:
(A ∩B) ∩ C = A ∩ (B ∩ C) ∩ associative(A ∪B) ∪ C = A ∪ (B ∪ C) ∪ associative
A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)∩ distributes over ∪
A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)∪ distributes over ∩
Compare with arithmetic and Boolean logic
Discrete Mathematics I – p. 94/292
![Page 361: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/361.jpg)
SetsMore laws of set operations:
(A ∩B) ∩ C = A ∩ (B ∩ C) ∩ associative(A ∪B) ∪ C = A ∪ (B ∪ C) ∪ associative
A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)∩ distributes over ∪
A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)∪ distributes over ∩
Compare with arithmetic and Boolean logic
Discrete Mathematics I – p. 94/292
![Page 362: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/362.jpg)
SetsMore laws of set operations:
(A ∩B) ∩ C = A ∩ (B ∩ C) ∩ associative(A ∪B) ∪ C = A ∪ (B ∪ C) ∪ associative
A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)∩ distributes over ∪
A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)∪ distributes over ∩
Compare with arithmetic and Boolean logic
Discrete Mathematics I – p. 94/292
![Page 363: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/363.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C B ∪ C A ∩ (B ∪ C)
A ∩B A ∩ C (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 364: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/364.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C
B ∪ C A ∩ (B ∪ C)
A ∩B A ∩ C (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 365: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/365.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C B ∪ C
A ∩ (B ∪ C)
A ∩B A ∩ C (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 366: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/366.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C B ∪ C A ∩ (B ∪ C)
A ∩B A ∩ C (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 367: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/367.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C B ∪ C A ∩ (B ∪ C)
A ∩B
A ∩ C (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 368: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/368.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C B ∪ C A ∩ (B ∪ C)
A ∩B A ∩ C
(A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 369: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/369.jpg)
SetsA ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
A B
C B ∪ C A ∩ (B ∪ C)
A ∩B A ∩ C (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 95/292
![Page 370: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/370.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 371: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/371.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 372: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/372.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒
(x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 373: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/373.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒
(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 374: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/374.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒
(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 375: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/375.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒
(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 376: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/376.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒
x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 377: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/377.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 378: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/378.jpg)
SetsProve A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Proof. Consider any x.
x ∈ A∩(B∪C) ⇐⇒ (x ∈ A)∧(x ∈ (B∪C)) ⇐⇒(x ∈ A) ∧ (x ∈ B ∨ x ∈ C) ⇐⇒(x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C) ⇐⇒(x ∈ A∩B)∨(x ∈ A∩C) ⇐⇒ x ∈ (A∩B)∪(A∩C)
Hence A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
Discrete Mathematics I – p. 96/292
![Page 379: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/379.jpg)
SetsDe Morgan’s laws:
A ∩B = A ∪ B A ∪B = A ∩ B
Thus, A ∩B = A ∪ B,so ∩ can be expressed via , ∪Alternatively, A ∪B = A ∩ B,so ∪ can be expressed via , ∩(Cannot remove both ∩, ∪ at the same time!)
Discrete Mathematics I – p. 97/292
![Page 380: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/380.jpg)
SetsDe Morgan’s laws:
A ∩B = A ∪ B A ∪B = A ∩ B
Thus, A ∩B = A ∪ B,so ∩ can be expressed via , ∪
Alternatively, A ∪B = A ∩ B,so ∪ can be expressed via , ∩(Cannot remove both ∩, ∪ at the same time!)
Discrete Mathematics I – p. 97/292
![Page 381: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/381.jpg)
SetsDe Morgan’s laws:
A ∩B = A ∪ B A ∪B = A ∩ B
Thus, A ∩B = A ∪ B,so ∩ can be expressed via , ∪Alternatively, A ∪B = A ∩ B,so ∪ can be expressed via , ∩
(Cannot remove both ∩, ∪ at the same time!)
Discrete Mathematics I – p. 97/292
![Page 382: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/382.jpg)
SetsDe Morgan’s laws:
A ∩B = A ∪ B A ∪B = A ∩ B
Thus, A ∩B = A ∪ B,so ∩ can be expressed via , ∪Alternatively, A ∪B = A ∩ B,so ∪ can be expressed via , ∩(Cannot remove both ∩, ∪ at the same time!)
Discrete Mathematics I – p. 97/292
![Page 383: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/383.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B A ∩B
A B A ∪ B
Discrete Mathematics I – p. 98/292
![Page 384: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/384.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B A ∩B
A B A ∪ B
Discrete Mathematics I – p. 98/292
![Page 385: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/385.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B
A ∩B
A B A ∪ B
Discrete Mathematics I – p. 98/292
![Page 386: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/386.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B A ∩B
A B A ∪ B
Discrete Mathematics I – p. 98/292
![Page 387: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/387.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B A ∩B
A
B A ∪ B
Discrete Mathematics I – p. 98/292
![Page 388: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/388.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B A ∩B
A B
A ∪ B
Discrete Mathematics I – p. 98/292
![Page 389: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/389.jpg)
Sets
A ∩B = A ∪ B
S
A B
A ∩B A ∩B
A B A ∪ B
Discrete Mathematics I – p. 98/292
![Page 390: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/390.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 391: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/391.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 392: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/392.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒
x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 393: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/393.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒
¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 394: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/394.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒
(x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 395: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/395.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒
(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 396: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/396.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒
x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 397: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/397.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 398: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/398.jpg)
Sets
Prove A ∩B = A ∪ B
Proof. Consider any x ∈ S.
x ∈ A ∩B ⇐⇒ x 6∈ A ∩B ⇐⇒¬(x ∈ A ∧ x ∈ B) ⇐⇒ (x 6∈ A) ∨ (x 6∈ B) ⇐⇒(x ∈ A) ∨ (x ∈ B) ⇐⇒ x ∈ (A ∪ B)
Hence A ∩B = A ∪ B
Discrete Mathematics I – p. 99/292
![Page 399: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/399.jpg)
SetsStill more laws:
Let A ⊆ S
A ∩ S = A A ∪ ∅ = A identity laws
A ∩ ∅ = ∅ A ∪ S = S annihilation laws
A ∩ A = ∅ A ∪ A = Slaws of excluded middle
A ∩ (A ∪B) = A = A ∪ (A ∩B)absorption laws
Discrete Mathematics I – p. 100/292
![Page 400: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/400.jpg)
SetsStill more laws:
Let A ⊆ S
A ∩ S = A A ∪ ∅ = A identity laws
A ∩ ∅ = ∅ A ∪ S = S annihilation laws
A ∩ A = ∅ A ∪ A = Slaws of excluded middle
A ∩ (A ∪B) = A = A ∪ (A ∩B)absorption laws
Discrete Mathematics I – p. 100/292
![Page 401: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/401.jpg)
SetsStill more laws:
Let A ⊆ S
A ∩ S = A A ∪ ∅ = A identity laws
A ∩ ∅ = ∅ A ∪ S = S annihilation laws
A ∩ A = ∅ A ∪ A = Slaws of excluded middle
A ∩ (A ∪B) = A = A ∪ (A ∩B)absorption laws
Discrete Mathematics I – p. 100/292
![Page 402: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/402.jpg)
SetsStill more laws:
Let A ⊆ S
A ∩ S = A A ∪ ∅ = A identity laws
A ∩ ∅ = ∅ A ∪ S = S annihilation laws
A ∩ A = ∅ A ∪ A = Slaws of excluded middle
A ∩ (A ∪B) = A = A ∪ (A ∩B)absorption laws
Discrete Mathematics I – p. 100/292
![Page 403: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/403.jpg)
SetsA structure with such properties is called a Booleanalgebra
Examples:
B = {F, T}operations ∧, ∨, ¬ identities F , T
Set of all subsets of fixed S
operations ∩, ∪,¯ identities ∅, S
Discrete Mathematics I – p. 101/292
![Page 404: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/404.jpg)
SetsA structure with such properties is called a Booleanalgebra
Examples:
B = {F, T}operations ∧, ∨, ¬ identities F , T
Set of all subsets of fixed S
operations ∩, ∪,¯ identities ∅, S
Discrete Mathematics I – p. 101/292
![Page 405: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/405.jpg)
SetsA structure with such properties is called a Booleanalgebra
Examples:
B = {F, T}operations ∧, ∨, ¬ identities F , T
Set of all subsets of fixed S
operations ∩, ∪,¯ identities ∅, S
Discrete Mathematics I – p. 101/292
![Page 406: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/406.jpg)
SetsThe powerset of S is the set of all subsets of S
P(S) = {A | A ⊆ S}A ∈ P(S) ⇐⇒ A ⊆ S
Discrete Mathematics I – p. 102/292
![Page 407: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/407.jpg)
SetsExamples:
P(∅) =
{∅} (note: P(∅) 6= ∅!)
P({Bunty}) = {∅, {Bunty}}P({a, b, c}) =
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 408: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/408.jpg)
SetsExamples:
P(∅) = {∅}
(note: P(∅) 6= ∅!)
P({Bunty}) = {∅, {Bunty}}P({a, b, c}) =
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 409: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/409.jpg)
SetsExamples:
P(∅) = {∅} (note: P(∅) 6= ∅!)
P({Bunty}) = {∅, {Bunty}}P({a, b, c}) =
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 410: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/410.jpg)
SetsExamples:
P(∅) = {∅} (note: P(∅) 6= ∅!)
P({Bunty}) =
{∅, {Bunty}}P({a, b, c}) =
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 411: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/411.jpg)
SetsExamples:
P(∅) = {∅} (note: P(∅) 6= ∅!)
P({Bunty}) = {∅, {Bunty}}
P({a, b, c}) ={∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 412: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/412.jpg)
SetsExamples:
P(∅) = {∅} (note: P(∅) 6= ∅!)
P({Bunty}) = {∅, {Bunty}}P({a, b, c}) =
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 413: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/413.jpg)
SetsExamples:
P(∅) = {∅} (note: P(∅) 6= ∅!)
P({Bunty}) = {∅, {Bunty}}P({a, b, c}) =
{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Discrete Mathematics I – p. 103/292
![Page 414: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/414.jpg)
SetsIf S finite, P(S) finite
If S has n elements, P(S) has 2n elements
If S infinite, P(S) infinite
(Sometimes P(S) denoted 2S , even if S infinite)
Discrete Mathematics I – p. 104/292
![Page 415: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/415.jpg)
SetsIf S finite, P(S) finite
If S has n elements, P(S) has 2n elements
If S infinite, P(S) infinite
(Sometimes P(S) denoted 2S , even if S infinite)
Discrete Mathematics I – p. 104/292
![Page 416: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/416.jpg)
SetsIf S finite, P(S) finite
If S has n elements, P(S) has 2n elements
If S infinite, P(S) infinite
(Sometimes P(S) denoted 2S , even if S infinite)
Discrete Mathematics I – p. 104/292
![Page 417: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/417.jpg)
SetsIf S finite, P(S) finite
If S has n elements, P(S) has 2n elements
If S infinite, P(S) infinite
(Sometimes P(S) denoted 2S , even if S infinite)
Discrete Mathematics I – p. 104/292
![Page 418: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/418.jpg)
SetsProperties of P:
P(A ∩B) = P(A) ∩ P(B)
In general, P(A ∪B) 6= P(A) ∪ P(B)(but ⊇ holds)
In general, P(A \B) 6= P(A) \ P(B)(but ⊆ holds)
Discrete Mathematics I – p. 105/292
![Page 419: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/419.jpg)
SetsProperties of P:
P(A ∩B) = P(A) ∩ P(B)
In general, P(A ∪B) 6= P(A) ∪ P(B)(but ⊇ holds)
In general, P(A \B) 6= P(A) \ P(B)(but ⊆ holds)
Discrete Mathematics I – p. 105/292
![Page 420: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/420.jpg)
SetsProperties of P:
P(A ∩B) = P(A) ∩ P(B)
In general, P(A ∪B) 6= P(A) ∪ P(B)(but ⊇ holds)
In general, P(A \B) 6= P(A) \ P(B)(but ⊆ holds)
Discrete Mathematics I – p. 105/292
![Page 421: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/421.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B).
True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 422: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/422.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 423: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/423.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 424: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/424.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒
(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 425: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/425.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒
(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 426: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/426.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒
(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 427: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/427.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒
∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 428: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/428.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒
∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 429: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/429.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒
X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 430: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/430.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒
X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 431: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/431.jpg)
SetsProve or disprove: For all A, B,P(A) ∪ P(B) ⊆ P(A ∪B). True
Proof. Consider any X .
X ∈ P(A) ∪ P(B) ⇐⇒(X ∈ P(A)) ∨ (X ∈ P(B)) ⇐⇒(X ⊆ A) ∨ (X ⊆ B) ⇐⇒(∀x ∈ X : x ∈ A) ∨ (∀x ∈ X : x ∈ B) =⇒∀x ∈ X : (x ∈ A) ∨ (x ∈ B) ⇐⇒∀x ∈ X : (x ∈ A ∪B) ⇐⇒X ⊆ A ∪B ⇐⇒ X ∈ P(A ∪B)
Discrete Mathematics I – p. 106/292
![Page 432: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/432.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B).
False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 433: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/433.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 434: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/434.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 435: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/435.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}.
Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 436: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/436.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 437: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/437.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 438: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/438.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}}
X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 439: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/439.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}
=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 440: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/440.jpg)
SetsProve or disprove: For all A, B,P(A ∪B) ⊆ P(A) ∪ P(B). False
Proof. Need to find A, B, X such thatX ∈ P(A ∪B), X 6∈ P(A) ∪ P(B).
Let A = {0}, B = {1}. Let X = {0, 1}.
X ⊆ A ∪B = {0, 1} =⇒ X ∈ P(A ∪B)
X 6∈ P(A) = {∅, {0}} X 6∈ P(B) = {∅, {1}}=⇒ X 6∈ P(A) ∪ P(B)
Discrete Mathematics I – p. 107/292
![Page 441: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/441.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 442: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/442.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 443: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/443.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 444: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/444.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 445: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/445.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 446: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/446.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 447: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/447.jpg)
SetsLet x1, x2, . . . , xn be any elements (n ∈ N)
A (finite) sequence: (x1, x2, . . . , xn)
JunkSeq1 = (239, banana, ace of spades)
JunkSeq2 = (banana, 239, ace of spades, 239)
JunkSeq1 6= JunkSeq2
A sequence is not a set!
(. . . and not a basic concept, will be defined later)
Discrete Mathematics I – p. 108/292
![Page 448: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/448.jpg)
SetsFor sequences, repetitions and order matter
(x, y) 6= (y, x) 6= (y, x, x)
Number n is sequence length
length(JunkSeq1 ) = 3 length((x, y)) = 2
Sequence of length 2 is called an ordered pair
A direct definition: (x, y) means {{x, y}, x}
Discrete Mathematics I – p. 109/292
![Page 449: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/449.jpg)
SetsFor sequences, repetitions and order matter
(x, y) 6= (y, x) 6= (y, x, x)
Number n is sequence length
length(JunkSeq1 ) = 3 length((x, y)) = 2
Sequence of length 2 is called an ordered pair
A direct definition: (x, y) means {{x, y}, x}
Discrete Mathematics I – p. 109/292
![Page 450: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/450.jpg)
SetsFor sequences, repetitions and order matter
(x, y) 6= (y, x) 6= (y, x, x)
Number n is sequence length
length(JunkSeq1 ) = 3 length((x, y)) = 2
Sequence of length 2 is called an ordered pair
A direct definition: (x, y) means {{x, y}, x}
Discrete Mathematics I – p. 109/292
![Page 451: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/451.jpg)
SetsFor sequences, repetitions and order matter
(x, y) 6= (y, x) 6= (y, x, x)
Number n is sequence length
length(JunkSeq1 ) = 3 length((x, y)) = 2
Sequence of length 2 is called an ordered pair
A direct definition: (x, y) means {{x, y}, x}
Discrete Mathematics I – p. 109/292
![Page 452: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/452.jpg)
SetsFor sequences, repetitions and order matter
(x, y) 6= (y, x) 6= (y, x, x)
Number n is sequence length
length(JunkSeq1 ) = 3 length((x, y)) = 2
Sequence of length 2 is called an ordered pair
A direct definition: (x, y) means {{x, y}, x}
Discrete Mathematics I – p. 109/292
![Page 453: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/453.jpg)
SetsFor sequences, repetitions and order matter
(x, y) 6= (y, x) 6= (y, x, x)
Number n is sequence length
length(JunkSeq1 ) = 3 length((x, y)) = 2
Sequence of length 2 is called an ordered pair
A direct definition: (x, y) means {{x, y}, x}
Discrete Mathematics I – p. 109/292
![Page 454: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/454.jpg)
SetsThe Cartesian product of sets A, B is the set of allordered pairs (a, b), where a ∈ A, b ∈ B
(After R. Descartes, 1596–1650)
A×B = {(a, b) | (a ∈ A) ∧ (b ∈ B)}A2 = A× A the Cartesian square of A
Discrete Mathematics I – p. 110/292
![Page 455: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/455.jpg)
SetsThe Cartesian product of sets A, B is the set of allordered pairs (a, b), where a ∈ A, b ∈ B
(After R. Descartes, 1596–1650)
A×B = {(a, b) | (a ∈ A) ∧ (b ∈ B)}A2 = A× A the Cartesian square of A
Discrete Mathematics I – p. 110/292
![Page 456: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/456.jpg)
SetsThe Cartesian product of sets A, B is the set of allordered pairs (a, b), where a ∈ A, b ∈ B
(After R. Descartes, 1596–1650)
A×B = {(a, b) | (a ∈ A) ∧ (b ∈ B)}
A2 = A× A the Cartesian square of A
Discrete Mathematics I – p. 110/292
![Page 457: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/457.jpg)
SetsThe Cartesian product of sets A, B is the set of allordered pairs (a, b), where a ∈ A, b ∈ B
(After R. Descartes, 1596–1650)
A×B = {(a, b) | (a ∈ A) ∧ (b ∈ B)}A2 = A× A the Cartesian square of A
Discrete Mathematics I – p. 110/292
![Page 458: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/458.jpg)
SetsExamples:
∅ × A =
A× ∅ = ∅ for any set A
{Bunty} × {Fowler} = {(Bunty, Fowler)}{Fowler} × {Bunty} = {(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 459: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/459.jpg)
SetsExamples:
∅ × A = A× ∅ =
∅ for any set A
{Bunty} × {Fowler} = {(Bunty, Fowler)}{Fowler} × {Bunty} = {(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 460: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/460.jpg)
SetsExamples:
∅ × A = A× ∅ = ∅ for any set A
{Bunty} × {Fowler} = {(Bunty, Fowler)}{Fowler} × {Bunty} = {(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 461: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/461.jpg)
SetsExamples:
∅ × A = A× ∅ = ∅ for any set A
{Bunty} × {Fowler} =
{(Bunty, Fowler)}{Fowler} × {Bunty} = {(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 462: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/462.jpg)
SetsExamples:
∅ × A = A× ∅ = ∅ for any set A
{Bunty} × {Fowler} = {(Bunty, Fowler)}
{Fowler} × {Bunty} = {(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 463: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/463.jpg)
SetsExamples:
∅ × A = A× ∅ = ∅ for any set A
{Bunty} × {Fowler} = {(Bunty, Fowler)}{Fowler} × {Bunty} =
{(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 464: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/464.jpg)
SetsExamples:
∅ × A = A× ∅ = ∅ for any set A
{Bunty} × {Fowler} = {(Bunty, Fowler)}{Fowler} × {Bunty} = {(Fowler, Bunty)}
Discrete Mathematics I – p. 111/292
![Page 465: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/465.jpg)
SetsMore examples:
{a, b, c} × {d, e} =
{(a, d), (a, e), (b, d), (b, e), (c, d), (c, e)}N×Planets = {(n, x) | (n ∈ N)∧ (x ∈ Planets)} ={(5, Saturn), (239, Earth), . . .}N2 = N× N = {(m, n) | m, n ∈ N}
Discrete Mathematics I – p. 112/292
![Page 466: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/466.jpg)
SetsMore examples:
{a, b, c} × {d, e} ={(a, d), (a, e), (b, d), (b, e), (c, d), (c, e)}
N×Planets = {(n, x) | (n ∈ N)∧ (x ∈ Planets)} ={(5, Saturn), (239, Earth), . . .}N2 = N× N = {(m, n) | m, n ∈ N}
Discrete Mathematics I – p. 112/292
![Page 467: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/467.jpg)
SetsMore examples:
{a, b, c} × {d, e} ={(a, d), (a, e), (b, d), (b, e), (c, d), (c, e)}N×Planets =
{(n, x) | (n ∈ N)∧ (x ∈ Planets)} ={(5, Saturn), (239, Earth), . . .}N2 = N× N = {(m, n) | m, n ∈ N}
Discrete Mathematics I – p. 112/292
![Page 468: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/468.jpg)
SetsMore examples:
{a, b, c} × {d, e} ={(a, d), (a, e), (b, d), (b, e), (c, d), (c, e)}N×Planets = {(n, x) | (n ∈ N)∧ (x ∈ Planets)} =
{(5, Saturn), (239, Earth), . . .}N2 = N× N = {(m, n) | m, n ∈ N}
Discrete Mathematics I – p. 112/292
![Page 469: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/469.jpg)
SetsMore examples:
{a, b, c} × {d, e} ={(a, d), (a, e), (b, d), (b, e), (c, d), (c, e)}N×Planets = {(n, x) | (n ∈ N)∧ (x ∈ Planets)} ={(5, Saturn), (239, Earth), . . .}
N2 = N× N = {(m, n) | m, n ∈ N}
Discrete Mathematics I – p. 112/292
![Page 470: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/470.jpg)
SetsMore examples:
{a, b, c} × {d, e} ={(a, d), (a, e), (b, d), (b, e), (c, d), (c, e)}N×Planets = {(n, x) | (n ∈ N)∧ (x ∈ Planets)} ={(5, Saturn), (239, Earth), . . .}N2 = N× N = {(m, n) | m, n ∈ N}
Discrete Mathematics I – p. 112/292
![Page 471: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/471.jpg)
SetsIf A, B finite, A×B finite
If A has m elements, B has n elements, then A×Bhas m · n elements
(. . . hence the “×” sign)
If A infinite, B nonempty, then A×B, B ×A infinite
Discrete Mathematics I – p. 113/292
![Page 472: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/472.jpg)
SetsIf A, B finite, A×B finite
If A has m elements, B has n elements, then A×Bhas m · n elements
(. . . hence the “×” sign)
If A infinite, B nonempty, then A×B, B ×A infinite
Discrete Mathematics I – p. 113/292
![Page 473: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/473.jpg)
SetsIf A, B finite, A×B finite
If A has m elements, B has n elements, then A×Bhas m · n elements
(. . . hence the “×” sign)
If A infinite, B nonempty, then A×B, B ×A infinite
Discrete Mathematics I – p. 113/292
![Page 474: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/474.jpg)
SetsIf A, B finite, A×B finite
If A has m elements, B has n elements, then A×Bhas m · n elements
(. . . hence the “×” sign)
If A infinite, B nonempty, then A×B, B ×A infinite
Discrete Mathematics I – p. 113/292
![Page 475: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/475.jpg)
SetsProperties of ×:
In general, A×B 6= B × A
In general, (A×B)× C 6= A× (B × C)
. . . but = holds if we identify ((a, b), c) and (a, (b, c))
Discrete Mathematics I – p. 114/292
![Page 476: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/476.jpg)
SetsProperties of ×:
In general, A×B 6= B × A
In general, (A×B)× C 6= A× (B × C)
. . . but = holds if we identify ((a, b), c) and (a, (b, c))
Discrete Mathematics I – p. 114/292
![Page 477: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/477.jpg)
SetsProperties of ×:
In general, A×B 6= B × A
In general, (A×B)× C 6= A× (B × C)
. . . but = holds if we identify ((a, b), c) and (a, (b, c))
Discrete Mathematics I – p. 114/292
![Page 478: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/478.jpg)
SetsProperties of ×:
In general, A×B 6= B × A
In general, (A×B)× C 6= A× (B × C)
. . . but = holds if we identify ((a, b), c) and (a, (b, c))
Discrete Mathematics I – p. 114/292
![Page 479: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/479.jpg)
SetsA× (B ∩ C) = (A×B) ∩ (A× C)(A ∩B)× C = (A× C) ∩ (B × C)
A× (B ∪ C) = (A×B) ∪ (A× C)(A ∪B)× C = (A× C) ∪ (B × C)
A× (B \ C) = (A×B) \ (A× C)(A \B)× C = (A× C) \ (B × C)
× distributes over ∩,∪, \
Discrete Mathematics I – p. 115/292
![Page 480: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/480.jpg)
SetsA× (B ∩ C) = (A×B) ∩ (A× C)(A ∩B)× C = (A× C) ∩ (B × C)
A× (B ∪ C) = (A×B) ∪ (A× C)(A ∪B)× C = (A× C) ∪ (B × C)
A× (B \ C) = (A×B) \ (A× C)(A \B)× C = (A× C) \ (B × C)
× distributes over ∩,∪, \
Discrete Mathematics I – p. 115/292
![Page 481: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/481.jpg)
SetsA× (B ∩ C) = (A×B) ∩ (A× C)(A ∩B)× C = (A× C) ∩ (B × C)
A× (B ∪ C) = (A×B) ∪ (A× C)(A ∪B)× C = (A× C) ∪ (B × C)
A× (B \ C) = (A×B) \ (A× C)(A \B)× C = (A× C) \ (B × C)
× distributes over ∩,∪, \
Discrete Mathematics I – p. 115/292
![Page 482: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/482.jpg)
SetsA× (B ∩ C) = (A×B) ∩ (A× C)(A ∩B)× C = (A× C) ∩ (B × C)
A× (B ∪ C) = (A×B) ∪ (A× C)(A ∪B)× C = (A× C) ∪ (B × C)
A× (B \ C) = (A×B) \ (A× C)(A \B)× C = (A× C) \ (B × C)
× distributes over ∩,∪, \
Discrete Mathematics I – p. 115/292
![Page 483: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/483.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 484: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/484.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).
(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 485: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/485.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒
(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 486: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/486.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒
(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 487: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/487.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒
(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 488: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/488.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒
((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 489: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/489.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒
(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 490: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/490.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 491: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/491.jpg)
SetsProve A× (B ∪ C) = (A×B) ∪ (A× C)
Proof. Consider any (x, y).(x, y) ∈ A× (B ∪ C) ⇐⇒(x ∈ A) ∧ (y ∈ (B ∪ C)) ⇐⇒(x ∈ A) ∧ (y ∈ B ∨ y ∈ C) ⇐⇒(x ∈ A ∧ y ∈ B) ∨ (x ∈ A ∧ y ∈ C) ⇐⇒((x, y) ∈ A×B) ∨ ((x, y) ∈ A× C) ⇐⇒(x, y) ∈ (A×B) ∪ (A× C)
Hence A× (B ∪ C) = (A×B) ∪ (A× C).
Discrete Mathematics I – p. 116/292
![Page 492: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/492.jpg)
SetsThe Cartesian product of sets A1, A2, . . . , An is theset of all ordered sequences (a1, a2, . . . , an), whereai ∈ Ai for all i ∈ {1, . . . , n}
A1 × · · · × An ={(a1, . . . , an) | ∀i ∈ {1, . . . , n} : ai ∈ Ai}
An = A× A× · · · × A (n times)the n-th Cartesian power of A
Discrete Mathematics I – p. 117/292
![Page 493: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/493.jpg)
SetsThe Cartesian product of sets A1, A2, . . . , An is theset of all ordered sequences (a1, a2, . . . , an), whereai ∈ Ai for all i ∈ {1, . . . , n}A1 × · · · × An =
{(a1, . . . , an) | ∀i ∈ {1, . . . , n} : ai ∈ Ai}
An = A× A× · · · × A (n times)the n-th Cartesian power of A
Discrete Mathematics I – p. 117/292
![Page 494: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/494.jpg)
SetsThe Cartesian product of sets A1, A2, . . . , An is theset of all ordered sequences (a1, a2, . . . , an), whereai ∈ Ai for all i ∈ {1, . . . , n}A1 × · · · × An =
{(a1, . . . , an) | ∀i ∈ {1, . . . , n} : ai ∈ Ai}An = A× A× · · · × A (n times)
the n-th Cartesian power of A
Discrete Mathematics I – p. 117/292
![Page 495: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/495.jpg)
SetsIf A1, A2, . . . , An finite, A1 × A2 × · · · × An finite
If for all i, Ai has ni elements, A1 × · · · × Ak hasn1 · . . . · nk elements
If one of A1, A2, . . . , An infinite, A1 × A2 × · · · × An
infinite (unless one of them is empty)
Discrete Mathematics I – p. 118/292
![Page 496: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/496.jpg)
SetsIf A1, A2, . . . , An finite, A1 × A2 × · · · × An finite
If for all i, Ai has ni elements, A1 × · · · × Ak hasn1 · . . . · nk elements
If one of A1, A2, . . . , An infinite, A1 × A2 × · · · × An
infinite (unless one of them is empty)
Discrete Mathematics I – p. 118/292
![Page 497: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/497.jpg)
SetsIf A1, A2, . . . , An finite, A1 × A2 × · · · × An finite
If for all i, Ai has ni elements, A1 × · · · × Ak hasn1 · . . . · nk elements
If one of A1, A2, . . . , An infinite, A1 × A2 × · · · × An
infinite (unless one of them is empty)
Discrete Mathematics I – p. 118/292
![Page 498: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/498.jpg)
SetsTherefore:
If A finite, An finite
If A has k elements, An has kn elements
If A infinite, An infinite
Discrete Mathematics I – p. 119/292
![Page 499: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/499.jpg)
Relations
Discrete Mathematics I – p. 120/292
![Page 500: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/500.jpg)
RelationsConsider P (x, y) = x ≤ y x, y ∈ N
{(x, y) | x ≤ y} ⊆ N× N = N2
Discrete Mathematics I – p. 121/292
![Page 501: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/501.jpg)
RelationsConsider P (x, y) = x ≤ y x, y ∈ N
{(x, y) | x ≤ y} ⊆ N× N = N2
Discrete Mathematics I – p. 121/292
![Page 502: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/502.jpg)
RelationsA relation between sets A, B is a subset of A×B
Rp : A ↔ B ⇐⇒ Rp ⊆ A×B
Example:
R≤ = {(a, b) ∈ N× N | a ≤ b} ⊆ N× N = N2
Write a p b for (a, b) ∈ Rp
For example a ≤ b instead of (a, b) ∈ R≤
Discrete Mathematics I – p. 122/292
![Page 503: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/503.jpg)
RelationsA relation between sets A, B is a subset of A×B
Rp : A ↔ B ⇐⇒ Rp ⊆ A×B
Example:
R≤ = {(a, b) ∈ N× N | a ≤ b} ⊆ N× N = N2
Write a p b for (a, b) ∈ Rp
For example a ≤ b instead of (a, b) ∈ R≤
Discrete Mathematics I – p. 122/292
![Page 504: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/504.jpg)
RelationsA relation between sets A, B is a subset of A×B
Rp : A ↔ B ⇐⇒ Rp ⊆ A×B
Example:
R≤ = {(a, b) ∈ N× N | a ≤ b} ⊆ N× N = N2
Write a p b for (a, b) ∈ Rp
For example a ≤ b instead of (a, b) ∈ R≤
Discrete Mathematics I – p. 122/292
![Page 505: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/505.jpg)
RelationsA relation between sets A, B is a subset of A×B
Rp : A ↔ B ⇐⇒ Rp ⊆ A×B
Example:
R≤ = {(a, b) ∈ N× N | a ≤ b} ⊆ N× N = N2
Write a p b for (a, b) ∈ Rp
For example a ≤ b instead of (a, b) ∈ R≤
Discrete Mathematics I – p. 122/292
![Page 506: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/506.jpg)
RelationsExamples of relations:
Equality relation R=A: A ↔ A
R=A= {(a, a) | a ∈ A}
Usually drop A: a = a
Empty relation ∅ : A ↔ A
Complete relation A2 : A ↔ A
Discrete Mathematics I – p. 123/292
![Page 507: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/507.jpg)
RelationsExamples of relations:
Equality relation R=A: A ↔ A
R=A= {(a, a) | a ∈ A}
Usually drop A: a = a
Empty relation ∅ : A ↔ A
Complete relation A2 : A ↔ A
Discrete Mathematics I – p. 123/292
![Page 508: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/508.jpg)
RelationsExamples of relations:
Equality relation R=A: A ↔ A
R=A= {(a, a) | a ∈ A}
Usually drop A: a = a
Empty relation ∅ : A ↔ A
Complete relation A2 : A ↔ A
Discrete Mathematics I – p. 123/292
![Page 509: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/509.jpg)
RelationsExamples of relations:
Equality relation R=A: A ↔ A
R=A= {(a, a) | a ∈ A}
Usually drop A: a = a
Empty relation ∅ : A ↔ A
Complete relation A2 : A ↔ A
Discrete Mathematics I – p. 123/292
![Page 510: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/510.jpg)
RelationsExamples of relations:
Equality relation R=A: A ↔ A
R=A= {(a, a) | a ∈ A}
Usually drop A: a = a
Empty relation ∅ : A ↔ A
Complete relation A2 : A ↔ A
Discrete Mathematics I – p. 123/292
![Page 511: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/511.jpg)
RelationsMore examples of relations:
R<, R≤, R>, R≥ N ↔ N
R| : N ↔ N m | n ⇐⇒ m divides n
m | n ⇐⇒ ∃k ∈ N : k ·m = n
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Discrete Mathematics I – p. 124/292
![Page 512: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/512.jpg)
RelationsMore examples of relations:
R<, R≤, R>, R≥ N ↔ N
R| : N ↔ N m | n ⇐⇒ m divides n
m | n ⇐⇒ ∃k ∈ N : k ·m = n
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Discrete Mathematics I – p. 124/292
![Page 513: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/513.jpg)
RelationsMore examples of relations:
R<, R≤, R>, R≥ N ↔ N
R| : N ↔ N m | n ⇐⇒ m divides n
m | n ⇐⇒ ∃k ∈ N : k ·m = n
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Discrete Mathematics I – p. 124/292
![Page 514: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/514.jpg)
RelationsMore examples of relations:
R<, R≤, R>, R≥ N ↔ N
R| : N ↔ N m | n ⇐⇒ m divides n
m | n ⇐⇒ ∃k ∈ N : k ·m = n
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Discrete Mathematics I – p. 124/292
![Page 515: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/515.jpg)
RelationsMore examples of relations:
R<, R≤, R>, R≥ N ↔ N
R| : N ↔ N m | n ⇐⇒ m divides n
m | n ⇐⇒ ∃k ∈ N : k ·m = n
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Discrete Mathematics I – p. 124/292
![Page 516: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/516.jpg)
RelationsMore examples of relations:
Rs : N ↔ N m s n ⇐⇒ m2 = n
Rm : People ↔ Peoplex m y ⇐⇒ the mother of x is y
In these relations, for every x ∈ A, there is a uniquey ∈ B, such that x is related to y
Such relations are called functions
Discrete Mathematics I – p. 125/292
![Page 517: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/517.jpg)
RelationsMore examples of relations:
Rs : N ↔ N m s n ⇐⇒ m2 = n
Rm : People ↔ Peoplex m y ⇐⇒ the mother of x is y
In these relations, for every x ∈ A, there is a uniquey ∈ B, such that x is related to y
Such relations are called functions
Discrete Mathematics I – p. 125/292
![Page 518: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/518.jpg)
RelationsMore examples of relations:
Rs : N ↔ N m s n ⇐⇒ m2 = n
Rm : People ↔ Peoplex m y ⇐⇒ the mother of x is y
In these relations, for every x ∈ A, there is a uniquey ∈ B, such that x is related to y
Such relations are called functions
Discrete Mathematics I – p. 125/292
![Page 519: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/519.jpg)
RelationsMore examples of relations:
Rs : N ↔ N m s n ⇐⇒ m2 = n
Rm : People ↔ Peoplex m y ⇐⇒ the mother of x is y
In these relations, for every x ∈ A, there is a uniquey ∈ B, such that x is related to y
Such relations are called functions
Discrete Mathematics I – p. 125/292
![Page 520: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/520.jpg)
RelationsRp, Rq : A ↔ B
Rp ∩Rq, Rp ∪Rq, Rp \Rq : A ↔ B
Discrete Mathematics I – p. 126/292
![Page 521: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/521.jpg)
RelationsRp, Rq : A ↔ B
Rp ∩Rq, Rp ∪Rq, Rp \Rq : A ↔ B
Discrete Mathematics I – p. 126/292
![Page 522: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/522.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
The composition of p and q Rp ◦ q : A ↔ C
∀(a, c) ∈ A×C : a(p ◦ q)c ⇔ ∃b ∈ B : (a p b)∧(b q c)
Discrete Mathematics I – p. 127/292
![Page 523: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/523.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
The composition of p and q Rp ◦ q : A ↔ C
∀(a, c) ∈ A×C : a(p ◦ q)c ⇔ ∃b ∈ B : (a p b)∧(b q c)
Discrete Mathematics I – p. 127/292
![Page 524: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/524.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
The composition of p and q Rp ◦ q : A ↔ C
∀(a, c) ∈ A×C : a(p ◦ q)c ⇔ ∃b ∈ B : (a p b)∧(b q c)
Discrete Mathematics I – p. 127/292
![Page 525: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/525.jpg)
RelationsExamples:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Rq ◦ t : People ↔ Animals
x(q ◦ t)z ⇐⇒ x has a parent with pet z
Rq ◦ q : People ↔ People
x(q ◦ q)z ⇐⇒ x is a grandchild of z
Discrete Mathematics I – p. 128/292
![Page 526: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/526.jpg)
RelationsExamples:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Rq ◦ t : People ↔ Animals
x(q ◦ t)z ⇐⇒ x has a parent with pet z
Rq ◦ q : People ↔ People
x(q ◦ q)z ⇐⇒ x is a grandchild of z
Discrete Mathematics I – p. 128/292
![Page 527: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/527.jpg)
RelationsExamples:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rt : People ↔ Animalsx t y ⇐⇒ x has y as a pet
Rq ◦ t : People ↔ Animals
x(q ◦ t)z ⇐⇒ x has a parent with pet z
Rq ◦ q : People ↔ People
x(q ◦ q)z ⇐⇒ x is a grandchild of z
Discrete Mathematics I – p. 128/292
![Page 528: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/528.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 529: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/529.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 530: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/530.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 531: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/531.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 532: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/532.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 533: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/533.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 534: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/534.jpg)
RelationsRp : A ↔ B Rq : B ↔ C
Prove: if Rp, Rq functions, then Rp ◦ q a function
Proof. Consider any x ∈ A.
Rp function =⇒ ∃!y ∈ B : x p y
Rq function =⇒ ∃!z ∈ C : y q z
Hence ∃!z ∈ C : x(p ◦ q)z
Therefore, Rp ◦ q is a function.
Discrete Mathematics I – p. 129/292
![Page 535: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/535.jpg)
RelationsRp : A ↔ B
The inverse of p Rp−1 : B ↔ A
∀(b, a) ∈ B × A : b(p−1)a ⇐⇒ a p b
Example:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rq−1 : People ↔ People
x q−1 y ⇐⇒ x is a parent of y
Discrete Mathematics I – p. 130/292
![Page 536: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/536.jpg)
RelationsRp : A ↔ B
The inverse of p Rp−1 : B ↔ A
∀(b, a) ∈ B × A : b(p−1)a ⇐⇒ a p b
Example:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rq−1 : People ↔ People
x q−1 y ⇐⇒ x is a parent of y
Discrete Mathematics I – p. 130/292
![Page 537: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/537.jpg)
RelationsRp : A ↔ B
The inverse of p Rp−1 : B ↔ A
∀(b, a) ∈ B × A : b(p−1)a ⇐⇒ a p b
Example:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rq−1 : People ↔ People
x q−1 y ⇐⇒ x is a parent of y
Discrete Mathematics I – p. 130/292
![Page 538: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/538.jpg)
RelationsRp : A ↔ B
The inverse of p Rp−1 : B ↔ A
∀(b, a) ∈ B × A : b(p−1)a ⇐⇒ a p b
Example:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rq−1 : People ↔ People
x q−1 y ⇐⇒ x is a parent of y
Discrete Mathematics I – p. 130/292
![Page 539: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/539.jpg)
RelationsRp : A ↔ B
The inverse of p Rp−1 : B ↔ A
∀(b, a) ∈ B × A : b(p−1)a ⇐⇒ a p b
Example:
Rq : People ↔ People x q y ⇐⇒ x is a child of y
Rq−1 : People ↔ People
x q−1 y ⇐⇒ x is a parent of y
Discrete Mathematics I – p. 130/292
![Page 540: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/540.jpg)
RelationsRelation Rp : A ↔ A is reflexive, if ∀a ∈ A : a p a
Examples: R= A2 R≤ R|
Rp reflexive iff R= ⊆ Rp
Discrete Mathematics I – p. 131/292
![Page 541: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/541.jpg)
RelationsRelation Rp : A ↔ A is reflexive, if ∀a ∈ A : a p a
Examples: R=
A2 R≤ R|
Rp reflexive iff R= ⊆ Rp
Discrete Mathematics I – p. 131/292
![Page 542: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/542.jpg)
RelationsRelation Rp : A ↔ A is reflexive, if ∀a ∈ A : a p a
Examples: R= A2
R≤ R|
Rp reflexive iff R= ⊆ Rp
Discrete Mathematics I – p. 131/292
![Page 543: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/543.jpg)
RelationsRelation Rp : A ↔ A is reflexive, if ∀a ∈ A : a p a
Examples: R= A2 R≤
R|
Rp reflexive iff R= ⊆ Rp
Discrete Mathematics I – p. 131/292
![Page 544: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/544.jpg)
RelationsRelation Rp : A ↔ A is reflexive, if ∀a ∈ A : a p a
Examples: R= A2 R≤ R|
Rp reflexive iff R= ⊆ Rp
Discrete Mathematics I – p. 131/292
![Page 545: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/545.jpg)
RelationsRelation Rp : A ↔ A is reflexive, if ∀a ∈ A : a p a
Examples: R= A2 R≤ R|
Rp reflexive iff R= ⊆ Rp
Discrete Mathematics I – p. 131/292
![Page 546: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/546.jpg)
RelationsRelation Rp : A ↔ A is symmetric, if
∀a, b ∈ A : a p b ⇒ b p a
Examples: R= A2 ∅R∗ : N ↔ N x ∗ y ⇔ (x + y = 10)
Rp symmetric iff Rp−1 = Rp
Discrete Mathematics I – p. 132/292
![Page 547: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/547.jpg)
RelationsRelation Rp : A ↔ A is symmetric, if
∀a, b ∈ A : a p b ⇒ b p a
Examples: R=
A2 ∅R∗ : N ↔ N x ∗ y ⇔ (x + y = 10)
Rp symmetric iff Rp−1 = Rp
Discrete Mathematics I – p. 132/292
![Page 548: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/548.jpg)
RelationsRelation Rp : A ↔ A is symmetric, if
∀a, b ∈ A : a p b ⇒ b p a
Examples: R= A2
∅R∗ : N ↔ N x ∗ y ⇔ (x + y = 10)
Rp symmetric iff Rp−1 = Rp
Discrete Mathematics I – p. 132/292
![Page 549: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/549.jpg)
RelationsRelation Rp : A ↔ A is symmetric, if
∀a, b ∈ A : a p b ⇒ b p a
Examples: R= A2 ∅
R∗ : N ↔ N x ∗ y ⇔ (x + y = 10)
Rp symmetric iff Rp−1 = Rp
Discrete Mathematics I – p. 132/292
![Page 550: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/550.jpg)
RelationsRelation Rp : A ↔ A is symmetric, if
∀a, b ∈ A : a p b ⇒ b p a
Examples: R= A2 ∅R∗ : N ↔ N x ∗ y ⇔ (x + y = 10)
Rp symmetric iff Rp−1 = Rp
Discrete Mathematics I – p. 132/292
![Page 551: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/551.jpg)
RelationsRelation Rp : A ↔ A is symmetric, if
∀a, b ∈ A : a p b ⇒ b p a
Examples: R= A2 ∅R∗ : N ↔ N x ∗ y ⇔ (x + y = 10)
Rp symmetric iff Rp−1 = Rp
Discrete Mathematics I – p. 132/292
![Page 552: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/552.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R= ∅ R≤ R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 553: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/553.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R=
∅ R≤ R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 554: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/554.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R= ∅
R≤ R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 555: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/555.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R= ∅ R≤
R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 556: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/556.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R= ∅ R≤ R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 557: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/557.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R= ∅ R≤ R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 558: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/558.jpg)
RelationsRelation Rp : A ↔ A is antisymmetric, if
∀a, b ∈ A : (a p b ∧ b p a) ⇒ a = b
Examples: R= ∅ R≤ R<
Rp antisymmetric iff Rp ∩Rp−1 ⊆ R=
Note non-symmetric 6⇔ antisymmetric (e.g. R=)
Discrete Mathematics I – p. 133/292
![Page 559: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/559.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R= A2 ∅ R≤ R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 560: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/560.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R=
A2 ∅ R≤ R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 561: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/561.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R= A2
∅ R≤ R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 562: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/562.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R= A2 ∅
R≤ R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 563: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/563.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R= A2 ∅ R≤
R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 564: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/564.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R= A2 ∅ R≤ R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 565: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/565.jpg)
RelationsRelation Rp : A ↔ A is transitive, if
∀a, b, c ∈ A : (a p b ∧ b p c) ⇒ a p c
Examples: R= A2 ∅ R≤ R<
Rp transitive iff Rp ◦ p ⊆ Rp
Discrete Mathematics I – p. 134/292
![Page 566: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/566.jpg)
RelationsRelation R∼ : A ↔ A is an equivalence relation,if it is reflexive, symmetric and transitive
Relation R� : A ↔ A is a partial order,if it is reflexive, antisymmetric and transitive
Discrete Mathematics I – p. 135/292
![Page 567: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/567.jpg)
RelationsRelation R∼ : A ↔ A is an equivalence relation,if it is reflexive, symmetric and transitive
Relation R� : A ↔ A is a partial order,if it is reflexive, antisymmetric and transitive
Discrete Mathematics I – p. 135/292
![Page 568: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/568.jpg)
RelationsExamples of equivalence relations:
R=
Rp : People ↔ People
a p b ⇐⇒ a and b share a birthday
Discrete Mathematics I – p. 136/292
![Page 569: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/569.jpg)
RelationsExamples of equivalence relations:
R=
Rp : People ↔ People
a p b ⇐⇒ a and b share a birthday
Discrete Mathematics I – p. 136/292
![Page 570: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/570.jpg)
Relations{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
R| : N ↔ Z a divides b
a | b ⇐⇒ ∃k ∈ Z : k · a = b
R≡n: Z ↔ Z a ≡n b ⇐⇒ n|(a− b)
R≡nis called congruence modulo n
Discrete Mathematics I – p. 137/292
![Page 571: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/571.jpg)
Relations{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
R| : N ↔ Z a divides b
a | b ⇐⇒ ∃k ∈ Z : k · a = b
R≡n: Z ↔ Z a ≡n b ⇐⇒ n|(a− b)
R≡nis called congruence modulo n
Discrete Mathematics I – p. 137/292
![Page 572: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/572.jpg)
Relations{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
R| : N ↔ Z a divides b
a | b ⇐⇒ ∃k ∈ Z : k · a = b
R≡n: Z ↔ Z a ≡n b ⇐⇒ n|(a− b)
R≡nis called congruence modulo n
Discrete Mathematics I – p. 137/292
![Page 573: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/573.jpg)
Relations{. . . ,−3,−2,−1, 0, 1, 2, 3, . . . } = Z integers
R| : N ↔ Z a divides b
a | b ⇐⇒ ∃k ∈ Z : k · a = b
R≡n: Z ↔ Z a ≡n b ⇐⇒ n|(a− b)
R≡nis called congruence modulo n
Discrete Mathematics I – p. 137/292
![Page 574: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/574.jpg)
RelationsProve: R≡n
is an equivalence for all n ∈ N, n ≥ 1.
Proof. Let n ∈ N, n ≥ 1.
Let x ∈ Z.
x ≡n x ⇐⇒ n | (x− x) ⇐⇒ n | 0 ⇐⇒∃k : k · n = 0 ⇐⇒ T (since 0 · n = 0)
Hence R≡nreflexive.
Discrete Mathematics I – p. 138/292
![Page 575: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/575.jpg)
RelationsProve: R≡n
is an equivalence for all n ∈ N, n ≥ 1.
Proof. Let n ∈ N, n ≥ 1.
Let x ∈ Z.
x ≡n x ⇐⇒ n | (x− x) ⇐⇒ n | 0 ⇐⇒∃k : k · n = 0 ⇐⇒ T (since 0 · n = 0)
Hence R≡nreflexive.
Discrete Mathematics I – p. 138/292
![Page 576: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/576.jpg)
RelationsProve: R≡n
is an equivalence for all n ∈ N, n ≥ 1.
Proof. Let n ∈ N, n ≥ 1.
Let x ∈ Z.
x ≡n x ⇐⇒ n | (x− x) ⇐⇒ n | 0 ⇐⇒∃k : k · n = 0 ⇐⇒ T (since 0 · n = 0)
Hence R≡nreflexive.
Discrete Mathematics I – p. 138/292
![Page 577: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/577.jpg)
RelationsProve: R≡n
is an equivalence for all n ∈ N, n ≥ 1.
Proof. Let n ∈ N, n ≥ 1.
Let x ∈ Z.
x ≡n x ⇐⇒ n | (x− x) ⇐⇒ n | 0 ⇐⇒∃k : k · n = 0 ⇐⇒ T (since 0 · n = 0)
Hence R≡nreflexive.
Discrete Mathematics I – p. 138/292
![Page 578: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/578.jpg)
RelationsProve: R≡n
is an equivalence for all n ∈ N, n ≥ 1.
Proof. Let n ∈ N, n ≥ 1.
Let x ∈ Z.
x ≡n x ⇐⇒ n | (x− x) ⇐⇒ n | 0 ⇐⇒∃k : k · n = 0 ⇐⇒ T (since 0 · n = 0)
Hence R≡nreflexive.
Discrete Mathematics I – p. 138/292
![Page 579: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/579.jpg)
RelationsLet x, y ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
(−k) · n = −(x− y) = y − x =⇒n | (y − x) =⇒ y ≡n x
Hence R≡nsymmetric.
Discrete Mathematics I – p. 139/292
![Page 580: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/580.jpg)
RelationsLet x, y ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
(−k) · n = −(x− y) = y − x =⇒n | (y − x) =⇒ y ≡n x
Hence R≡nsymmetric.
Discrete Mathematics I – p. 139/292
![Page 581: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/581.jpg)
RelationsLet x, y ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
(−k) · n = −(x− y) = y − x =⇒n | (y − x) =⇒ y ≡n x
Hence R≡nsymmetric.
Discrete Mathematics I – p. 139/292
![Page 582: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/582.jpg)
RelationsLet x, y ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
(−k) · n = −(x− y) = y − x =⇒n | (y − x) =⇒ y ≡n x
Hence R≡nsymmetric.
Discrete Mathematics I – p. 139/292
![Page 583: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/583.jpg)
RelationsLet x, y, z ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
y ≡n z =⇒ n | (y − z) =⇒ ∃l : l · n = y − z
(k + l) · n = (x− y) + (y − z) = x− z =⇒n | (x− z) =⇒ x ≡n z
Hence R≡ntransitive.
R≡nreflexive, symmetric and transitive, therefore
R≡nis an equivalence relation.
Discrete Mathematics I – p. 140/292
![Page 584: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/584.jpg)
RelationsLet x, y, z ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
y ≡n z =⇒ n | (y − z) =⇒ ∃l : l · n = y − z
(k + l) · n = (x− y) + (y − z) = x− z =⇒n | (x− z) =⇒ x ≡n z
Hence R≡ntransitive.
R≡nreflexive, symmetric and transitive, therefore
R≡nis an equivalence relation.
Discrete Mathematics I – p. 140/292
![Page 585: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/585.jpg)
RelationsLet x, y, z ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
y ≡n z =⇒ n | (y − z) =⇒ ∃l : l · n = y − z
(k + l) · n = (x− y) + (y − z) = x− z =⇒n | (x− z) =⇒ x ≡n z
Hence R≡ntransitive.
R≡nreflexive, symmetric and transitive, therefore
R≡nis an equivalence relation.
Discrete Mathematics I – p. 140/292
![Page 586: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/586.jpg)
RelationsLet x, y, z ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
y ≡n z =⇒ n | (y − z) =⇒ ∃l : l · n = y − z
(k + l) · n = (x− y) + (y − z) = x− z =⇒n | (x− z) =⇒ x ≡n z
Hence R≡ntransitive.
R≡nreflexive, symmetric and transitive, therefore
R≡nis an equivalence relation.
Discrete Mathematics I – p. 140/292
![Page 587: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/587.jpg)
RelationsLet x, y, z ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
y ≡n z =⇒ n | (y − z) =⇒ ∃l : l · n = y − z
(k + l) · n = (x− y) + (y − z) = x− z =⇒n | (x− z) =⇒ x ≡n z
Hence R≡ntransitive.
R≡nreflexive, symmetric and transitive, therefore
R≡nis an equivalence relation.
Discrete Mathematics I – p. 140/292
![Page 588: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/588.jpg)
RelationsLet x, y, z ∈ Z.
x ≡n y =⇒ n | (x− y) =⇒ ∃k : k · n = x− y
y ≡n z =⇒ n | (y − z) =⇒ ∃l : l · n = y − z
(k + l) · n = (x− y) + (y − z) = x− z =⇒n | (x− z) =⇒ x ≡n z
Hence R≡ntransitive.
R≡nreflexive, symmetric and transitive, therefore
R≡nis an equivalence relation.
Discrete Mathematics I – p. 140/292
![Page 589: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/589.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 590: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/590.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 591: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/591.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}
By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 592: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/592.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 593: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/593.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 594: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/594.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 595: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/595.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
For any a ∈ A, the equivalence class of a is the set ofall elements related to a
[a]∼ = {x ∈ A | x ∼ a}By reflexivity, a ∈ [a]∼
Every a is a representative of [a]∼
The set of all equivalence classes of R∼ is the quotientset of A with respect to R∼
A/R∼ = {[a]∼ | a ∈ A}
Discrete Mathematics I – p. 141/292
![Page 596: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/596.jpg)
RelationsExample:
Rp : People ↔ People
x p y ⇐⇒ x and y share a birthday
Size ofPeople/Rp = 366
∀x, y ∈ People : ([x]p = [y]p) ∨ ([x]p ∩ [y]p = ∅)
Discrete Mathematics I – p. 142/292
![Page 597: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/597.jpg)
RelationsExample:
Rp : People ↔ People
x p y ⇐⇒ x and y share a birthday
Size ofPeople/Rp =
366
∀x, y ∈ People : ([x]p = [y]p) ∨ ([x]p ∩ [y]p = ∅)
Discrete Mathematics I – p. 142/292
![Page 598: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/598.jpg)
RelationsExample:
Rp : People ↔ People
x p y ⇐⇒ x and y share a birthday
Size ofPeople/Rp = 366
∀x, y ∈ People : ([x]p = [y]p) ∨ ([x]p ∩ [y]p = ∅)
Discrete Mathematics I – p. 142/292
![Page 599: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/599.jpg)
RelationsExample:
Rp : People ↔ People
x p y ⇐⇒ x and y share a birthday
Size ofPeople/Rp = 366
∀x, y ∈ People : ([x]p = [y]p) ∨ ([x]p ∩ [y]p = ∅)
Discrete Mathematics I – p. 142/292
![Page 600: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/600.jpg)
RelationsLet Fred, George ∈ People
Suppose Fred was born on 1 November
[Fred]p = {x ∈ People | x born on 1 November}Size of [Fred]p ≈ 6bln/365.25 ≈ 16mln
Suppose George was born on 29 February
[George]p = {x ∈ People | x born on 29 February}Size of [George]p ≈ 6bln/(4 ∗ 365.25) ≈ 4mln
Discrete Mathematics I – p. 143/292
![Page 601: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/601.jpg)
RelationsLet Fred, George ∈ People
Suppose Fred was born on 1 November
[Fred]p = {x ∈ People | x born on 1 November}
Size of [Fred]p ≈ 6bln/365.25 ≈ 16mln
Suppose George was born on 29 February
[George]p = {x ∈ People | x born on 29 February}Size of [George]p ≈ 6bln/(4 ∗ 365.25) ≈ 4mln
Discrete Mathematics I – p. 143/292
![Page 602: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/602.jpg)
RelationsLet Fred, George ∈ People
Suppose Fred was born on 1 November
[Fred]p = {x ∈ People | x born on 1 November}Size of [Fred]p ≈ 6bln/365.25 ≈ 16mln
Suppose George was born on 29 February
[George]p = {x ∈ People | x born on 29 February}Size of [George]p ≈ 6bln/(4 ∗ 365.25) ≈ 4mln
Discrete Mathematics I – p. 143/292
![Page 603: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/603.jpg)
RelationsLet Fred, George ∈ People
Suppose Fred was born on 1 November
[Fred]p = {x ∈ People | x born on 1 November}Size of [Fred]p ≈ 6bln/365.25 ≈ 16mln
Suppose George was born on 29 February
[George]p = {x ∈ People | x born on 29 February}Size of [George]p ≈ 6bln/(4 ∗ 365.25) ≈ 4mln
Discrete Mathematics I – p. 143/292
![Page 604: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/604.jpg)
RelationsLet Fred, George ∈ People
Suppose Fred was born on 1 November
[Fred]p = {x ∈ People | x born on 1 November}Size of [Fred]p ≈ 6bln/365.25 ≈ 16mln
Suppose George was born on 29 February
[George]p = {x ∈ People | x born on 29 February}
Size of [George]p ≈ 6bln/(4 ∗ 365.25) ≈ 4mln
Discrete Mathematics I – p. 143/292
![Page 605: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/605.jpg)
RelationsLet Fred, George ∈ People
Suppose Fred was born on 1 November
[Fred]p = {x ∈ People | x born on 1 November}Size of [Fred]p ≈ 6bln/365.25 ≈ 16mln
Suppose George was born on 29 February
[George]p = {x ∈ People | x born on 29 February}Size of [George]p ≈ 6bln/(4 ∗ 365.25) ≈ 4mln
Discrete Mathematics I – p. 143/292
![Page 606: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/606.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 607: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/607.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5=
{. . . ,−10,−5, 0, 5, 10, . . . }[1]≡5
= {. . . ,−9,−4, 1, 6, 11, . . . }[2]≡5
= {. . . ,−8,−3, 2, 7, 12, . . . }[3]≡5
= {. . . ,−7,−2, 3, 8, 13, . . . }[4]≡5
= {. . . ,−6,−1, 4, 9, 14, . . . }[a]≡5
called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 608: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/608.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 609: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/609.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5=
{. . . ,−9,−4, 1, 6, 11, . . . }[2]≡5
= {. . . ,−8,−3, 2, 7, 12, . . . }[3]≡5
= {. . . ,−7,−2, 3, 8, 13, . . . }[4]≡5
= {. . . ,−6,−1, 4, 9, 14, . . . }[a]≡5
called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 610: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/610.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 611: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/611.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 612: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/612.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 613: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/613.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 614: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/614.jpg)
RelationsAnother example:
R≡5: Z ↔ Z a ≡5 b ⇐⇒ 5 | (a− b)
[0]≡5= {. . . ,−10,−5, 0, 5, 10, . . . }
[1]≡5= {. . . ,−9,−4, 1, 6, 11, . . . }
[2]≡5= {. . . ,−8,−3, 2, 7, 12, . . . }
[3]≡5= {. . . ,−7,−2, 3, 8, 13, . . . }
[4]≡5= {. . . ,−6,−1, 4, 9, 14, . . . }
[a]≡5called residue classes modulo 5 (can be any n)
Discrete Mathematics I – p. 144/292
![Page 615: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/615.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
Theorem.
The equivalence classes of R∼ are pairwise disjoint.
∀a, b ∈ A : ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)The union of all equivalence classes is the whole A.⋃
a∈A[a]∼ = A
A is partitioned by R∼ into a disjoint union ofequivalence classes
Discrete Mathematics I – p. 145/292
![Page 616: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/616.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
Theorem.
The equivalence classes of R∼ are pairwise disjoint.
∀a, b ∈ A : ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)
The union of all equivalence classes is the whole A.⋃
a∈A[a]∼ = A
A is partitioned by R∼ into a disjoint union ofequivalence classes
Discrete Mathematics I – p. 145/292
![Page 617: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/617.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
Theorem.
The equivalence classes of R∼ are pairwise disjoint.
∀a, b ∈ A : ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)The union of all equivalence classes is the whole A.⋃
a∈A[a]∼ = A
A is partitioned by R∼ into a disjoint union ofequivalence classes
Discrete Mathematics I – p. 145/292
![Page 618: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/618.jpg)
RelationsR∼ : A ↔ A — an equivalence relation
Theorem.
The equivalence classes of R∼ are pairwise disjoint.
∀a, b ∈ A : ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)The union of all equivalence classes is the whole A.⋃
a∈A[a]∼ = A
A is partitioned by R∼ into a disjoint union ofequivalence classes
Discrete Mathematics I – p. 145/292
![Page 619: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/619.jpg)
RelationsProof. For all a, b, we need:
([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)
Consider two cases: a ∼ b, a 6∼ b.
Discrete Mathematics I – p. 146/292
![Page 620: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/620.jpg)
RelationsProof. For all a, b, we need:
([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)Consider two cases: a ∼ b, a 6∼ b.
Discrete Mathematics I – p. 146/292
![Page 621: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/621.jpg)
RelationsCase a ∼ b. Take any x ∈ [a]∼.
(x ∼ a) ∧ (a ∼ b) =⇒ x ∼ b =⇒ x ∈ [b]∼
Hence [a]∼ ⊆ [b]∼. Similarly [b]∼ ⊆ [a]∼.
Therefore [a]∼ = [b]∼.
Discrete Mathematics I – p. 147/292
![Page 622: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/622.jpg)
RelationsCase a ∼ b. Take any x ∈ [a]∼.
(x ∼ a) ∧ (a ∼ b) =⇒ x ∼ b =⇒ x ∈ [b]∼
Hence [a]∼ ⊆ [b]∼. Similarly [b]∼ ⊆ [a]∼.
Therefore [a]∼ = [b]∼.
Discrete Mathematics I – p. 147/292
![Page 623: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/623.jpg)
RelationsCase a ∼ b. Take any x ∈ [a]∼.
(x ∼ a) ∧ (a ∼ b) =⇒ x ∼ b =⇒ x ∈ [b]∼
Hence [a]∼ ⊆ [b]∼. Similarly [b]∼ ⊆ [a]∼.
Therefore [a]∼ = [b]∼.
Discrete Mathematics I – p. 147/292
![Page 624: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/624.jpg)
RelationsCase a ∼ b. Take any x ∈ [a]∼.
(x ∼ a) ∧ (a ∼ b) =⇒ x ∼ b =⇒ x ∈ [b]∼
Hence [a]∼ ⊆ [b]∼. Similarly [b]∼ ⊆ [a]∼.
Therefore [a]∼ = [b]∼.
Discrete Mathematics I – p. 147/292
![Page 625: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/625.jpg)
RelationsCase a 6∼ b. Suppose ∃x : x ∈ [a]∼ ∩ [b]∼.
(x ∼ a) ∧ (x ∼ b) =⇒ a ∼ b — contradiction.
Therefore [a]∼ ∩ [b]∼ = ∅.
Discrete Mathematics I – p. 148/292
![Page 626: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/626.jpg)
RelationsCase a 6∼ b. Suppose ∃x : x ∈ [a]∼ ∩ [b]∼.
(x ∼ a) ∧ (x ∼ b) =⇒ a ∼ b — contradiction.
Therefore [a]∼ ∩ [b]∼ = ∅.
Discrete Mathematics I – p. 148/292
![Page 627: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/627.jpg)
RelationsCase a 6∼ b. Suppose ∃x : x ∈ [a]∼ ∩ [b]∼.
(x ∼ a) ∧ (x ∼ b) =⇒ a ∼ b — contradiction.
Therefore [a]∼ ∩ [b]∼ = ∅.
Discrete Mathematics I – p. 148/292
![Page 628: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/628.jpg)
RelationsCase a ∼ b =⇒ [a]∼ = [b]∼
Case a 6∼ b =⇒ [a]∼ ∩ [b]∼ = ∅
Therefore ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)Finally, for all a ∈ A: a ∼ a =⇒ a ∈ [a]∼ ⊆ A.
Hence A ⊆ ⋃
a∈A[a]∼ ⊆ A.
Therefore⋃
a∈A[a]∼ = A.
Discrete Mathematics I – p. 149/292
![Page 629: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/629.jpg)
RelationsCase a ∼ b =⇒ [a]∼ = [b]∼
Case a 6∼ b =⇒ [a]∼ ∩ [b]∼ = ∅Therefore ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)
Finally, for all a ∈ A: a ∼ a =⇒ a ∈ [a]∼ ⊆ A.
Hence A ⊆ ⋃
a∈A[a]∼ ⊆ A.
Therefore⋃
a∈A[a]∼ = A.
Discrete Mathematics I – p. 149/292
![Page 630: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/630.jpg)
RelationsCase a ∼ b =⇒ [a]∼ = [b]∼
Case a 6∼ b =⇒ [a]∼ ∩ [b]∼ = ∅Therefore ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)
Finally, for all a ∈ A: a ∼ a =⇒ a ∈ [a]∼ ⊆ A.
Hence A ⊆ ⋃
a∈A[a]∼ ⊆ A.
Therefore⋃
a∈A[a]∼ = A.
Discrete Mathematics I – p. 149/292
![Page 631: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/631.jpg)
RelationsCase a ∼ b =⇒ [a]∼ = [b]∼
Case a 6∼ b =⇒ [a]∼ ∩ [b]∼ = ∅Therefore ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)
Finally, for all a ∈ A: a ∼ a =⇒ a ∈ [a]∼ ⊆ A.
Hence A ⊆ ⋃
a∈A[a]∼ ⊆ A.
Therefore⋃
a∈A[a]∼ = A.
Discrete Mathematics I – p. 149/292
![Page 632: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/632.jpg)
RelationsCase a ∼ b =⇒ [a]∼ = [b]∼
Case a 6∼ b =⇒ [a]∼ ∩ [b]∼ = ∅Therefore ([a]∼ = [b]∼) ∨ ([a]∼ ∩ [b]∼ = ∅)
Finally, for all a ∈ A: a ∼ a =⇒ a ∈ [a]∼ ⊆ A.
Hence A ⊆ ⋃
a∈A[a]∼ ⊆ A.
Therefore⋃
a∈A[a]∼ = A.
Discrete Mathematics I – p. 149/292
![Page 633: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/633.jpg)
RelationsIf A finite, A/R∼ finite
If A has n elements, and if every [a]∼ has m elements,then m | n, and A/R∼ has n/m elements
If A infinite, A/R∼ can be finite or infinite
Discrete Mathematics I – p. 150/292
![Page 634: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/634.jpg)
RelationsIf A finite, A/R∼ finite
If A has n elements, and if every [a]∼ has m elements,then m | n, and A/R∼ has n/m elements
If A infinite, A/R∼ can be finite or infinite
Discrete Mathematics I – p. 150/292
![Page 635: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/635.jpg)
RelationsIf A finite, A/R∼ finite
If A has n elements, and if every [a]∼ has m elements,then m | n, and A/R∼ has n/m elements
If A infinite, A/R∼ can be finite or infinite
Discrete Mathematics I – p. 150/292
![Page 636: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/636.jpg)
RelationsRelation R∼ : A ↔ A is an equivalence relation,if it is reflexive, symmetric and transitive
Relation R� : A ↔ A is a partial order,if it is reflexive, antisymmetric and transitive
Set A is partially ordered
Discrete Mathematics I – p. 151/292
![Page 637: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/637.jpg)
RelationsRelation R∼ : A ↔ A is an equivalence relation,if it is reflexive, symmetric and transitive
Relation R� : A ↔ A is a partial order,if it is reflexive, antisymmetric and transitive
Set A is partially ordered
Discrete Mathematics I – p. 151/292
![Page 638: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/638.jpg)
RelationsExamples:
R≤, R≥ : N ↔ N
Hasse diagram (illustration only):
0
1
2
3
4
R≤
0
1
2
3
4
R≥
Discrete Mathematics I – p. 152/292
![Page 639: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/639.jpg)
RelationsExamples:
R≤, R≥ : N ↔ N
Hasse diagram (illustration only):
��
��
�
0
1
2
3
4
R≤
0
1
2
3
4
R≥
Discrete Mathematics I – p. 152/292
![Page 640: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/640.jpg)
RelationsExamples:
R≤, R≥ : N ↔ N
Hasse diagram (illustration only):
��
��
�
0
1
2
3
4
R≤
��
��
�
0
1
2
3
4
R≥
Discrete Mathematics I – p. 152/292
![Page 641: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/641.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is an descendant of y
(Everyone is his/her own descendant)
Lamech Bitenosh
Noah Naamah
Shem Ham Japheth
Discrete Mathematics I – p. 153/292
![Page 642: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/642.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is an descendant of y
(Everyone is his/her own descendant)
Lamech Bitenosh
Noah Naamah
Shem Ham Japheth
Discrete Mathematics I – p. 153/292
![Page 643: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/643.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is an descendant of y
(Everyone is his/her own descendant)
�
Lamech
�
Bitenosh
�Noah � Naamah
�
Shem
�
Ham
�
Japheth
Discrete Mathematics I – p. 153/292
![Page 644: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/644.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ ∃k : k · x = y
1
2 3 5 7
4 6 9
810
0
Discrete Mathematics I – p. 154/292
![Page 645: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/645.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ ∃k : k · x = y
�
1�2 �3 � 5 � 7
�
4
�
6�
9
�
8
10
0
Discrete Mathematics I – p. 154/292
![Page 646: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/646.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 647: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/647.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 648: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/648.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 649: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/649.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 650: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/650.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 651: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/651.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 652: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/652.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 653: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/653.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.
Discrete Mathematics I – p. 155/292
![Page 654: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/654.jpg)
RelationsProve: R| is a partial order.
Proof. Let x ∈ N.
x | x ⇐⇒ ∃k : k · x = x ⇐⇒ T (since 1 · x = x)
Hence R| reflexive.
Let x, y ∈ N.
x | y =⇒ ∃k : k · x = y
y | x =⇒ ∃l : l · y = x
x = l · y = k · l · x =⇒ k · l = 1 =⇒k = l = 1 =⇒ x = y
Hence R| antisymmetric.Discrete Mathematics I – p. 155/292
![Page 655: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/655.jpg)
RelationsLet x, y, z ∈ N.
x | y =⇒ ∃k : k · x = y
y | z =⇒ ∃l : l · y = z
z = l · y = (k · l) · x =⇒ x | zHence R| transitive.
R| reflexive, antisymmetric and transitive, thereforeR| is a partial order.
Discrete Mathematics I – p. 156/292
![Page 656: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/656.jpg)
RelationsLet x, y, z ∈ N.
x | y =⇒ ∃k : k · x = y
y | z =⇒ ∃l : l · y = z
z = l · y = (k · l) · x =⇒ x | zHence R| transitive.
R| reflexive, antisymmetric and transitive, thereforeR| is a partial order.
Discrete Mathematics I – p. 156/292
![Page 657: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/657.jpg)
RelationsLet x, y, z ∈ N.
x | y =⇒ ∃k : k · x = y
y | z =⇒ ∃l : l · y = z
z = l · y = (k · l) · x =⇒ x | zHence R| transitive.
R| reflexive, antisymmetric and transitive, thereforeR| is a partial order.
Discrete Mathematics I – p. 156/292
![Page 658: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/658.jpg)
RelationsLet x, y, z ∈ N.
x | y =⇒ ∃k : k · x = y
y | z =⇒ ∃l : l · y = z
z = l · y = (k · l) · x =⇒ x | z
Hence R| transitive.
R| reflexive, antisymmetric and transitive, thereforeR| is a partial order.
Discrete Mathematics I – p. 156/292
![Page 659: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/659.jpg)
RelationsLet x, y, z ∈ N.
x | y =⇒ ∃k : k · x = y
y | z =⇒ ∃l : l · y = z
z = l · y = (k · l) · x =⇒ x | zHence R| transitive.
R| reflexive, antisymmetric and transitive, thereforeR| is a partial order.
Discrete Mathematics I – p. 156/292
![Page 660: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/660.jpg)
RelationsLet x, y, z ∈ N.
x | y =⇒ ∃k : k · x = y
y | z =⇒ ∃l : l · y = z
z = l · y = (k · l) · x =⇒ x | zHence R| transitive.
R| reflexive, antisymmetric and transitive, thereforeR| is a partial order.
Discrete Mathematics I – p. 156/292
![Page 661: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/661.jpg)
RelationsR⊆ : P(S) ↔ P(S) A ⊆ B
S = {0, 1, 2}
∅
01
2
1202
01
012
Discrete Mathematics I – p. 157/292
![Page 662: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/662.jpg)
RelationsR⊆ : P(S) ↔ P(S) A ⊆ B
S = {0, 1, 2}
∅
01
2
1202
01
012
Discrete Mathematics I – p. 157/292
![Page 663: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/663.jpg)
RelationsR⊆ : P(S) ↔ P(S) A ⊆ B
S = {0, 1, 2}
�
∅
�0 �
1
� 2�12 �
02
� 01�
012
Discrete Mathematics I – p. 157/292
![Page 664: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/664.jpg)
RelationsR� : A ↔ A — a partial order
R� is called a total order, if for all a, b ∈ A,either a � b or b � a
Set A is called totally ordered
Discrete Mathematics I – p. 158/292
![Page 665: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/665.jpg)
RelationsR� : A ↔ A — a partial order
R� is called a total order, if for all a, b ∈ A,either a � b or b � a
Set A is called totally ordered
Discrete Mathematics I – p. 158/292
![Page 666: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/666.jpg)
RelationsExamples:
R≤ R≥ — total (but still a partial order!)
RE R| R⊆ — not total
Discrete Mathematics I – p. 159/292
![Page 667: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/667.jpg)
RelationsExamples:
R≤ R≥ — total (but still a partial order!)
RE R| R⊆ — not total
Discrete Mathematics I – p. 159/292
![Page 668: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/668.jpg)
RelationsR� : A ↔ A — a partial order (need not be total)
a ∈ A
c ∈ A is an upper bound of a, if a � c
d ∈ A is a lower bound of a, if d � a
Discrete Mathematics I – p. 160/292
![Page 669: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/669.jpg)
RelationsR� : A ↔ A — a partial order (need not be total)
a ∈ A
c ∈ A is an upper bound of a, if a � c
d ∈ A is a lower bound of a, if d � a
Discrete Mathematics I – p. 160/292
![Page 670: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/670.jpg)
RelationsR� : A ↔ A — a partial order (need not be total)
a ∈ A
c ∈ A is an upper bound of a, if a � c
d ∈ A is a lower bound of a, if d � a
Discrete Mathematics I – p. 160/292
![Page 671: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/671.jpg)
Relationsa, b ∈ A
c ∈ A is an upper bound of a, b, if (a � c) ∧ (b � c)
d ∈ A is a lower bound of a, b, if (d � a) ∧ (d � b)
Discrete Mathematics I – p. 161/292
![Page 672: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/672.jpg)
Relationsa, b ∈ A
c ∈ A is an upper bound of a, b, if (a � c) ∧ (b � c)
d ∈ A is a lower bound of a, b, if (d � a) ∧ (d � b)
Discrete Mathematics I – p. 161/292
![Page 673: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/673.jpg)
Relationsa, b ∈ A
c ∈ A is the least upper bound of a, b, if
• c is an upper bound of a, b
• for all x ∈ A, (a � x) ∧ (b � x) ⇒ (c � x)
c = lub(a, b) (may not exist!)
Discrete Mathematics I – p. 162/292
![Page 674: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/674.jpg)
Relationsa, b ∈ A
c ∈ A is the least upper bound of a, b, if
• c is an upper bound of a, b
• for all x ∈ A, (a � x) ∧ (b � x) ⇒ (c � x)
c = lub(a, b) (may not exist!)
Discrete Mathematics I – p. 162/292
![Page 675: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/675.jpg)
Relationsa, b ∈ A
c ∈ A is the least upper bound of a, b, if
• c is an upper bound of a, b
• for all x ∈ A, (a � x) ∧ (b � x) ⇒ (c � x)
c = lub(a, b) (may not exist!)
Discrete Mathematics I – p. 162/292
![Page 676: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/676.jpg)
Relationsa, b ∈ A
c ∈ A is the least upper bound of a, b, if
• c is an upper bound of a, b
• for all x ∈ A, (a � x) ∧ (b � x) ⇒ (c � x)
c = lub(a, b) (may not exist!)
Discrete Mathematics I – p. 162/292
![Page 677: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/677.jpg)
Relationsa, b ∈ A
c ∈ A is the least upper bound of a, b, if
• c is an upper bound of a, b
• for all x ∈ A, (a � x) ∧ (b � x) ⇒ (c � x)
c = lub(a, b) (may not exist!)
Discrete Mathematics I – p. 162/292
![Page 678: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/678.jpg)
Relationsa, b ∈ A
d ∈ A is the greatest lower bound of a, b, if
• d is a lower bound of a, b
• for all x ∈ A, (x � a) ∧ (x � b) ⇒ (x � d)
d = glb(a, b) (may not exist!)
Discrete Mathematics I – p. 163/292
![Page 679: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/679.jpg)
Relationsa, b ∈ A
d ∈ A is the greatest lower bound of a, b, if
• d is a lower bound of a, b
• for all x ∈ A, (x � a) ∧ (x � b) ⇒ (x � d)
d = glb(a, b) (may not exist!)
Discrete Mathematics I – p. 163/292
![Page 680: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/680.jpg)
Relationsa, b ∈ A
d ∈ A is the greatest lower bound of a, b, if
• d is a lower bound of a, b
• for all x ∈ A, (x � a) ∧ (x � b) ⇒ (x � d)
d = glb(a, b) (may not exist!)
Discrete Mathematics I – p. 163/292
![Page 681: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/681.jpg)
Relationsa, b ∈ A
d ∈ A is the greatest lower bound of a, b, if
• d is a lower bound of a, b
• for all x ∈ A, (x � a) ∧ (x � b) ⇒ (x � d)
d = glb(a, b) (may not exist!)
Discrete Mathematics I – p. 163/292
![Page 682: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/682.jpg)
Relationsa, b ∈ A
d ∈ A is the greatest lower bound of a, b, if
• d is a lower bound of a, b
• for all x ∈ A, (x � a) ∧ (x � b) ⇒ (x � d)
d = glb(a, b) (may not exist!)
Discrete Mathematics I – p. 163/292
![Page 683: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/683.jpg)
RelationsExample:
RE : People ↔ PeoplexE y ⇐⇒ x is a descendant of y
lub(x, y) = youngest common ancestor(x, y)
glb(x, y) = oldest common descendant(x, y)
May not exist, e.g. if x, y are not relatives
Discrete Mathematics I – p. 164/292
![Page 684: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/684.jpg)
RelationsExample:
RE : People ↔ PeoplexE y ⇐⇒ x is a descendant of y
lub(x, y) = youngest common ancestor(x, y)
glb(x, y) = oldest common descendant(x, y)
May not exist, e.g. if x, y are not relatives
Discrete Mathematics I – p. 164/292
![Page 685: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/685.jpg)
RelationsExample:
RE : People ↔ PeoplexE y ⇐⇒ x is a descendant of y
lub(x, y) = youngest common ancestor(x, y)
glb(x, y) = oldest common descendant(x, y)
May not exist, e.g. if x, y are not relatives
Discrete Mathematics I – p. 164/292
![Page 686: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/686.jpg)
RelationsExample:
RE : People ↔ PeoplexE y ⇐⇒ x is a descendant of y
lub(x, y) = youngest common ancestor(x, y)
glb(x, y) = oldest common descendant(x, y)
May not exist, e.g. if x, y are not relatives
Discrete Mathematics I – p. 164/292
![Page 687: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/687.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) = least common multiple(a, b)always exists
glb(a, b) = greatest common divisor(a, b)always exists
Discrete Mathematics I – p. 165/292
![Page 688: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/688.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) =
least common multiple(a, b)always exists
glb(a, b) = greatest common divisor(a, b)always exists
Discrete Mathematics I – p. 165/292
![Page 689: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/689.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) = least common multiple(a, b)
always exists
glb(a, b) = greatest common divisor(a, b)always exists
Discrete Mathematics I – p. 165/292
![Page 690: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/690.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) = least common multiple(a, b)always exists
glb(a, b) = greatest common divisor(a, b)always exists
Discrete Mathematics I – p. 165/292
![Page 691: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/691.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) = least common multiple(a, b)always exists
glb(a, b) =
greatest common divisor(a, b)always exists
Discrete Mathematics I – p. 165/292
![Page 692: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/692.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) = least common multiple(a, b)always exists
glb(a, b) = greatest common divisor(a, b)
always exists
Discrete Mathematics I – p. 165/292
![Page 693: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/693.jpg)
RelationsR| : N ↔ N x | y ⇐⇒ x divides y
lub(a, b) = least common multiple(a, b)always exists
glb(a, b) = greatest common divisor(a, b)always exists
Discrete Mathematics I – p. 165/292
![Page 694: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/694.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 695: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/695.jpg)
RelationsR≤ : N ↔ N
lub(a, b) =
max(a, b) always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 696: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/696.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b)
always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 697: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/697.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 698: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/698.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) =
min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 699: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/699.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) = min(a, b)
always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 700: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/700.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 701: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/701.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}
Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 702: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/702.jpg)
RelationsR≤ : N ↔ N
lub(a, b) = max(a, b) always exists
glb(a, b) = min(a, b) always exists
{lub(a, b), glb(a, b)} = {a, b}Same holds for any total order
Discrete Mathematics I – p. 166/292
![Page 703: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/703.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) = A ∪B always exists
glb(A, B) = A ∩B always exists
Discrete Mathematics I – p. 167/292
![Page 704: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/704.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) =
A ∪B always exists
glb(A, B) = A ∩B always exists
Discrete Mathematics I – p. 167/292
![Page 705: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/705.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) = A ∪B
always exists
glb(A, B) = A ∩B always exists
Discrete Mathematics I – p. 167/292
![Page 706: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/706.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) = A ∪B always exists
glb(A, B) = A ∩B always exists
Discrete Mathematics I – p. 167/292
![Page 707: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/707.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) = A ∪B always exists
glb(A, B) =
A ∩B always exists
Discrete Mathematics I – p. 167/292
![Page 708: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/708.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) = A ∪B always exists
glb(A, B) = A ∩B
always exists
Discrete Mathematics I – p. 167/292
![Page 709: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/709.jpg)
RelationsR⊆ : P(S) ↔ P(S)
lub(A, B) = A ∪B always exists
glb(A, B) = A ∩B always exists
Discrete Mathematics I – p. 167/292
![Page 710: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/710.jpg)
RelationsR� : A ↔ A — a partial order
R� is called a lattice, if for all a, b ∈ A,lub(a, b) and glb(a, b) exist
(Sometimes A itself called a lattice)
Discrete Mathematics I – p. 168/292
![Page 711: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/711.jpg)
RelationsR� : A ↔ A — a partial order
R� is called a lattice, if for all a, b ∈ A,lub(a, b) and glb(a, b) exist
(Sometimes A itself called a lattice)
Discrete Mathematics I – p. 168/292
![Page 712: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/712.jpg)
RelationsR� : A ↔ A — a partial order
R� is called a lattice, if for all a, b ∈ A,lub(a, b) and glb(a, b) exist
(Sometimes A itself called a lattice)
Discrete Mathematics I – p. 168/292
![Page 713: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/713.jpg)
RelationsExamples:
any total order (e.g. R≤, R≥)
R| R⊆ for any S
Discrete Mathematics I – p. 169/292
![Page 714: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/714.jpg)
RelationsExamples:
any total order (e.g. R≤, R≥)
R|
R⊆ for any S
Discrete Mathematics I – p. 169/292
![Page 715: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/715.jpg)
RelationsExamples:
any total order (e.g. R≤, R≥)
R| R⊆ for any S
Discrete Mathematics I – p. 169/292
![Page 716: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/716.jpg)
RelationsR� : A ↔ A — a partial order
a ∈ A is maximal: ∀x ∈ A : (a � x) ⇒ (a = x)
b ∈ A is minimal: ∀x ∈ A : (x � b) ⇒ (b = x)
Can have many maximal/minimal elements
a ∈ A is the greatest: ∀x ∈ A : x � a
b ∈ A is the least: ∀x ∈ A : b � x
Can have at most one greatest/least element
Discrete Mathematics I – p. 170/292
![Page 717: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/717.jpg)
RelationsR� : A ↔ A — a partial order
a ∈ A is maximal: ∀x ∈ A : (a � x) ⇒ (a = x)
b ∈ A is minimal: ∀x ∈ A : (x � b) ⇒ (b = x)
Can have many maximal/minimal elements
a ∈ A is the greatest: ∀x ∈ A : x � a
b ∈ A is the least: ∀x ∈ A : b � x
Can have at most one greatest/least element
Discrete Mathematics I – p. 170/292
![Page 718: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/718.jpg)
RelationsR� : A ↔ A — a partial order
a ∈ A is maximal: ∀x ∈ A : (a � x) ⇒ (a = x)
b ∈ A is minimal: ∀x ∈ A : (x � b) ⇒ (b = x)
Can have many maximal/minimal elements
a ∈ A is the greatest: ∀x ∈ A : x � a
b ∈ A is the least: ∀x ∈ A : b � x
Can have at most one greatest/least element
Discrete Mathematics I – p. 170/292
![Page 719: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/719.jpg)
RelationsR� : A ↔ A — a partial order
a ∈ A is maximal: ∀x ∈ A : (a � x) ⇒ (a = x)
b ∈ A is minimal: ∀x ∈ A : (x � b) ⇒ (b = x)
Can have many maximal/minimal elements
a ∈ A is the greatest: ∀x ∈ A : x � a
b ∈ A is the least: ∀x ∈ A : b � x
Can have at most one greatest/least element
Discrete Mathematics I – p. 170/292
![Page 720: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/720.jpg)
RelationsR� : A ↔ A — a partial order
a ∈ A is maximal: ∀x ∈ A : (a � x) ⇒ (a = x)
b ∈ A is minimal: ∀x ∈ A : (x � b) ⇒ (b = x)
Can have many maximal/minimal elements
a ∈ A is the greatest: ∀x ∈ A : x � a
b ∈ A is the least: ∀x ∈ A : b � x
Can have at most one greatest/least element
Discrete Mathematics I – p. 170/292
![Page 721: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/721.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S)
∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 722: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/722.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least
S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 723: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/723.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 724: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/724.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N
0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 725: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/725.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least
no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 726: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/726.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 727: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/727.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N
no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 728: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/728.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least
0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 729: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/729.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 730: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/730.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N
1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 731: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/731.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least
0 greatest
Discrete Mathematics I – p. 171/292
![Page 732: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/732.jpg)
RelationsExamples:
R⊆ : P(S) ↔ P(S) ∅ least S greatest
R≤ : N ↔ N 0 least no greatest
R≥ : N ↔ N no least 0 greatest
R| : N ↔ N 1 least 0 greatest
Discrete Mathematics I – p. 171/292
![Page 733: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/733.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is a descendant of y
minimal elements: childless people
no least elements
no maximal elements (Adam and Eve? GE?)
no greatest elements
Discrete Mathematics I – p. 172/292
![Page 734: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/734.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is a descendant of y
minimal elements: childless people
no least elements
no maximal elements (Adam and Eve? GE?)
no greatest elements
Discrete Mathematics I – p. 172/292
![Page 735: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/735.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is a descendant of y
minimal elements: childless people
no least elements
no maximal elements (Adam and Eve? GE?)
no greatest elements
Discrete Mathematics I – p. 172/292
![Page 736: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/736.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is a descendant of y
minimal elements: childless people
no least elements
no maximal elements (Adam and Eve? GE?)
no greatest elements
Discrete Mathematics I – p. 172/292
![Page 737: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/737.jpg)
RelationsRE : People ↔ People
xE y ⇐⇒ x is a descendant of y
minimal elements: childless people
no least elements
no maximal elements (Adam and Eve? GE?)
no greatest elements
Discrete Mathematics I – p. 172/292
![Page 738: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/738.jpg)
RelationsR| : N \ {0, 1} ↔ N \ {0, 1}
minimal elements: prime numbers
no least elements
no maximal ⇒ no greatest
Discrete Mathematics I – p. 173/292
![Page 739: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/739.jpg)
RelationsR| : N \ {0, 1} ↔ N \ {0, 1}minimal elements: prime numbers
no least elements
no maximal ⇒ no greatest
Discrete Mathematics I – p. 173/292
![Page 740: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/740.jpg)
RelationsR| : N \ {0, 1} ↔ N \ {0, 1}minimal elements: prime numbers
no least elements
no maximal ⇒ no greatest
Discrete Mathematics I – p. 173/292
![Page 741: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/741.jpg)
RelationsR| : N \ {0, 1} ↔ N \ {0, 1}minimal elements: prime numbers
no least elements
no maximal ⇒ no greatest
Discrete Mathematics I – p. 173/292
![Page 742: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/742.jpg)
RelationsR⊆ : P(S) \ {∅, S} ↔ P(S) \ {∅, S}
minimal elements: singletons
no least elements
maximal elements: singleton complements
no greatest elements
Discrete Mathematics I – p. 174/292
![Page 743: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/743.jpg)
RelationsR⊆ : P(S) \ {∅, S} ↔ P(S) \ {∅, S}minimal elements: singletons
no least elements
maximal elements: singleton complements
no greatest elements
Discrete Mathematics I – p. 174/292
![Page 744: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/744.jpg)
RelationsR⊆ : P(S) \ {∅, S} ↔ P(S) \ {∅, S}minimal elements: singletons
no least elements
maximal elements: singleton complements
no greatest elements
Discrete Mathematics I – p. 174/292
![Page 745: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/745.jpg)
RelationsR⊆ : P(S) \ {∅, S} ↔ P(S) \ {∅, S}minimal elements: singletons
no least elements
maximal elements: singleton complements
no greatest elements
Discrete Mathematics I – p. 174/292
![Page 746: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/746.jpg)
RelationsR⊆ : P(S) \ {∅, S} ↔ P(S) \ {∅, S}minimal elements: singletons
no least elements
maximal elements: singleton complements
no greatest elements
Discrete Mathematics I – p. 174/292
![Page 747: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/747.jpg)
RelationsA partially ordered set may have no greatest or leastelement (even if the set is finite)
A finite, totally ordered set must have the greatest andthe least elements
A finite, partially ordered set must have maximal andminimal elements (but may not have the greatest andthe least)
A maximal or minimal element may not be unique
If a greatest or least element exists, it is unique
Discrete Mathematics I – p. 175/292
![Page 748: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/748.jpg)
RelationsA partially ordered set may have no greatest or leastelement (even if the set is finite)
A finite, totally ordered set must have the greatest andthe least elements
A finite, partially ordered set must have maximal andminimal elements (but may not have the greatest andthe least)
A maximal or minimal element may not be unique
If a greatest or least element exists, it is unique
Discrete Mathematics I – p. 175/292
![Page 749: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/749.jpg)
RelationsA partially ordered set may have no greatest or leastelement (even if the set is finite)
A finite, totally ordered set must have the greatest andthe least elements
A finite, partially ordered set must have maximal andminimal elements (but may not have the greatest andthe least)
A maximal or minimal element may not be unique
If a greatest or least element exists, it is unique
Discrete Mathematics I – p. 175/292
![Page 750: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/750.jpg)
RelationsA partially ordered set may have no greatest or leastelement (even if the set is finite)
A finite, totally ordered set must have the greatest andthe least elements
A finite, partially ordered set must have maximal andminimal elements (but may not have the greatest andthe least)
A maximal or minimal element may not be unique
If a greatest or least element exists, it is unique
Discrete Mathematics I – p. 175/292
![Page 751: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/751.jpg)
RelationsA partially ordered set may have no greatest or leastelement (even if the set is finite)
A finite, totally ordered set must have the greatest andthe least elements
A finite, partially ordered set must have maximal andminimal elements (but may not have the greatest andthe least)
A maximal or minimal element may not be unique
If a greatest or least element exists, it is unique
Discrete Mathematics I – p. 175/292
![Page 752: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/752.jpg)
Functions
Discrete Mathematics I – p. 176/292
![Page 753: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/753.jpg)
FunctionsA function from set A to set B is a relationRf : A ↔ B, where for every a ∈ A, there is a uniqueb ∈ B, such that afb
A B
f
Discrete Mathematics I – p. 177/292
![Page 754: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/754.jpg)
FunctionsA function from set A to set B is a relationRf : A ↔ B, where for every a ∈ A, there is a uniqueb ∈ B, such that afb
��
��
�
A
��
��
B
f
Discrete Mathematics I – p. 177/292
![Page 755: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/755.jpg)
Functionsf : A → B ⇐⇒
(Rf : A ↔ B) ∧ (∀a ∈ A : ∃!b ∈ B : afb)
f maps A into B
Instead of afb, write f(a) = b or f : a 7→ b
f maps a to b
Discrete Mathematics I – p. 178/292
![Page 756: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/756.jpg)
Functionsf : A → B ⇐⇒
(Rf : A ↔ B) ∧ (∀a ∈ A : ∃!b ∈ B : afb)
f maps A into B
Instead of afb, write f(a) = b or f : a 7→ b
f maps a to b
Discrete Mathematics I – p. 178/292
![Page 757: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/757.jpg)
Functionsf : A → B ⇐⇒
(Rf : A ↔ B) ∧ (∀a ∈ A : ∃!b ∈ B : afb)
f maps A into B
Instead of afb, write f(a) = b or f : a 7→ b
f maps a to b
Discrete Mathematics I – p. 178/292
![Page 758: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/758.jpg)
Functionsf : A → B ⇐⇒
(Rf : A ↔ B) ∧ (∀a ∈ A : ∃!b ∈ B : afb)
f maps A into B
Instead of afb, write f(a) = b or f : a 7→ b
f maps a to b
Discrete Mathematics I – p. 178/292
![Page 759: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/759.jpg)
Functionsf : A → B
A is the domain of f Domf = A
B is the co-domain of f Codomf = B
f : a 7→ b a ∈ A b ∈ B
b is the image of a
a is the pre-image of b
Discrete Mathematics I – p. 179/292
![Page 760: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/760.jpg)
Functionsf : A → B
A is the domain of f Domf = A
B is the co-domain of f Codomf = B
f : a 7→ b a ∈ A b ∈ B
b is the image of a
a is the pre-image of b
Discrete Mathematics I – p. 179/292
![Page 761: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/761.jpg)
Functionsf : A → B
A is the domain of f Domf = A
B is the co-domain of f Codomf = B
f : a 7→ b a ∈ A b ∈ B
b is the image of a
a is the pre-image of b
Discrete Mathematics I – p. 179/292
![Page 762: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/762.jpg)
Functionsf : A → B
A is the domain of f Domf = A
B is the co-domain of f Codomf = B
f : a 7→ b a ∈ A b ∈ B
b is the image of a
a is the pre-image of b
Discrete Mathematics I – p. 179/292
![Page 763: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/763.jpg)
FunctionsExamples:
Identity function idA : A → AidA = {(a, a) | a ∈ A} = R=A
A A
idA
Discrete Mathematics I – p. 180/292
![Page 764: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/764.jpg)
FunctionsExamples:
Identity function idA : A → AidA = {(a, a) | a ∈ A} = R=A
��
��
�
A
��
��
A
idA
Discrete Mathematics I – p. 180/292
![Page 765: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/765.jpg)
Functionsf : N → N f = {(m, n) ∈ N2 | m2 = n}
= {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), . . .}
g : {0, 1, 2} → N g = {(0, 0), (1, 1), (2, 4)}g ⊆ f Domg ⊆ Domf Codomg ⊆ Codomf
Function g called a restriction of f
Function f called an extension of g
Discrete Mathematics I – p. 181/292
![Page 766: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/766.jpg)
Functionsf : N → N f = {(m, n) ∈ N2 | m2 = n}
= {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), . . .}g : {0, 1, 2} → N g = {(0, 0), (1, 1), (2, 4)}
g ⊆ f Domg ⊆ Domf Codomg ⊆ Codomf
Function g called a restriction of f
Function f called an extension of g
Discrete Mathematics I – p. 181/292
![Page 767: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/767.jpg)
Functionsf : N → N f = {(m, n) ∈ N2 | m2 = n}
= {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), . . .}g : {0, 1, 2} → N g = {(0, 0), (1, 1), (2, 4)}g ⊆ f
Domg ⊆ Domf Codomg ⊆ Codomf
Function g called a restriction of f
Function f called an extension of g
Discrete Mathematics I – p. 181/292
![Page 768: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/768.jpg)
Functionsf : N → N f = {(m, n) ∈ N2 | m2 = n}
= {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), . . .}g : {0, 1, 2} → N g = {(0, 0), (1, 1), (2, 4)}g ⊆ f Domg ⊆ Domf Codomg ⊆ Codomf
Function g called a restriction of f
Function f called an extension of g
Discrete Mathematics I – p. 181/292
![Page 769: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/769.jpg)
Functionsf : N → N f = {(m, n) ∈ N2 | m2 = n}
= {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), . . .}g : {0, 1, 2} → N g = {(0, 0), (1, 1), (2, 4)}g ⊆ f Domg ⊆ Domf Codomg ⊆ Codomf
Function g called a restriction of f
Function f called an extension of g
Discrete Mathematics I – p. 181/292
![Page 770: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/770.jpg)
Functionsf : N → N f = {(m, n) ∈ N2 | m2 = n}
= {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), . . .}g : {0, 1, 2} → N g = {(0, 0), (1, 1), (2, 4)}g ⊆ f Domg ⊆ Domf Codomg ⊆ Codomf
Function g called a restriction of f
Function f called an extension of g
Discrete Mathematics I – p. 181/292
![Page 771: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/771.jpg)
FunctionsAn infinite sequence of elements of set A is anyfunction a : N → A
Integer sequence N → Z, Boolean sequence N → B,etc.
Instead of a(k) write ak: a = (a0, a1, a2, a3, . . . )
Examples:
a : N → N a = (0, 1, 4, 9, 16, . . . )
b : N → B b = (F, T, F, T, F, . . . )
Discrete Mathematics I – p. 182/292
![Page 772: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/772.jpg)
FunctionsAn infinite sequence of elements of set A is anyfunction a : N → A
Integer sequence N → Z, Boolean sequence N → B,etc.
Instead of a(k) write ak: a = (a0, a1, a2, a3, . . . )
Examples:
a : N → N a = (0, 1, 4, 9, 16, . . . )
b : N → B b = (F, T, F, T, F, . . . )
Discrete Mathematics I – p. 182/292
![Page 773: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/773.jpg)
FunctionsAn infinite sequence of elements of set A is anyfunction a : N → A
Integer sequence N → Z, Boolean sequence N → B,etc.
Instead of a(k) write ak: a = (a0, a1, a2, a3, . . . )
Examples:
a : N → N a = (0, 1, 4, 9, 16, . . . )
b : N → B b = (F, T, F, T, F, . . . )
Discrete Mathematics I – p. 182/292
![Page 774: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/774.jpg)
FunctionsAn infinite sequence of elements of set A is anyfunction a : N → A
Integer sequence N → Z, Boolean sequence N → B,etc.
Instead of a(k) write ak: a = (a0, a1, a2, a3, . . . )
Examples:
a : N → N a = (0, 1, 4, 9, 16, . . . )
b : N → B b = (F, T, F, T, F, . . . )
Discrete Mathematics I – p. 182/292
![Page 775: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/775.jpg)
FunctionsAn infinite sequence of elements of set A is anyfunction a : N → A
Integer sequence N → Z, Boolean sequence N → B,etc.
Instead of a(k) write ak: a = (a0, a1, a2, a3, . . . )
Examples:
a : N → N a = (0, 1, 4, 9, 16, . . . )
b : N → B b = (F, T, F, T, F, . . . )
Discrete Mathematics I – p. 182/292
![Page 776: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/776.jpg)
FunctionsLet n ∈ N
Nn = {x ∈ N | x < n} = {0, 1, 2, . . . , n− 1}
A finite sequence of elements of set A is any functiona : Nn → A
Example: g = (0, 1, 4)
Number n is the sequence length
Discrete Mathematics I – p. 183/292
![Page 777: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/777.jpg)
FunctionsLet n ∈ N
Nn = {x ∈ N | x < n} = {0, 1, 2, . . . , n− 1}A finite sequence of elements of set A is any functiona : Nn → A
Example: g = (0, 1, 4)
Number n is the sequence length
Discrete Mathematics I – p. 183/292
![Page 778: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/778.jpg)
FunctionsLet n ∈ N
Nn = {x ∈ N | x < n} = {0, 1, 2, . . . , n− 1}A finite sequence of elements of set A is any functiona : Nn → A
Example: g = (0, 1, 4)
Number n is the sequence length
Discrete Mathematics I – p. 183/292
![Page 779: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/779.jpg)
FunctionsLet n ∈ N
Nn = {x ∈ N | x < n} = {0, 1, 2, . . . , n− 1}A finite sequence of elements of set A is any functiona : Nn → A
Example: g = (0, 1, 4)
Number n is the sequence length
Discrete Mathematics I – p. 183/292
![Page 780: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/780.jpg)
Functionsf : A → B g : B → C
The composition of f and g f ◦ g : A → C
Same as relation composition Rf◦g
Rf , Rg functions =⇒ Rf◦g a function
∀a ∈ A : (f ◦ g)(a) = g(f(a))
Discrete Mathematics I – p. 184/292
![Page 781: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/781.jpg)
Functionsf : A → B g : B → C
The composition of f and g f ◦ g : A → C
Same as relation composition Rf◦g
Rf , Rg functions =⇒ Rf◦g a function
∀a ∈ A : (f ◦ g)(a) = g(f(a))
Discrete Mathematics I – p. 184/292
![Page 782: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/782.jpg)
Functionsf : A → B g : B → C
The composition of f and g f ◦ g : A → C
Same as relation composition Rf◦g
Rf , Rg functions =⇒ Rf◦g a function
∀a ∈ A : (f ◦ g)(a) = g(f(a))
Discrete Mathematics I – p. 184/292
![Page 783: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/783.jpg)
Functionsf : A → B g : B → C
The composition of f and g f ◦ g : A → C
Same as relation composition Rf◦g
Rf , Rg functions =⇒ Rf◦g a function
∀a ∈ A : (f ◦ g)(a) = g(f(a))
Discrete Mathematics I – p. 184/292
![Page 784: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/784.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 785: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/785.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) =
(n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 786: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/786.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 787: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/787.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) =
(n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 788: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/788.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 789: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/789.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) =
n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 790: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/790.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 791: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/791.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) =
(n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 792: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/792.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 793: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/793.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
g : Z → Z n 7→ n2
(f ◦ f)(n) = (n + 1) + 1 = n + 2
(f ◦ g)(n) = (n + 1)2 = n2 + 2n + 1
(g ◦ f)(n) = n2 + 1
(g ◦ g)(n) = (n2)2 = n4
Note (f ◦ g) 6= (g ◦ f)
Discrete Mathematics I – p. 185/292
![Page 794: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/794.jpg)
Functionsf : A → B
The (functional) inverse of f f−1 : B → A
Same as relation inverse Rf−1, but may not be afunction
(we say “functional inverse may not exist”)
∀a ∈ A, b ∈ B : (f(a) = b) ⇔ (f−1(b) = a)
Discrete Mathematics I – p. 186/292
![Page 795: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/795.jpg)
Functionsf : A → B
The (functional) inverse of f f−1 : B → A
Same as relation inverse Rf−1, but may not be afunction
(we say “functional inverse may not exist”)
∀a ∈ A, b ∈ B : (f(a) = b) ⇔ (f−1(b) = a)
Discrete Mathematics I – p. 186/292
![Page 796: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/796.jpg)
Functionsf : A → B
The (functional) inverse of f f−1 : B → A
Same as relation inverse Rf−1, but may not be afunction
(we say “functional inverse may not exist”)
∀a ∈ A, b ∈ B : (f(a) = b) ⇔ (f−1(b) = a)
Discrete Mathematics I – p. 186/292
![Page 797: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/797.jpg)
Functionsf : A → B
The (functional) inverse of f f−1 : B → A
Same as relation inverse Rf−1, but may not be afunction
(we say “functional inverse may not exist”)
∀a ∈ A, b ∈ B : (f(a) = b) ⇔ (f−1(b) = a)
Discrete Mathematics I – p. 186/292
![Page 798: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/798.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
f−1(n) = n− 1
g : Z → Z n 7→ n2
g−1does not exist (i.e. Rg−1 is not a function)
Discrete Mathematics I – p. 187/292
![Page 799: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/799.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
f−1(n) =
n− 1
g : Z → Z n 7→ n2
g−1does not exist (i.e. Rg−1 is not a function)
Discrete Mathematics I – p. 187/292
![Page 800: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/800.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
f−1(n) = n− 1
g : Z → Z n 7→ n2
g−1does not exist (i.e. Rg−1 is not a function)
Discrete Mathematics I – p. 187/292
![Page 801: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/801.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
f−1(n) = n− 1
g : Z → Z n 7→ n2
g−1does not exist (i.e. Rg−1 is not a function)
Discrete Mathematics I – p. 187/292
![Page 802: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/802.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
f−1(n) = n− 1
g : Z → Z n 7→ n2
g−1
does not exist (i.e. Rg−1 is not a function)
Discrete Mathematics I – p. 187/292
![Page 803: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/803.jpg)
FunctionsExamples:
f : Z → Z n 7→ n + 1
f−1(n) = n− 1
g : Z → Z n 7→ n2
g−1 does not exist (i.e. Rg−1 is not a function)
Discrete Mathematics I – p. 187/292
![Page 804: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/804.jpg)
Functionsf : A → B
The range of f is the set of all elements in B that havea pre-image in A
f(A) = {b ∈ B | ∃a ∈ A : f(a) = b}
f(A)
A B
f
Discrete Mathematics I – p. 188/292
![Page 805: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/805.jpg)
Functionsf : A → B
The range of f is the set of all elements in B that havea pre-image in A
f(A) = {b ∈ B | ∃a ∈ A : f(a) = b}
f(A)
A B
f
Discrete Mathematics I – p. 188/292
![Page 806: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/806.jpg)
Functionsf : A → B
The range of f is the set of all elements in B that havea pre-image in A
f(A) = {b ∈ B | ∃a ∈ A : f(a) = b}
f(A)
A B
f
Discrete Mathematics I – p. 188/292
![Page 807: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/807.jpg)
Functionsf : A → B
The range of f is the set of all elements in B that havea pre-image in A
f(A) = {b ∈ B | ∃a ∈ A : f(a) = b}
f(A)
��
��
�A
��
��
B
f
Discrete Mathematics I – p. 188/292
![Page 808: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/808.jpg)
FunctionsExamples:
f : N → N n 7→ n + 1
f(N) = N \ {0} = {1, 2, 3, 4, . . . }g : {0, 1, 2} → N n 7→ n2
g({0, 1, 2}) = {0, 1, 4}
Discrete Mathematics I – p. 189/292
![Page 809: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/809.jpg)
FunctionsExamples:
f : N → N n 7→ n + 1
f(N) =
N \ {0} = {1, 2, 3, 4, . . . }g : {0, 1, 2} → N n 7→ n2
g({0, 1, 2}) = {0, 1, 4}
Discrete Mathematics I – p. 189/292
![Page 810: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/810.jpg)
FunctionsExamples:
f : N → N n 7→ n + 1
f(N) = N \ {0} = {1, 2, 3, 4, . . . }
g : {0, 1, 2} → N n 7→ n2
g({0, 1, 2}) = {0, 1, 4}
Discrete Mathematics I – p. 189/292
![Page 811: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/811.jpg)
FunctionsExamples:
f : N → N n 7→ n + 1
f(N) = N \ {0} = {1, 2, 3, 4, . . . }g : {0, 1, 2} → N n 7→ n2
g({0, 1, 2}) = {0, 1, 4}
Discrete Mathematics I – p. 189/292
![Page 812: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/812.jpg)
FunctionsExamples:
f : N → N n 7→ n + 1
f(N) = N \ {0} = {1, 2, 3, 4, . . . }g : {0, 1, 2} → N n 7→ n2
g({0, 1, 2}) =
{0, 1, 4}
Discrete Mathematics I – p. 189/292
![Page 813: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/813.jpg)
FunctionsExamples:
f : N → N n 7→ n + 1
f(N) = N \ {0} = {1, 2, 3, 4, . . . }g : {0, 1, 2} → N n 7→ n2
g({0, 1, 2}) = {0, 1, 4}
Discrete Mathematics I – p. 189/292
![Page 814: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/814.jpg)
FunctionsFunction called surjective, if its range is the wholeco-domain
f : A� B ⇐⇒ ∀b ∈ B : ∃a ∈ A : f(a) = b
A B
f
Also say f maps A onto B
Discrete Mathematics I – p. 190/292
![Page 815: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/815.jpg)
FunctionsFunction called surjective, if its range is the wholeco-domain
f : A� B ⇐⇒ ∀b ∈ B : ∃a ∈ A : f(a) = b
A B
f
Also say f maps A onto B
Discrete Mathematics I – p. 190/292
![Page 816: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/816.jpg)
FunctionsFunction called surjective, if its range is the wholeco-domain
f : A� B ⇐⇒ ∀b ∈ B : ∃a ∈ A : f(a) = b
��
��
�
A
��
�
B
f
Also say f maps A onto B
Discrete Mathematics I – p. 190/292
![Page 817: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/817.jpg)
FunctionsFunction called surjective, if its range is the wholeco-domain
f : A� B ⇐⇒ ∀b ∈ B : ∃a ∈ A : f(a) = b
��
��
�
A
��
�
B
f
Also say f maps A onto B
Discrete Mathematics I – p. 190/292
![Page 818: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/818.jpg)
FunctionsFunction called injective, if it maps different elementsto different elements
f : A� B ⇐⇒ ∀x, y : (f(x) = f(y)) ⇒ (x = y)
A B
f
Also say f maps A into B one-to-one
Discrete Mathematics I – p. 191/292
![Page 819: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/819.jpg)
FunctionsFunction called injective, if it maps different elementsto different elements
f : A� B ⇐⇒ ∀x, y : (f(x) = f(y)) ⇒ (x = y)
A B
f
Also say f maps A into B one-to-one
Discrete Mathematics I – p. 191/292
![Page 820: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/820.jpg)
FunctionsFunction called injective, if it maps different elementsto different elements
f : A� B ⇐⇒ ∀x, y : (f(x) = f(y)) ⇒ (x = y)
��
�
A
��
��
�
B
f
Also say f maps A into B one-to-one
Discrete Mathematics I – p. 191/292
![Page 821: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/821.jpg)
FunctionsFunction called injective, if it maps different elementsto different elements
f : A� B ⇐⇒ ∀x, y : (f(x) = f(y)) ⇒ (x = y)
��
�
A
��
��
�
B
f
Also say f maps A into B one-to-one
Discrete Mathematics I – p. 191/292
![Page 822: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/822.jpg)
FunctionsExamples:
f : Cards � {♠,♥,♣,♦} f : x 7→ suit of x
Function f surjective, but not injective
g : N� N g : m 7→ m2
Function g injective, but not surjective
Discrete Mathematics I – p. 192/292
![Page 823: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/823.jpg)
FunctionsExamples:
f : Cards � {♠,♥,♣,♦} f : x 7→ suit of x
Function f surjective, but not injective
g : N� N g : m 7→ m2
Function g injective, but not surjective
Discrete Mathematics I – p. 192/292
![Page 824: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/824.jpg)
FunctionsExamples:
f : Cards � {♠,♥,♣,♦} f : x 7→ suit of x
Function f surjective, but not injective
g : N� N g : m 7→ m2
Function g injective, but not surjective
Discrete Mathematics I – p. 192/292
![Page 825: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/825.jpg)
FunctionsExamples:
f : Cards � {♠,♥,♣,♦} f : x 7→ suit of x
Function f surjective, but not injective
g : N� N g : m 7→ m2
Function g injective, but not surjective
Discrete Mathematics I – p. 192/292
![Page 826: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/826.jpg)
FunctionsFunction called bijective, if it is both surjective andinjective
f : A��B ⇐⇒ ∀b ∈ B : ∃!a ∈ A : f(a) = b
A B
f
Also say f is a one-to-one correspondence between Aand B
Discrete Mathematics I – p. 193/292
![Page 827: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/827.jpg)
FunctionsFunction called bijective, if it is both surjective andinjective
f : A��B ⇐⇒ ∀b ∈ B : ∃!a ∈ A : f(a) = b
A B
f
Also say f is a one-to-one correspondence between Aand B
Discrete Mathematics I – p. 193/292
![Page 828: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/828.jpg)
FunctionsFunction called bijective, if it is both surjective andinjective
f : A��B ⇐⇒ ∀b ∈ B : ∃!a ∈ A : f(a) = b
��
��
A
��
��
B
f
Also say f is a one-to-one correspondence between Aand B
Discrete Mathematics I – p. 193/292
![Page 829: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/829.jpg)
FunctionsFunction called bijective, if it is both surjective andinjective
f : A��B ⇐⇒ ∀b ∈ B : ∃!a ∈ A : f(a) = b
��
��
A
��
��
B
f
Also say f is a one-to-one correspondence between Aand B
Discrete Mathematics I – p. 193/292
![Page 830: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/830.jpg)
FunctionsExamples:
idA : A��A a 7→ a
Function idA bijective for any set A
id−1
A = idA
f : Z��Z n 7→ n + 5
g : Z��Z n 7→ −n
Discrete Mathematics I – p. 194/292
![Page 831: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/831.jpg)
FunctionsExamples:
idA : A��A a 7→ a
Function idA bijective for any set A
id−1
A = idA
f : Z��Z n 7→ n + 5
g : Z��Z n 7→ −n
Discrete Mathematics I – p. 194/292
![Page 832: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/832.jpg)
FunctionsExamples:
idA : A��A a 7→ a
Function idA bijective for any set A
id−1
A = idA
f : Z��Z n 7→ n + 5
g : Z��Z n 7→ −n
Discrete Mathematics I – p. 194/292
![Page 833: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/833.jpg)
FunctionsExamples:
idA : A��A a 7→ a
Function idA bijective for any set A
id−1
A = idA
f : Z��Z n 7→ n + 5
g : Z��Z n 7→ −n
Discrete Mathematics I – p. 194/292
![Page 834: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/834.jpg)
FunctionsExamples:
idA : A��A a 7→ a
Function idA bijective for any set A
id−1
A = idA
f : Z��Z n 7→ n + 5
g : Z��Z n 7→ −n
Discrete Mathematics I – p. 194/292
![Page 835: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/835.jpg)
Functionsf : Z��Z n 7→ n + 5
Function f bijective
f−1 : Z��Z m 7→ m− 5
f ◦ f−1 = f−1 ◦ f = idZ
For any set A, bijective f : A��A is a permutation
Discrete Mathematics I – p. 195/292
![Page 836: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/836.jpg)
Functionsf : Z��Z n 7→ n + 5
Function f bijective
f−1 : Z��Z m 7→ m− 5
f ◦ f−1 = f−1 ◦ f = idZ
For any set A, bijective f : A��A is a permutation
Discrete Mathematics I – p. 195/292
![Page 837: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/837.jpg)
Functionsf : Z��Z n 7→ n + 5
Function f bijective
f−1 : Z��Z m 7→ m− 5
f ◦ f−1 = f−1 ◦ f = idZ
For any set A, bijective f : A��A is a permutation
Discrete Mathematics I – p. 195/292
![Page 838: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/838.jpg)
Functionsf : Z��Z n 7→ n + 5
Function f bijective
f−1 : Z��Z m 7→ m− 5
f ◦ f−1 = f−1 ◦ f = idZ
For any set A, bijective f : A��A is a permutation
Discrete Mathematics I – p. 195/292
![Page 839: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/839.jpg)
Functionsf : Z��Z n 7→ n + 5
Function f bijective
f−1 : Z��Z m 7→ m− 5
f ◦ f−1 = f−1 ◦ f = idZ
For any set A, bijective f : A��A is a permutation
Discrete Mathematics I – p. 195/292
![Page 840: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/840.jpg)
Functionsg : Z��Z n 7→ −n
Function g bijective
g−1 : Z��Z g−1 = g
g ◦ g−1 = g−1 ◦ g = g ◦ g = idZ
For any set A, a permutation g : A��A with g−1 = gis an involution
Discrete Mathematics I – p. 196/292
![Page 841: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/841.jpg)
Functionsg : Z��Z n 7→ −n
Function g bijective
g−1 : Z��Z g−1 = g
g ◦ g−1 = g−1 ◦ g = g ◦ g = idZ
For any set A, a permutation g : A��A with g−1 = gis an involution
Discrete Mathematics I – p. 196/292
![Page 842: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/842.jpg)
Functionsg : Z��Z n 7→ −n
Function g bijective
g−1 : Z��Z g−1 = g
g ◦ g−1 = g−1 ◦ g = g ◦ g = idZ
For any set A, a permutation g : A��A with g−1 = gis an involution
Discrete Mathematics I – p. 196/292
![Page 843: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/843.jpg)
Functionsg : Z��Z n 7→ −n
Function g bijective
g−1 : Z��Z g−1 = g
g ◦ g−1 = g−1 ◦ g = g ◦ g = idZ
For any set A, a permutation g : A��A with g−1 = gis an involution
Discrete Mathematics I – p. 196/292
![Page 844: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/844.jpg)
Functionsg : Z��Z n 7→ −n
Function g bijective
g−1 : Z��Z g−1 = g
g ◦ g−1 = g−1 ◦ g = g ◦ g = idZ
For any set A, a permutation g : A��A with g−1 = gis an involution
Discrete Mathematics I – p. 196/292
![Page 845: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/845.jpg)
Functionsf : A → B g : B → C
If f , g surjective, then f ◦ g surjective
If f , g injective, then f ◦ g injective
If f , g bijective, then f ◦ g bijective
Discrete Mathematics I – p. 197/292
![Page 846: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/846.jpg)
Functionsf : A → B g : B → C
If f , g surjective, then f ◦ g surjective
If f , g injective, then f ◦ g injective
If f , g bijective, then f ◦ g bijective
Discrete Mathematics I – p. 197/292
![Page 847: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/847.jpg)
Functionsf : A → B g : B → C
If f , g surjective, then f ◦ g surjective
If f , g injective, then f ◦ g injective
If f , g bijective, then f ◦ g bijective
Discrete Mathematics I – p. 197/292
![Page 848: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/848.jpg)
Functionsf : A → B g : B → C
If f , g surjective, then f ◦ g surjective
If f , g injective, then f ◦ g injective
If f , g bijective, then f ◦ g bijective
Discrete Mathematics I – p. 197/292
![Page 849: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/849.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 850: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/850.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 851: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/851.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒
∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 852: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/852.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒
∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 853: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/853.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒
f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 854: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/854.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 855: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/855.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒
∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 856: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/856.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒
∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 857: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/857.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒
f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 858: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/858.jpg)
FunctionsProve: if f bijective, then f−1 bijective.
Proof. Consider Rf : A ↔ B, Rf−1 : B ↔ A
f bijective =⇒ ∀b ∈ B : ∃!a ∈ A : afb =⇒∀b ∈ B : ∃!a ∈ A : b(f−1)a =⇒ f−1 a function
f a function =⇒ ∀a ∈ A : ∃!b ∈ B : afb =⇒∀a ∈ A : ∃!b ∈ B : b(f−1)a =⇒ f−1 bijective
Discrete Mathematics I – p. 198/292
![Page 859: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/859.jpg)
FunctionsProve: if both f and f−1 are functions, then both fand f−1 are bijective
Proof (sketch). Let f : A → B, f−1 : B → A.
f a function ⇒ f−1 surjective
f a function ⇒ f−1 injective
f = (f−1)−1 ⇒ f surjective and injective
To prove f : A��B, only need f−1 : B → A
f ◦ f−1 = idA f−1 ◦ f = idB
Discrete Mathematics I – p. 199/292
![Page 860: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/860.jpg)
FunctionsProve: if both f and f−1 are functions, then both fand f−1 are bijective
Proof (sketch). Let f : A → B, f−1 : B → A.
f a function ⇒ f−1 surjective
f a function ⇒ f−1 injective
f = (f−1)−1 ⇒ f surjective and injective
To prove f : A��B, only need f−1 : B → A
f ◦ f−1 = idA f−1 ◦ f = idB
Discrete Mathematics I – p. 199/292
![Page 861: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/861.jpg)
FunctionsProve: if both f and f−1 are functions, then both fand f−1 are bijective
Proof (sketch). Let f : A → B, f−1 : B → A.
f a function ⇒ f−1 surjective
f a function ⇒ f−1 injective
f = (f−1)−1 ⇒ f surjective and injective
To prove f : A��B, only need f−1 : B → A
f ◦ f−1 = idA f−1 ◦ f = idB
Discrete Mathematics I – p. 199/292
![Page 862: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/862.jpg)
FunctionsProve: if both f and f−1 are functions, then both fand f−1 are bijective
Proof (sketch). Let f : A → B, f−1 : B → A.
f a function ⇒ f−1 surjective
f a function ⇒ f−1 injective
f = (f−1)−1 ⇒ f surjective and injective
To prove f : A��B, only need f−1 : B → A
f ◦ f−1 = idA f−1 ◦ f = idB
Discrete Mathematics I – p. 199/292
![Page 863: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/863.jpg)
FunctionsProve: if both f and f−1 are functions, then both fand f−1 are bijective
Proof (sketch). Let f : A → B, f−1 : B → A.
f a function ⇒ f−1 surjective
f a function ⇒ f−1 injective
f = (f−1)−1 ⇒ f surjective and injective
To prove f : A��B, only need f−1 : B → A
f ◦ f−1 = idA f−1 ◦ f = idB
Discrete Mathematics I – p. 199/292
![Page 864: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/864.jpg)
FunctionsProve: if both f and f−1 are functions, then both fand f−1 are bijective
Proof (sketch). Let f : A → B, f−1 : B → A.
f a function ⇒ f−1 surjective
f a function ⇒ f−1 injective
f = (f−1)−1 ⇒ f surjective and injective
To prove f : A��B, only need f−1 : B → A
f ◦ f−1 = idA f−1 ◦ f = idB
Discrete Mathematics I – p. 199/292
![Page 865: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/865.jpg)
FunctionsS — any set A ⊆ S B = {F, T}
Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}Set of all Boolean functions on S:
B(S) = {f | f : S → B}χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 866: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/866.jpg)
FunctionsS — any set A ⊆ S B = {F, T}Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}Set of all Boolean functions on S:
B(S) = {f | f : S → B}χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 867: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/867.jpg)
FunctionsS — any set A ⊆ S B = {F, T}Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}Set of all Boolean functions on S:
B(S) = {f | f : S → B}χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 868: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/868.jpg)
FunctionsS — any set A ⊆ S B = {F, T}Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}
Set of all Boolean functions on S:B(S) = {f | f : S → B}
χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 869: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/869.jpg)
FunctionsS — any set A ⊆ S B = {F, T}Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}Set of all Boolean functions on S:
B(S) = {f | f : S → B}
χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 870: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/870.jpg)
FunctionsS — any set A ⊆ S B = {F, T}Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}Set of all Boolean functions on S:
B(S) = {f | f : S → B}χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 871: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/871.jpg)
FunctionsS — any set A ⊆ S B = {F, T}Indicator function of A χA : S → B
∀x ∈ S : χA(x) =
{
T if x ∈ A
F if x 6∈ A
Set of all subsets of S P(S) = {A | A ⊆ S}Set of all Boolean functions on S:
B(S) = {f | f : S → B}χ : P(S)��B(S) ∀A ⊆ S : A 7→ χA
Subsets of S�� Boolean functions on S
Discrete Mathematics I – p. 200/292
![Page 872: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/872.jpg)
FunctionsSets A, B are called equinumerous, if there is abijection between A and B
A ∼= B ⇐⇒ ∃f : A��B
A, B ⊆ S =⇒ R∼= an equivalence on P(S)
Discrete Mathematics I – p. 201/292
![Page 873: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/873.jpg)
FunctionsSets A, B are called equinumerous, if there is abijection between A and B
A ∼= B ⇐⇒ ∃f : A��B
A, B ⊆ S =⇒ R∼= an equivalence on P(S)
Discrete Mathematics I – p. 201/292
![Page 874: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/874.jpg)
FunctionsSets A, B are called equinumerous, if there is abijection between A and B
A ∼= B ⇐⇒ ∃f : A��B
A, B ⊆ S =⇒ R∼= an equivalence on P(S)
Discrete Mathematics I – p. 201/292
![Page 875: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/875.jpg)
Functionsn ∈ N Nn = {x ∈ N | x < n}
N0 = ∅ N1 = {0} N2 = {0, 1} . . .
Nn = {0, 1, 2, . . . , n− 1}Set A called finite, if A ∼= Nn for some n ∈ N
Otherwise, the set is called infinite
Discrete Mathematics I – p. 202/292
![Page 876: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/876.jpg)
Functionsn ∈ N Nn = {x ∈ N | x < n}N0 = ∅ N1 = {0} N2 = {0, 1} . . .
Nn = {0, 1, 2, . . . , n− 1}Set A called finite, if A ∼= Nn for some n ∈ N
Otherwise, the set is called infinite
Discrete Mathematics I – p. 202/292
![Page 877: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/877.jpg)
Functionsn ∈ N Nn = {x ∈ N | x < n}N0 = ∅ N1 = {0} N2 = {0, 1} . . .
Nn = {0, 1, 2, . . . , n− 1}
Set A called finite, if A ∼= Nn for some n ∈ N
Otherwise, the set is called infinite
Discrete Mathematics I – p. 202/292
![Page 878: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/878.jpg)
Functionsn ∈ N Nn = {x ∈ N | x < n}N0 = ∅ N1 = {0} N2 = {0, 1} . . .
Nn = {0, 1, 2, . . . , n− 1}Set A called finite, if A ∼= Nn for some n ∈ N
Otherwise, the set is called infinite
Discrete Mathematics I – p. 202/292
![Page 879: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/879.jpg)
Functionsn ∈ N Nn = {x ∈ N | x < n}N0 = ∅ N1 = {0} N2 = {0, 1} . . .
Nn = {0, 1, 2, . . . , n− 1}Set A called finite, if A ∼= Nn for some n ∈ N
Otherwise, the set is called infinite
Discrete Mathematics I – p. 202/292
![Page 880: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/880.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 881: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/881.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 882: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/882.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 883: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/883.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 884: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/884.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 885: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/885.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A
|A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 886: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/886.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 887: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/887.jpg)
FunctionsFor every finite set A, there is a unique n ∈ N, suchthat A ∼= Nn
Proof.
Let f : A��Nk. Then f−1 : Nk��A.
Let g : A��Nl.
f−1 ◦ g : Nk��Nl. Therefore k = l.
Number n called the cardinality of A |A| = n
A, B finite, A ∼= B =⇒ |A| = |B|
Discrete Mathematics I – p. 203/292
![Page 888: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/888.jpg)
FunctionsN, Z, N2, N3, P(N), Neven — infinite
An infinite set is called countable, if it isequinumerous with N
In particular, N itself is countable
Discrete Mathematics I – p. 204/292
![Page 889: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/889.jpg)
FunctionsN, Z, N2, N3, P(N), Neven — infinite
An infinite set is called countable, if it isequinumerous with N
In particular, N itself is countable
Discrete Mathematics I – p. 204/292
![Page 890: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/890.jpg)
FunctionsN, Z, N2, N3, P(N), Neven — infinite
An infinite set is called countable, if it isequinumerous with N
In particular, N itself is countable
Discrete Mathematics I – p. 204/292
![Page 891: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/891.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 892: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/892.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 893: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/893.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 894: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/894.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 895: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/895.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 896: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/896.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 897: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/897.jpg)
FunctionsProve: set N+ = N \ {0} is countable.
Proof. Let f : N → N+ ∀n : n 7→ n + 1
∀n ∈ N+ : n = (n− 1) + 1 = f(n− 1)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (m + 1 6= n + 1)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 205/292
![Page 898: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/898.jpg)
FunctionsN+ ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l1 2 3 4 5 6 7 8 · · ·
N+ ∼= N
A part is of the same size as the whole!
Discrete Mathematics I – p. 206/292
![Page 899: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/899.jpg)
FunctionsN+ ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l1 2 3 4 5 6 7 8 · · ·
N+ ∼= N
A part is of the same size as the whole!
Discrete Mathematics I – p. 206/292
![Page 900: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/900.jpg)
FunctionsN+ ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l1 2 3 4 5 6 7 8 · · ·
N+ ∼= N
A part is of the same size as the whole!
Discrete Mathematics I – p. 206/292
![Page 901: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/901.jpg)
FunctionsN+ ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l1 2 3 4 5 6 7 8 · · ·
N+ ∼= N
A part is of the same size as the whole!
Discrete Mathematics I – p. 206/292
![Page 902: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/902.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 903: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/903.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 904: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/904.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 905: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/905.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 906: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/906.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 907: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/907.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 908: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/908.jpg)
FunctionsProve: set Neven = {0, 2, 4, 6, . . . } is countable.
Proof. Let f : N → Neven ∀n : n 7→ 2n
∀n ∈ Neven : n = 2 · n/2 = f(n/2)
Hence f surjective
∀m, n ∈ N : (m 6= n) ⇒ (2m 6= 2n)
Hence f injective
f surjective and injective =⇒ f bijective
Discrete Mathematics I – p. 207/292
![Page 909: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/909.jpg)
FunctionsNeven ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l0 2 4 6 8 10 12 14 · · ·
Neven∼= N
In general, any subset of a countable set is finite orcountable. Any quotient set of a countable set is finiteor countable.
Discrete Mathematics I – p. 208/292
![Page 910: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/910.jpg)
FunctionsNeven ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l0 2 4 6 8 10 12 14 · · ·
Neven∼= N
In general, any subset of a countable set is finite orcountable. Any quotient set of a countable set is finiteor countable.
Discrete Mathematics I – p. 208/292
![Page 911: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/911.jpg)
FunctionsNeven ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l0 2 4 6 8 10 12 14 · · ·
Neven∼= N
In general, any subset of a countable set is finite orcountable. Any quotient set of a countable set is finiteor countable.
Discrete Mathematics I – p. 208/292
![Page 912: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/912.jpg)
FunctionsNeven ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l0 2 4 6 8 10 12 14 · · ·
Neven∼= N
In general, any subset of a countable set is finite orcountable.
Any quotient set of a countable set is finiteor countable.
Discrete Mathematics I – p. 208/292
![Page 913: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/913.jpg)
FunctionsNeven ⊆ N
0 1 2 3 4 5 6 7 · · ·l l l l l l l l0 2 4 6 8 10 12 14 · · ·
Neven∼= N
In general, any subset of a countable set is finite orcountable. Any quotient set of a countable set is finiteor countable.
Discrete Mathematics I – p. 208/292
![Page 914: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/914.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2 0 1 3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 915: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/915.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2 0 1 3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 916: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/916.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2 0 1 3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 917: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/917.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2
0
1 3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 918: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/918.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2
0 1
3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 919: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/919.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4
2 0 1
3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 920: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/920.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4
2 0 1 3
5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 921: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/921.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6
4 2 0 1 3
5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 922: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/922.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2 0 1 3 5 7 · · ·
f bijective
Discrete Mathematics I – p. 209/292
![Page 923: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/923.jpg)
FunctionsProve: set Z is countable.
Proof. Let f : N → Z
∀n : n 7→{
(n + 1)/2 if n odd−n/2 if n even
· · · −4 −3 −2 −1 0 1 2 3 4 · · ·l l l l l l l l l
· · · 8 6 4 2 0 1 3 5 7 · · ·f bijective
Discrete Mathematics I – p. 209/292
![Page 924: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/924.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0 1 3 6 10
1 2 4 7 11 ·2 5 8 12 · ·3 9 13 · · ·4 14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 925: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/925.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0
0 1 3 6 10
1
2 4 7 11 ·
2
5 8 12 · ·
3
9 13 · · ·
4
14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 926: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/926.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0
1 3 6 10
1
2 4 7 11 ·
2
5 8 12 · ·
3
9 13 · · ·
4
14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 927: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/927.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0 1
3 6 10
1 2
4 7 11 ·
2
5 8 12 · ·
3
9 13 · · ·
4
14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 928: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/928.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0 1 3
6 10
1 2 4
7 11 ·
2 5
8 12 · ·
3
9 13 · · ·
4
14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 929: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/929.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0 1 3 6
10
1 2 4 7
11 ·
2 5 8
12 · ·
3 9
13 · · ·
4
14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 930: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/930.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0 1 3 6 10
1 2 4 7 11
·
2 5 8 12
· ·
3 9 13
· · ·
4 14
· · · ·
Discrete Mathematics I – p. 210/292
![Page 931: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/931.jpg)
FunctionsProve: Set N2 = N× N is countable.
Proof idea:
0 1 2 3 4
0 0 1 3 6 10
1 2 4 7 11 ·2 5 8 12 · ·3 9 13 · · ·4 14 · · · ·
Discrete Mathematics I – p. 210/292
![Page 932: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/932.jpg)
FunctionsProve: Set N3 = N× N× N is countable.
Proof. N3 = (N× N)× N ∼= N× N ∼= N
In general, the Cartesian product of a finite number ofcountable sets is countable
Not true for an infinite Cartesian product!
Discrete Mathematics I – p. 211/292
![Page 933: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/933.jpg)
FunctionsProve: Set N3 = N× N× N is countable.
Proof. N3 = (N× N)× N ∼= N× N ∼= N
In general, the Cartesian product of a finite number ofcountable sets is countable
Not true for an infinite Cartesian product!
Discrete Mathematics I – p. 211/292
![Page 934: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/934.jpg)
FunctionsProve: Set N3 = N× N× N is countable.
Proof. N3 = (N× N)× N ∼= N× N ∼= N
In general, the Cartesian product of a finite number ofcountable sets is countable
Not true for an infinite Cartesian product!
Discrete Mathematics I – p. 211/292
![Page 935: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/935.jpg)
FunctionsProve: Set N3 = N× N× N is countable.
Proof. N3 = (N× N)× N ∼= N× N ∼= N
In general, the Cartesian product of a finite number ofcountable sets is countable
Not true for an infinite Cartesian product!
Discrete Mathematics I – p. 211/292
![Page 936: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/936.jpg)
FunctionsA digression on rational numbers
Fractions 1/2, −3/4, 355/113
Similar to Z2, but:
1/2 = 3/6 −5 = −10/2 = 30/(−6) . . .
Discrete Mathematics I – p. 212/292
![Page 937: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/937.jpg)
FunctionsA digression on rational numbers
Fractions 1/2, −3/4, 355/113
Similar to Z2, but:
1/2 = 3/6 −5 = −10/2 = 30/(−6) . . .
Discrete Mathematics I – p. 212/292
![Page 938: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/938.jpg)
FunctionsA digression on rational numbers
Fractions 1/2, −3/4, 355/113
Similar to Z2, but:
1/2 = 3/6 −5 = −10/2 = 30/(−6) . . .
Discrete Mathematics I – p. 212/292
![Page 939: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/939.jpg)
Functionsa/b = c/d ⇐⇒ a · d = b · c b, d 6= 0
R∼ : Z2 ↔ Z2 (a, b) ∼ (c, d) ⇐⇒ a · d = b · cThe rational numbers: Q = Z2/R∼
Discrete Mathematics I – p. 213/292
![Page 940: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/940.jpg)
Functionsa/b = c/d ⇐⇒ a · d = b · c b, d 6= 0
R∼ : Z2 ↔ Z2 (a, b) ∼ (c, d) ⇐⇒ a · d = b · c
The rational numbers: Q = Z2/R∼
Discrete Mathematics I – p. 213/292
![Page 941: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/941.jpg)
Functionsa/b = c/d ⇐⇒ a · d = b · c b, d 6= 0
R∼ : Z2 ↔ Z2 (a, b) ∼ (c, d) ⇐⇒ a · d = b · cThe rational numbers: Q = Z2/R∼
Discrete Mathematics I – p. 213/292
![Page 942: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/942.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
0 11/21/3
5/12
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 943: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/943.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
1/21/3
5/12
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 944: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/944.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
1/3
5/12
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 945: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/945.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
�
1/3
5/12
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 946: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/946.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
�
1/3
�
5/12
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 947: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/947.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
�
1/3
�
5/12
� � � � � � � � � � � �
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 948: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/948.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
�
1/3
�
5/12
� � � � � � � � � � � �
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 949: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/949.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
�
1/3
�
5/12
� � � � � � � � � � � �
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 950: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/950.jpg)
FunctionsSet Q is infinite, and also dense: for any two distinctrationals, there is a rational in between
�
0
�
1
�
1/2
�
1/3
�
5/12
� � � � � � � � � � � �
∀a, b ∈ Q :(
(a < b) ⇒ ∃x ∈ Q : a < x < b)
Still, set Q is countable
Z2 ∼= N =⇒ Q = Z2/R∼ ∼= N
Discrete Mathematics I – p. 214/292
![Page 951: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/951.jpg)
FunctionsSets N, Z, Q countable: N ∼= Z ∼= Q
N ∼= N2 ∼= N3 ∼= N× Z×Q ∼= · · ·Are there any uncountable sets?
Discrete Mathematics I – p. 215/292
![Page 952: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/952.jpg)
FunctionsSets N, Z, Q countable: N ∼= Z ∼= Q
N ∼= N2 ∼= N3 ∼= N× Z×Q ∼= · · ·
Are there any uncountable sets?
Discrete Mathematics I – p. 215/292
![Page 953: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/953.jpg)
FunctionsSets N, Z, Q countable: N ∼= Z ∼= Q
N ∼= N2 ∼= N3 ∼= N× Z×Q ∼= · · ·Are there any uncountable sets?
Discrete Mathematics I – p. 215/292
![Page 954: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/954.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 955: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/955.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 956: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/956.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}
D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 957: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/957.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 958: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/958.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 959: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/959.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = D
Case d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 960: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/960.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 961: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/961.jpg)
FunctionsCantor’s Theorem. For all sets A, A 6∼= P(A).
Proof. Suppose ∃f : A��P(A).
Let D = {a ∈ A | a 6∈ f(a)}D ⊆ A =⇒ D ∈ P(A) =⇒ ∃d ∈ A : f(d) = D
d ∈ D — true or false?
Case d ∈ D =⇒ d 6∈ f(d) = DCase d 6∈ D = f(d) =⇒ d ∈ D
Contradiction! Bijection f cannot exist.
Discrete Mathematics I – p. 216/292
![Page 962: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/962.jpg)
FunctionsCorollary. Set P(N) is uncountable.
P(N) ∼= B(N) = {f : N → B} =⇒ B(N) 6∼= N
B = {F, T} ∼= {0, 1} = N2 ⊆ N
B(N) ∼= {f : N → N2} ⊆{f : N → N} = N× N× · · ·
B(N) uncountable =⇒ N× N× · · · uncountable
Discrete Mathematics I – p. 217/292
![Page 963: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/963.jpg)
FunctionsCorollary. Set P(N) is uncountable.
P(N) ∼= B(N) = {f : N → B} =⇒ B(N) 6∼= N
B = {F, T} ∼= {0, 1} = N2 ⊆ N
B(N) ∼= {f : N → N2} ⊆{f : N → N} = N× N× · · ·
B(N) uncountable =⇒ N× N× · · · uncountable
Discrete Mathematics I – p. 217/292
![Page 964: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/964.jpg)
FunctionsCorollary. Set P(N) is uncountable.
P(N) ∼= B(N) = {f : N → B} =⇒ B(N) 6∼= N
B = {F, T} ∼= {0, 1} = N2 ⊆ N
B(N) ∼= {f : N → N2} ⊆{f : N → N} = N× N× · · ·
B(N) uncountable =⇒ N× N× · · · uncountable
Discrete Mathematics I – p. 217/292
![Page 965: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/965.jpg)
FunctionsCorollary. Set P(N) is uncountable.
P(N) ∼= B(N) = {f : N → B} =⇒ B(N) 6∼= N
B = {F, T} ∼= {0, 1} = N2 ⊆ N
B(N) ∼= {f : N → N2} ⊆{f : N → N} = N× N× · · ·
B(N) uncountable =⇒ N× N× · · · uncountable
Discrete Mathematics I – p. 217/292
![Page 966: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/966.jpg)
FunctionsCorollary. Set P(N) is uncountable.
P(N) ∼= B(N) = {f : N → B} =⇒ B(N) 6∼= N
B = {F, T} ∼= {0, 1} = N2 ⊆ N
B(N) ∼= {f : N → N2} ⊆{f : N → N} = N× N× · · ·
B(N) uncountable =⇒ N× N× · · · uncountable
Discrete Mathematics I – p. 217/292
![Page 967: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/967.jpg)
FunctionsA digression on real numbers
Rationals (20, −3/5, . . . ), irrationals (√
2, π, . . . )
Definition: Dedekind cuts of Q
Q = Q0 ∪Q1 ∀x ∈ Q0, y ∈ Q1 : x < y
Q
Q0 Q1
Every Dedekind cut defines a real number
Discrete Mathematics I – p. 218/292
![Page 968: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/968.jpg)
FunctionsA digression on real numbers
Rationals (20, −3/5, . . . ), irrationals (√
2, π, . . . )
Definition: Dedekind cuts of Q
Q = Q0 ∪Q1 ∀x ∈ Q0, y ∈ Q1 : x < y
Q
Q0 Q1
Every Dedekind cut defines a real number
Discrete Mathematics I – p. 218/292
![Page 969: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/969.jpg)
FunctionsA digression on real numbers
Rationals (20, −3/5, . . . ), irrationals (√
2, π, . . . )
Definition: Dedekind cuts of Q
Q = Q0 ∪Q1 ∀x ∈ Q0, y ∈ Q1 : x < y
Q
Q0 Q1
Every Dedekind cut defines a real number
Discrete Mathematics I – p. 218/292
![Page 970: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/970.jpg)
FunctionsA digression on real numbers
Rationals (20, −3/5, . . . ), irrationals (√
2, π, . . . )
Definition: Dedekind cuts of Q
Q = Q0 ∪Q1 ∀x ∈ Q0, y ∈ Q1 : x < y
Q
Q0 Q1
Every Dedekind cut defines a real number
Discrete Mathematics I – p. 218/292
![Page 971: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/971.jpg)
FunctionsA digression on real numbers
Rationals (20, −3/5, . . . ), irrationals (√
2, π, . . . )
Definition: Dedekind cuts of Q
Q = Q0 ∪Q1 ∀x ∈ Q0, y ∈ Q1 : x < y
� � � � � � � � � � � � � � � � � � � � � � � �
Q
Q0 Q1
Every Dedekind cut defines a real number
Discrete Mathematics I – p. 218/292
![Page 972: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/972.jpg)
FunctionsA digression on real numbers
Rationals (20, −3/5, . . . ), irrationals (√
2, π, . . . )
Definition: Dedekind cuts of Q
Q = Q0 ∪Q1 ∀x ∈ Q0, y ∈ Q1 : x < y
� � � � � � � � � � � � � � � � � � � � � � � �
Q
Q0 Q1
Every Dedekind cut defines a real number
Discrete Mathematics I – p. 218/292
![Page 973: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/973.jpg)
FunctionsThe real numbers R = {all Dedekind cuts of Q}
Irrationals are “gaps between rationals”
Example: π = 3.1415926536 . . .
Number π defined by
Q0 = {3, 3.1, 3.14, 3.141, 3.1415, 3.141592, . . .}Q1 = {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, . . .}We approximate real by rationals using a positionalnumber system (decimal, binary, etc.)
Discrete Mathematics I – p. 219/292
![Page 974: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/974.jpg)
FunctionsThe real numbers R = {all Dedekind cuts of Q}Irrationals are “gaps between rationals”
Example: π = 3.1415926536 . . .
Number π defined by
Q0 = {3, 3.1, 3.14, 3.141, 3.1415, 3.141592, . . .}Q1 = {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, . . .}We approximate real by rationals using a positionalnumber system (decimal, binary, etc.)
Discrete Mathematics I – p. 219/292
![Page 975: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/975.jpg)
FunctionsThe real numbers R = {all Dedekind cuts of Q}Irrationals are “gaps between rationals”
Example: π = 3.1415926536 . . .
Number π defined by
Q0 = {3, 3.1, 3.14, 3.141, 3.1415, 3.141592, . . .}Q1 = {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, . . .}We approximate real by rationals using a positionalnumber system (decimal, binary, etc.)
Discrete Mathematics I – p. 219/292
![Page 976: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/976.jpg)
FunctionsThe real numbers R = {all Dedekind cuts of Q}Irrationals are “gaps between rationals”
Example: π = 3.1415926536 . . .
Number π defined by
Q0 = {3, 3.1, 3.14, 3.141, 3.1415, 3.141592, . . .}Q1 = {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, . . .}We approximate real by rationals using a positionalnumber system (decimal, binary, etc.)
Discrete Mathematics I – p. 219/292
![Page 977: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/977.jpg)
FunctionsThe real numbers R = {all Dedekind cuts of Q}Irrationals are “gaps between rationals”
Example: π = 3.1415926536 . . .
Number π defined by
Q0 = {3, 3.1, 3.14, 3.141, 3.1415, 3.141592, . . .}Q1 = {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, . . .}
We approximate real by rationals using a positionalnumber system (decimal, binary, etc.)
Discrete Mathematics I – p. 219/292
![Page 978: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/978.jpg)
FunctionsThe real numbers R = {all Dedekind cuts of Q}Irrationals are “gaps between rationals”
Example: π = 3.1415926536 . . .
Number π defined by
Q0 = {3, 3.1, 3.14, 3.141, 3.1415, 3.141592, . . .}Q1 = {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, . . .}We approximate real by rationals using a positionalnumber system (decimal, binary, etc.)
Discrete Mathematics I – p. 219/292
![Page 979: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/979.jpg)
FunctionsIs set R countable?
Consider [0, 1] = {a ∈ R : 0 ≤ a < 1}Decimal representation of a: sequence N → N10
For example, π − 3 = .141592
f(0) = 1 f(1) = 4 f(2) = 1 f(3) = 5 . . .
(For simplicity, ignore .1415000 . . . = .1414999 . . .)
Discrete Mathematics I – p. 220/292
![Page 980: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/980.jpg)
FunctionsIs set R countable?
Consider [0, 1] = {a ∈ R : 0 ≤ a < 1}
Decimal representation of a: sequence N → N10
For example, π − 3 = .141592
f(0) = 1 f(1) = 4 f(2) = 1 f(3) = 5 . . .
(For simplicity, ignore .1415000 . . . = .1414999 . . .)
Discrete Mathematics I – p. 220/292
![Page 981: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/981.jpg)
FunctionsIs set R countable?
Consider [0, 1] = {a ∈ R : 0 ≤ a < 1}Decimal representation of a: sequence N → N10
For example, π − 3 = .141592
f(0) = 1 f(1) = 4 f(2) = 1 f(3) = 5 . . .
(For simplicity, ignore .1415000 . . . = .1414999 . . .)
Discrete Mathematics I – p. 220/292
![Page 982: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/982.jpg)
FunctionsIs set R countable?
Consider [0, 1] = {a ∈ R : 0 ≤ a < 1}Decimal representation of a: sequence N → N10
For example, π − 3 = .141592
f(0) = 1 f(1) = 4 f(2) = 1 f(3) = 5 . . .
(For simplicity, ignore .1415000 . . . = .1414999 . . .)
Discrete Mathematics I – p. 220/292
![Page 983: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/983.jpg)
FunctionsIs set R countable?
Consider [0, 1] = {a ∈ R : 0 ≤ a < 1}Decimal representation of a: sequence N → N10
For example, π − 3 = .141592
f(0) = 1 f(1) = 4 f(2) = 1 f(3) = 5 . . .
(For simplicity, ignore .1415000 . . . = .1414999 . . .)
Discrete Mathematics I – p. 220/292
![Page 984: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/984.jpg)
FunctionsIs set R countable?
Consider [0, 1] = {a ∈ R : 0 ≤ a < 1}Decimal representation of a: sequence N → N10
For example, π − 3 = .141592
f(0) = 1 f(1) = 4 f(2) = 1 f(3) = 5 . . .
(For simplicity, ignore .1415000 . . . = .1414999 . . .)
Discrete Mathematics I – p. 220/292
![Page 985: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/985.jpg)
FunctionsB = {F, T} ∼= {0, 1} = N2 ⊆ N10 =⇒
B(N) = {f : N → B} ∼={f : N → N2} ⊆ {f : N → N10} ∼= [0, 1]
B(N) uncountable, hence [0, 1] uncountable
Therefore, R uncountable
Discrete Mathematics I – p. 221/292
![Page 986: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/986.jpg)
FunctionsB = {F, T} ∼= {0, 1} = N2 ⊆ N10 =⇒B(N) = {f : N → B} ∼=
{f : N → N2} ⊆ {f : N → N10} ∼= [0, 1]
B(N) uncountable, hence [0, 1] uncountable
Therefore, R uncountable
Discrete Mathematics I – p. 221/292
![Page 987: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/987.jpg)
FunctionsB = {F, T} ∼= {0, 1} = N2 ⊆ N10 =⇒B(N) = {f : N → B} ∼=
{f : N → N2} ⊆ {f : N → N10} ∼= [0, 1]
B(N) uncountable, hence [0, 1] uncountable
Therefore, R uncountable
Discrete Mathematics I – p. 221/292
![Page 988: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/988.jpg)
FunctionsB = {F, T} ∼= {0, 1} = N2 ⊆ N10 =⇒B(N) = {f : N → B} ∼=
{f : N → N2} ⊆ {f : N → N10} ∼= [0, 1]
B(N) uncountable, hence [0, 1] uncountable
Therefore, R uncountable
Discrete Mathematics I – p. 221/292
![Page 989: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/989.jpg)
FunctionsFinite sets:
n elements, n− 1 gaps
Infinite sets:
R are gaps in Q, but R is much bigger than Q
Weird!
Discrete Mathematics I – p. 222/292
![Page 990: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/990.jpg)
FunctionsFinite sets:
n elements, n− 1 gaps
Infinite sets:
R are gaps in Q, but R is much bigger than Q
Weird!
Discrete Mathematics I – p. 222/292
![Page 991: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/991.jpg)
FunctionsFinite sets:
n elements, n− 1 gaps
Infinite sets:
R are gaps in Q, but R is much bigger than Q
Weird!
Discrete Mathematics I – p. 222/292
![Page 992: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/992.jpg)
FunctionsFinite sets:
n elements, n− 1 gaps
Infinite sets:
R are gaps in Q, but R is much bigger than Q
Weird!
Discrete Mathematics I – p. 222/292
![Page 993: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/993.jpg)
FunctionsFinite sets:
n elements, n− 1 gaps
Infinite sets:
R are gaps in Q, but R is much bigger than Q
Weird!
Discrete Mathematics I – p. 222/292
![Page 994: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/994.jpg)
FunctionsFinite sets:
n elements, n− 1 gaps
Infinite sets:
R are gaps in Q, but R is much bigger than Q
Weird!
Discrete Mathematics I – p. 222/292
![Page 995: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/995.jpg)
FunctionsN, P(N), P(P(N)), . . . — uncountable
All of different cardinalities — and there any manymore. . .
So much more they don’t even form a set!
Discrete Mathematics I – p. 223/292
![Page 996: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/996.jpg)
FunctionsN, P(N), P(P(N)), . . . — uncountable
All of different cardinalities — and there any manymore. . .
So much more they don’t even form a set!
Discrete Mathematics I – p. 223/292
![Page 997: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/997.jpg)
FunctionsN, P(N), P(P(N)), . . . — uncountable
All of different cardinalities — and there any manymore. . .
So much more they don’t even form a set!
Discrete Mathematics I – p. 223/292
![Page 998: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/998.jpg)
Induction
Discrete Mathematics I – p. 224/292
![Page 999: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/999.jpg)
InductionNatural numbers: N = {0, 1, 2, 3, 4, 5, 6, 7, . . . }
God created the natural numbers, all the restis the work of man.
L. Kronecker (1823–1891)
Discrete Mathematics I – p. 225/292
![Page 1000: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1000.jpg)
InductionNatural numbers: N = {0, 1, 2, 3, 4, 5, 6, 7, . . . }
God created the natural numbers, all the restis the work of man.
L. Kronecker (1823–1891)
Discrete Mathematics I – p. 225/292
![Page 1001: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1001.jpg)
InductionThe only possible definition of N is self-referential:
• 0 ∈ N
• for all x ∈ N next(x) ∈ N
• everything else 6∈ N
This is an inductive definition
Discrete Mathematics I – p. 226/292
![Page 1002: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1002.jpg)
InductionThe only possible definition of N is self-referential:
• 0 ∈ N
• for all x ∈ N next(x) ∈ N
• everything else 6∈ N
This is an inductive definition
Discrete Mathematics I – p. 226/292
![Page 1003: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1003.jpg)
InductionThe only possible definition of N is self-referential:
• 0 ∈ N
• for all x ∈ N next(x) ∈ N
• everything else 6∈ N
This is an inductive definition
Discrete Mathematics I – p. 226/292
![Page 1004: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1004.jpg)
InductionThe only possible definition of N is self-referential:
• 0 ∈ N
• for all x ∈ N next(x) ∈ N
• everything else 6∈ N
This is an inductive definition
Discrete Mathematics I – p. 226/292
![Page 1005: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1005.jpg)
InductionThe only possible definition of N is self-referential:
• 0 ∈ N
• for all x ∈ N next(x) ∈ N
• everything else 6∈ N
This is an inductive definition
Discrete Mathematics I – p. 226/292
![Page 1006: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1006.jpg)
InductionStructure of inductive definition:
• induction base
• inductive step(s)
• completeness (sometimes implicit)
Discrete Mathematics I – p. 227/292
![Page 1007: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1007.jpg)
InductionStructure of inductive definition:
• induction base
• inductive step(s)
• completeness (sometimes implicit)
Discrete Mathematics I – p. 227/292
![Page 1008: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1008.jpg)
InductionStructure of inductive definition:
• induction base
• inductive step(s)
• completeness (sometimes implicit)
Discrete Mathematics I – p. 227/292
![Page 1009: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1009.jpg)
InductionStructure of inductive definition:
• induction base
• inductive step(s)
• completeness (sometimes implicit)
Discrete Mathematics I – p. 227/292
![Page 1010: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1010.jpg)
InductionFor example, a queue:
• the empty set ∅ is a queue;
• a queue with a new person behind is still a queue
• every queue is formed in this way
If we know what “behind” means, all works!
Discrete Mathematics I – p. 228/292
![Page 1011: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1011.jpg)
InductionFor example, a queue:
• the empty set ∅ is a queue;
• a queue with a new person behind is still a queue
• every queue is formed in this way
If we know what “behind” means, all works!
Discrete Mathematics I – p. 228/292
![Page 1012: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1012.jpg)
InductionFor example, a queue:
• the empty set ∅ is a queue;
• a queue with a new person behind is still a queue
• every queue is formed in this way
If we know what “behind” means, all works!
Discrete Mathematics I – p. 228/292
![Page 1013: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1013.jpg)
InductionFor example, a queue:
• the empty set ∅ is a queue;
• a queue with a new person behind is still a queue
• every queue is formed in this way
If we know what “behind” means, all works!
Discrete Mathematics I – p. 228/292
![Page 1014: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1014.jpg)
InductionFor example, a queue:
• the empty set ∅ is a queue;
• a queue with a new person behind is still a queue
• every queue is formed in this way
If we know what “behind” means, all works!
Discrete Mathematics I – p. 228/292
![Page 1015: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1015.jpg)
InductionAnother example: Boolean statements
• F , T are statements
• if A, B are statements, then ¬A, A ∧B, A ∨B,A ⇒ B, A ⇔ B are statements
• there are no other statements
Discrete Mathematics I – p. 229/292
![Page 1016: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1016.jpg)
InductionAnother example: Boolean statements
• F , T are statements
• if A, B are statements, then ¬A, A ∧B, A ∨B,A ⇒ B, A ⇔ B are statements
• there are no other statements
Discrete Mathematics I – p. 229/292
![Page 1017: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1017.jpg)
InductionAnother example: Boolean statements
• F , T are statements
• if A, B are statements, then ¬A, A ∧B, A ∨B,A ⇒ B, A ⇔ B are statements
• there are no other statements
Discrete Mathematics I – p. 229/292
![Page 1018: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1018.jpg)
InductionAnother example: Boolean statements
• F , T are statements
• if A, B are statements, then ¬A, A ∧B, A ∨B,A ⇒ B, A ⇔ B are statements
• there are no other statements
Discrete Mathematics I – p. 229/292
![Page 1019: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1019.jpg)
InductionIn general:
• induction base: initial objects
• inductive step(s): ways to make new objects
• completeness: no other objects allowed!
Discrete Mathematics I – p. 230/292
![Page 1020: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1020.jpg)
InductionIn general:
• induction base: initial objects
• inductive step(s): ways to make new objects
• completeness: no other objects allowed!
Discrete Mathematics I – p. 230/292
![Page 1021: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1021.jpg)
InductionIn general:
• induction base: initial objects
• inductive step(s): ways to make new objects
• completeness: no other objects allowed!
Discrete Mathematics I – p. 230/292
![Page 1022: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1022.jpg)
InductionIn general:
• induction base: initial objects
• inductive step(s): ways to make new objects
• completeness: no other objects allowed!
Discrete Mathematics I – p. 230/292
![Page 1023: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1023.jpg)
InductionInductive proofs: “the domino principle”
Need to prove ∀x ∈ N : P (x) for some P
• induction base: P (0)
• inductive step: ∀x ∈ N : P (x) ⇒ P (next(x))(here P (x) is the induction hypothesis)
• by completeness, P (x) true for all x ∈ N
Discrete Mathematics I – p. 231/292
![Page 1024: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1024.jpg)
InductionInductive proofs: “the domino principle”
Need to prove ∀x ∈ N : P (x) for some P
• induction base: P (0)
• inductive step: ∀x ∈ N : P (x) ⇒ P (next(x))(here P (x) is the induction hypothesis)
• by completeness, P (x) true for all x ∈ N
Discrete Mathematics I – p. 231/292
![Page 1025: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1025.jpg)
InductionInductive proofs: “the domino principle”
Need to prove ∀x ∈ N : P (x) for some P
• induction base: P (0)
• inductive step: ∀x ∈ N : P (x) ⇒ P (next(x))(here P (x) is the induction hypothesis)
• by completeness, P (x) true for all x ∈ N
Discrete Mathematics I – p. 231/292
![Page 1026: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1026.jpg)
InductionInductive proofs: “the domino principle”
Need to prove ∀x ∈ N : P (x) for some P
• induction base: P (0)
• inductive step: ∀x ∈ N : P (x) ⇒ P (next(x))(here P (x) is the induction hypothesis)
• by completeness, P (x) true for all x ∈ N
Discrete Mathematics I – p. 231/292
![Page 1027: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1027.jpg)
InductionInductive proofs: “the domino principle”
Need to prove ∀x ∈ N : P (x) for some P
• induction base: P (0)
• inductive step: ∀x ∈ N : P (x) ⇒ P (next(x))(here P (x) is the induction hypothesis)
• by completeness, P (x) true for all x ∈ N
Discrete Mathematics I – p. 231/292
![Page 1028: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1028.jpg)
InductionExample: plane colouring
Consider n lines in the plane.
Can always colour regions like a chessboard.
Discrete Mathematics I – p. 232/292
![Page 1029: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1029.jpg)
InductionExample: plane colouring
Consider n lines in the plane.
Can always colour regions like a chessboard.
Discrete Mathematics I – p. 232/292
![Page 1030: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1030.jpg)
InductionExample: plane colouring
Consider n lines in the plane.
Can always colour regions like a chessboard.
Discrete Mathematics I – p. 232/292
![Page 1031: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1031.jpg)
InductionExample: plane colouring
Consider n lines in the plane.
Can always colour regions like a chessboard.
Discrete Mathematics I – p. 232/292
![Page 1032: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1032.jpg)
InductionExample: plane colouring
Consider n lines in the plane.
Can always colour regions like a chessboard.
Discrete Mathematics I – p. 232/292
![Page 1033: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1033.jpg)
InductionProof.
Induction base: n = 0.
Paint the plane white.
Discrete Mathematics I – p. 233/292
![Page 1034: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1034.jpg)
InductionProof.
Induction base: n = 0. Paint the plane white.
Discrete Mathematics I – p. 233/292
![Page 1035: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1035.jpg)
InductionProof.
Induction base: n = 0. Paint the plane white.
Discrete Mathematics I – p. 233/292
![Page 1036: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1036.jpg)
InductionInductive step. Suppose can colour for n lines.
Need to color for n + 1 lines.
Add another line, invert all colours on one side.
By induction, can colour for all n.
Discrete Mathematics I – p. 234/292
![Page 1037: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1037.jpg)
InductionInductive step. Suppose can colour for n lines.
Need to color for n + 1 lines.
Add another line, invert all colours on one side.
By induction, can colour for all n.
Discrete Mathematics I – p. 234/292
![Page 1038: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1038.jpg)
InductionInductive step. Suppose can colour for n lines.
Need to color for n + 1 lines.
Add another line, invert all colours on one side.
By induction, can colour for all n.
Discrete Mathematics I – p. 234/292
![Page 1039: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1039.jpg)
InductionInductive step. Suppose can colour for n lines.
Need to color for n + 1 lines.
Add another line, invert all colours on one side.
By induction, can colour for all n.
Discrete Mathematics I – p. 234/292
![Page 1040: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1040.jpg)
InductionInductive step. Suppose can colour for n lines.
Need to color for n + 1 lines.
Add another line, invert all colours on one side.
By induction, can colour for all n.
Discrete Mathematics I – p. 234/292
![Page 1041: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1041.jpg)
InductionInductive step. Suppose can colour for n lines.
Need to color for n + 1 lines.
Add another line, invert all colours on one side.
By induction, can colour for all n.
Discrete Mathematics I – p. 234/292
![Page 1042: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1042.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1043: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1043.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof.
Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1044: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1044.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base:
8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1045: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1045.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1046: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1046.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1047: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1047.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1048: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1048.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s.
Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1049: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1049.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s.
Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1050: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1050.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1051: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1051.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1052: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1052.jpg)
InductionProve: Any postage ≥ 8p can be paid by 3p and 5pstamps.
Proof. Induction base: 8 = 3 + 5.
Inductive step. Suppose can pay n (n ≥ 8).
We need: can pay n + 1
Case 1: have used a 5. Replace 5 → 3 + 3.
Case 2: have only used 3s. Since n ≥ 8, there are atleast three 3s. Replace 3 + 3 + 3 → 5 + 5.
In both cases have paid n + 1.
By induction, can pay any n ≥ 8.
Discrete Mathematics I – p. 235/292
![Page 1053: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1053.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒ A = ∅ =⇒P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1054: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1054.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof.
Induction base:
|A| = 0 =⇒ A = ∅ =⇒P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1055: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1055.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒ A = ∅ =⇒P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1056: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1056.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒
A = ∅ =⇒P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1057: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1057.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒ A = ∅ =⇒
P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1058: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1058.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒ A = ∅ =⇒P(A) = {∅} =⇒
|P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1059: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1059.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒ A = ∅ =⇒P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1060: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1060.jpg)
InductionProve: For any finite set A,
|A| = n =⇒ |P(A)| = 2n
Proof. Induction base:
|A| = 0 =⇒ A = ∅ =⇒P(A) = {∅} =⇒ |P(A)| = 1 = 20
Inductive step. Suppose it holds for a given n:for all B |B| = n =⇒ |P(B)| = 2n
We need:for all A |A| = n + 1 =⇒ |P(A)| = 2n+1
Discrete Mathematics I – p. 236/292
![Page 1061: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1061.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1062: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1062.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1063: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1063.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B)
Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1064: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1064.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}
P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1065: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1065.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q
P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1066: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1066.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅
|P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1067: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1067.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1068: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1068.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1069: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1069.jpg)
InductionLet a ∈ A B = A \ {a}.
We have |B| = n, therefore |P(B)| = 2n.
Let P = P(B) Q = {X ∪ {a} | X ∈ P}P(A) = P ∪Q P ∩Q = ∅ |P | = |Q| = 2n
Hence |P(A)| = |P |+ |Q| = 2n + 2n = 2n · 2 = 2n+1
By induction, statement true for all A
Discrete Mathematics I – p. 237/292
![Page 1070: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1070.jpg)
InductionConsider induction on n ∈ N
Inductive step: P (n) ⇒ P (n + 1)
Suppose can only prove:(
P (0) ∧ P (1) ∧ · · · ∧ P (n− 1))
⇒ P (n)
(Also covers T ⇒ P (0))
So-called “strong” induction(in fact, the implication has become weaker!)
Does P (n) still hold for all n?
Discrete Mathematics I – p. 238/292
![Page 1071: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1071.jpg)
InductionConsider induction on n ∈ N
Inductive step: P (n) ⇒ P (n + 1)
Suppose can only prove:(
P (0) ∧ P (1) ∧ · · · ∧ P (n− 1))
⇒ P (n)
(Also covers T ⇒ P (0))
So-called “strong” induction(in fact, the implication has become weaker!)
Does P (n) still hold for all n?
Discrete Mathematics I – p. 238/292
![Page 1072: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1072.jpg)
InductionConsider induction on n ∈ N
Inductive step: P (n) ⇒ P (n + 1)
Suppose can only prove:(
P (0) ∧ P (1) ∧ · · · ∧ P (n− 1))
⇒ P (n)
(Also covers T ⇒ P (0))
So-called “strong” induction(in fact, the implication has become weaker!)
Does P (n) still hold for all n?
Discrete Mathematics I – p. 238/292
![Page 1073: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1073.jpg)
InductionConsider induction on n ∈ N
Inductive step: P (n) ⇒ P (n + 1)
Suppose can only prove:(
P (0) ∧ P (1) ∧ · · · ∧ P (n− 1))
⇒ P (n)
(Also covers T ⇒ P (0))
So-called “strong” induction(in fact, the implication has become weaker!)
Does P (n) still hold for all n?
Discrete Mathematics I – p. 238/292
![Page 1074: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1074.jpg)
InductionConsider induction on n ∈ N
Inductive step: P (n) ⇒ P (n + 1)
Suppose can only prove:(
P (0) ∧ P (1) ∧ · · · ∧ P (n− 1))
⇒ P (n)
(Also covers T ⇒ P (0))
So-called “strong” induction(in fact, the implication has become weaker!)
Does P (n) still hold for all n?
Discrete Mathematics I – p. 238/292
![Page 1075: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1075.jpg)
InductionConsider induction on n ∈ N
Inductive step: P (n) ⇒ P (n + 1)
Suppose can only prove:(
P (0) ∧ P (1) ∧ · · · ∧ P (n− 1))
⇒ P (n)
(Also covers T ⇒ P (0))
So-called “strong” induction(in fact, the implication has become weaker!)
Does P (n) still hold for all n?
Discrete Mathematics I – p. 238/292
![Page 1076: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1076.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1077: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1077.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1078: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1078.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0)
induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1079: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1079.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1080: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1080.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)
(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1081: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1081.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1082: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1082.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1)
inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1083: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1083.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1084: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1084.jpg)
InductionLet Q(n) ⇐⇒ P (0) ∧ P (1) ∧ · · · ∧ P (n)
We need: ∀n ∈ N : Q(n) ( =⇒ ∀n ∈ N : P (n))
T ⇒ P (0) ⇐⇒ P (0) ⇐⇒ Q(0) induction base
Q(n) ⇐⇒ P (0)∧P (1)∧ · · · ∧P (n) =⇒ P (n+1)(
P (0) ∧ · · · ∧ P (n))
∧ P (n + 1) ⇐⇒ Q(n + 1)
Hence Q(n) ⇒ Q(n + 1) inductive step
By induction, ∀n : Q(n), therefore ∀n : P (n)
Discrete Mathematics I – p. 239/292
![Page 1085: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1085.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1086: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1086.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof.
Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1087: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1087.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base:
2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1088: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1088.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1089: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1089.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1090: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1090.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | n
Case 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1091: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1091.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | n
Then ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1092: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1092.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1093: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1093.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | n
In both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1094: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1094.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1095: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1095.jpg)
InductionProve: ∀n ∈ N : n ≥ 2 ⇒ n is divisible by a prime
Proof. Induction base: 2 | 2 prime
Inductive step. Suppose true ∀m < n. True for n?
Case 1: n prime n | nCase 2: ∃m < n : m | nThen ∃p : (p prime) ∧ (p | m)
(p | m) ∧ (m | n) =⇒ p | nIn both cases n divisible by a prime
By induction, all n ≥ 2 divisible by a prime
Discrete Mathematics I – p. 240/292
![Page 1096: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1096.jpg)
Graphs
Discrete Mathematics I – p. 241/292
![Page 1097: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1097.jpg)
Graphs
The Königsberg Bridges (L. Euler, 1707–1783)
A tour crossing every bridge exactly once?
Discrete Mathematics I – p. 242/292
![Page 1098: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1098.jpg)
Graphs
The Königsberg Bridges (L. Euler, 1707–1783)
A tour crossing every bridge exactly once?
Discrete Mathematics I – p. 242/292
![Page 1099: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1099.jpg)
Graphs
The Königsberg Bridges (L. Euler, 1707–1783)
A tour crossing every bridge exactly once?
Discrete Mathematics I – p. 242/292
![Page 1100: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1100.jpg)
Graphs
The Königsberg Bridges graph
��
�
�
�� ��
� �
� ��
��
4 • nodes (islands) 7 ◦ nodes (bridges)
Discrete Mathematics I – p. 243/292
![Page 1101: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1101.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
Discrete Mathematics I – p. 244/292
![Page 1102: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1102.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
Discrete Mathematics I – p. 244/292
![Page 1103: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1103.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
Discrete Mathematics I – p. 244/292
![Page 1104: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1104.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
Discrete Mathematics I – p. 244/292
![Page 1105: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1105.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
Discrete Mathematics I – p. 244/292
![Page 1106: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1106.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
Discrete Mathematics I – p. 244/292
![Page 1107: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1107.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG
Discrete Mathematics I – p. 244/292
![Page 1108: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1108.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
Discrete Mathematics I – p. 244/292
![Page 1109: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1109.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
�G
FWC
Discrete Mathematics I – p. 244/292
![Page 1110: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1110.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
�G
FWC
�
FG
WC
Discrete Mathematics I – p. 244/292
![Page 1111: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1111.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
�G
FWC
�
FG
WC
�
FWGC
Discrete Mathematics I – p. 244/292
![Page 1112: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1112.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
�G
FWC
�
FG
WC
�
FWGC
�
W
FGC
Discrete Mathematics I – p. 244/292
![Page 1113: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1113.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
�G
FWC
�
FG
WC
�
FWGC
�
W
FGC
FGW
C
Discrete Mathematics I – p. 244/292
![Page 1114: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1114.jpg)
Graphs
Wolf, goat and cabbage
Farmer wants to take W , G, C across the riverCan only take one item at a time
W eats G, G eats C — farmer must keep an eye
�
FWGC
�
WC
FG
� FWC
G
�
C
FWG �
FGC
W
�G
FWC
�
FG
WC
�
FWGC
�
W
FGC
FGW
C
Discrete Mathematics I – p. 244/292
![Page 1115: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1115.jpg)
Graphs
Houses and wells
Each of 3 houses needs a path to each of 3 wells
Paths must not cross
�
H1
�
H2�
H3
�
W1
�
W2
�
W3
Discrete Mathematics I – p. 245/292
![Page 1116: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1116.jpg)
Graphs
Houses and wells
Each of 3 houses needs a path to each of 3 wells
Paths must not cross
�
H1
�
H2�
H3
�
W1
�
W2
�
W3
Discrete Mathematics I – p. 245/292
![Page 1117: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1117.jpg)
Graphs
Houses and wells
Each of 3 houses needs a path to each of 3 wells
Paths must not cross
�
H1
�
H2�
H3
�
W1
�
W2
�
W3
Discrete Mathematics I – p. 245/292
![Page 1118: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1118.jpg)
Graphs
Houses and wells
Each of 3 houses needs a path to each of 3 wells
Paths must not cross
�
H1
�
H2�
H3
�
W1
�
W2
�
W3
Discrete Mathematics I – p. 245/292
![Page 1119: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1119.jpg)
Graphs
Houses and wells
Each of 3 houses needs a path to each of 3 wells
Paths must not cross
�
H1
�
H2�
H3
�
W1
�
W2
�
W3
Discrete Mathematics I – p. 245/292
![Page 1120: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1120.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1121: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1121.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1122: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1122.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1123: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1123.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1124: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1124.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1125: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1125.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1126: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1126.jpg)
Graphs
V — finite set, elements called nodes R⇀ : V ↔ V
R⇀ called irreflexive, if ∀a ∈ A : ¬(a ⇀ a)
R⇀ called symmetric, if ∀a, b ∈ A : a ⇀ b ⇒ b ⇀ a
A graph is an irreflexive, symmetric relationE = R⇀ : V ↔ V
Nodes u, v called adjacent, if u ⇀ v
Pairs in E called edges
Common notation: graph G = (V, E)
Discrete Mathematics I – p. 246/292
![Page 1127: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1127.jpg)
Graphs
The complete graph on V : K(V ) = (V, E)where E = {(u, v) ∈ V 2 | u 6= v}
0
1
2
3 4
K(N5)
The complete graph on n nodes: K(n)
Discrete Mathematics I – p. 247/292
![Page 1128: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1128.jpg)
Graphs
The complete graph on V : K(V ) = (V, E)where E = {(u, v) ∈ V 2 | u 6= v}
� 0
�
1
�2
�
3
�
4
K(N5)
The complete graph on n nodes: K(n)
Discrete Mathematics I – p. 247/292
![Page 1129: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1129.jpg)
Graphs
The complete graph on V : K(V ) = (V, E)where E = {(u, v) ∈ V 2 | u 6= v}
� 0
�
1
�2
�
3
�
4
K(N5)
The complete graph on n nodes: K(n)
Discrete Mathematics I – p. 247/292
![Page 1130: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1130.jpg)
Graphs
Different graphs may be “similar”
Two graphs are called isomorphic, if there is abijection between their node sets, which preserves theedges
G1 = (V1, E1) G2 = (V2, E2)
G1∼= G2 ⇐⇒ ∃f : V1��V2 :
∀u, v ∈ V1 : (u, v) ∈ E1 ⇔ (f(u), f(v)) ∈ E2
Bijection f is the isomorphism between G1, G2
Discrete Mathematics I – p. 248/292
![Page 1131: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1131.jpg)
Graphs
Different graphs may be “similar”
Two graphs are called isomorphic, if there is abijection between their node sets, which preserves theedges
G1 = (V1, E1) G2 = (V2, E2)
G1∼= G2 ⇐⇒ ∃f : V1��V2 :
∀u, v ∈ V1 : (u, v) ∈ E1 ⇔ (f(u), f(v)) ∈ E2
Bijection f is the isomorphism between G1, G2
Discrete Mathematics I – p. 248/292
![Page 1132: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1132.jpg)
Graphs
Different graphs may be “similar”
Two graphs are called isomorphic, if there is abijection between their node sets, which preserves theedges
G1 = (V1, E1) G2 = (V2, E2)
G1∼= G2 ⇐⇒ ∃f : V1��V2 :
∀u, v ∈ V1 : (u, v) ∈ E1 ⇔ (f(u), f(v)) ∈ E2
Bijection f is the isomorphism between G1, G2
Discrete Mathematics I – p. 248/292
![Page 1133: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1133.jpg)
Graphs
Different graphs may be “similar”
Two graphs are called isomorphic, if there is abijection between their node sets, which preserves theedges
G1 = (V1, E1) G2 = (V2, E2)
G1∼= G2 ⇐⇒ ∃f : V1��V2 :
∀u, v ∈ V1 : (u, v) ∈ E1 ⇔ (f(u), f(v)) ∈ E2
Bijection f is the isomorphism between G1, G2
Discrete Mathematics I – p. 248/292
![Page 1134: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1134.jpg)
Graphs
Different graphs may be “similar”
Two graphs are called isomorphic, if there is abijection between their node sets, which preserves theedges
G1 = (V1, E1) G2 = (V2, E2)
G1∼= G2 ⇐⇒ ∃f : V1��V2 :
∀u, v ∈ V1 : (u, v) ∈ E1 ⇔ (f(u), f(v)) ∈ E2
Bijection f is the isomorphism between G1, G2
Discrete Mathematics I – p. 248/292
![Page 1135: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1135.jpg)
Graphs
Example:
�
0
�
2
�
1
�
3
�
4
�
0
�
2
�
1
�
3
4
0
�
1
�
2
3
�
4
Discrete Mathematics I – p. 249/292
![Page 1136: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1136.jpg)
Graphs
Example:
�
0
�2 �1
�
3
�
4
�0
�2 �1�
3
4
0
�
1
�
2
3
�4
Discrete Mathematics I – p. 249/292
![Page 1137: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1137.jpg)
Graphs
G = (V, E) V = V1 ∪ V2 V1 ∩ V2 = ∅
G called bipartite (or two-coloured), if
• for all u, v ∈ V1, (u, v) 6∈ E
• for all u, v ∈ V2, (u, v) 6∈ E
Sets V1, V2 called colour classes of G
Discrete Mathematics I – p. 250/292
![Page 1138: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1138.jpg)
Graphs
G = (V, E) V = V1 ∪ V2 V1 ∩ V2 = ∅G called bipartite (or two-coloured), if
• for all u, v ∈ V1, (u, v) 6∈ E
• for all u, v ∈ V2, (u, v) 6∈ E
Sets V1, V2 called colour classes of G
Discrete Mathematics I – p. 250/292
![Page 1139: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1139.jpg)
Graphs
G = (V, E) V = V1 ∪ V2 V1 ∩ V2 = ∅G called bipartite (or two-coloured), if
• for all u, v ∈ V1, (u, v) 6∈ E
• for all u, v ∈ V2, (u, v) 6∈ E
Sets V1, V2 called colour classes of G
Discrete Mathematics I – p. 250/292
![Page 1140: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1140.jpg)
Graphs
Example:
��
�
�
�� ��
� �
� ��
��
The Königsberg Bridges graph is bipartite
Discrete Mathematics I – p. 251/292
![Page 1141: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1141.jpg)
Graphs
Example:
��
�
�
�� ��
� �
� ��
��
The Königsberg Bridges graph is bipartite
Discrete Mathematics I – p. 251/292
![Page 1142: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1142.jpg)
Graphs
Example:
� �� �
�� �
�� �
� �
The wolf/goat/cabbage graph is bipartite
Discrete Mathematics I – p. 252/292
![Page 1143: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1143.jpg)
Graphs
Example:
� �� �
�� �
�� �
� �
The wolf/goat/cabbage graph is bipartite
Discrete Mathematics I – p. 252/292
![Page 1144: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1144.jpg)
Graphs
V1 ∩ V2 = ∅
Bipartite graph with all possible edges between V1, V2
is called complete bipartite
K(V1, V2) = (V1 ∪ V2, (V1 × V2) ∪ (V2 × V1))
The complete bipartite graph on m + n nodes:K(m, n)
Can also define n-partite (n-coloured), and completen-partite graph
Discrete Mathematics I – p. 253/292
![Page 1145: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1145.jpg)
Graphs
V1 ∩ V2 = ∅Bipartite graph with all possible edges between V1, V2
is called complete bipartite
K(V1, V2) = (V1 ∪ V2, (V1 × V2) ∪ (V2 × V1))
The complete bipartite graph on m + n nodes:K(m, n)
Can also define n-partite (n-coloured), and completen-partite graph
Discrete Mathematics I – p. 253/292
![Page 1146: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1146.jpg)
Graphs
V1 ∩ V2 = ∅Bipartite graph with all possible edges between V1, V2
is called complete bipartite
K(V1, V2) = (V1 ∪ V2, (V1 × V2) ∪ (V2 × V1))
The complete bipartite graph on m + n nodes:K(m, n)
Can also define n-partite (n-coloured), and completen-partite graph
Discrete Mathematics I – p. 253/292
![Page 1147: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1147.jpg)
Graphs
V1 ∩ V2 = ∅Bipartite graph with all possible edges between V1, V2
is called complete bipartite
K(V1, V2) = (V1 ∪ V2, (V1 × V2) ∪ (V2 × V1))
The complete bipartite graph on m + n nodes:K(m, n)
Can also define n-partite (n-coloured), and completen-partite graph
Discrete Mathematics I – p. 253/292
![Page 1148: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1148.jpg)
Graphs
V1 ∩ V2 = ∅Bipartite graph with all possible edges between V1, V2
is called complete bipartite
K(V1, V2) = (V1 ∪ V2, (V1 × V2) ∪ (V2 × V1))
The complete bipartite graph on m + n nodes:K(m, n)
Can also define n-partite (n-coloured), and completen-partite graph
Discrete Mathematics I – p. 253/292
![Page 1149: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1149.jpg)
Graphs
Example:
�
H1
�
H2
�
H3
��
W1
��W2
��
W3
The houses/wells graph is complete bipartite
K({H1, H2, H3}, {W1, W2, W3})
Discrete Mathematics I – p. 254/292
![Page 1150: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1150.jpg)
Graphs
Example:
�
H1
�
H2
�
H3
��
W1
��W2
��
W3
The houses/wells graph is complete bipartite
K({H1, H2, H3}, {W1, W2, W3})
Discrete Mathematics I – p. 254/292
![Page 1151: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1151.jpg)
Graphs
Example:
�
H1
�
H2
�
H3
��
W1
��W2
��
W3
The houses/wells graph is complete bipartite
K({H1, H2, H3}, {W1, W2, W3})
Discrete Mathematics I – p. 254/292
![Page 1152: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1152.jpg)
Graphs
G = (V, E)
A walk: sequence (u, u1, . . . , uk−1, v), such that
(u ⇀ u1) ∧ (u1 ⇀ u2) ∧ · · · ∧ (uk−1 ⇀ v)
Nodes u, v connected by a walk: u# v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
A tour is a walk from a node to itself: u# u
Discrete Mathematics I – p. 255/292
![Page 1153: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1153.jpg)
Graphs
G = (V, E)
A walk: sequence (u, u1, . . . , uk−1, v), such that
(u ⇀ u1) ∧ (u1 ⇀ u2) ∧ · · · ∧ (uk−1 ⇀ v)
Nodes u, v connected by a walk: u# v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
A tour is a walk from a node to itself: u# u
Discrete Mathematics I – p. 255/292
![Page 1154: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1154.jpg)
Graphs
G = (V, E)
A walk: sequence (u, u1, . . . , uk−1, v), such that
(u ⇀ u1) ∧ (u1 ⇀ u2) ∧ · · · ∧ (uk−1 ⇀ v)
Nodes u, v connected by a walk: u# v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
A tour is a walk from a node to itself: u# u
Discrete Mathematics I – p. 255/292
![Page 1155: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1155.jpg)
Graphs
G = (V, E)
A walk: sequence (u, u1, . . . , uk−1, v), such that
(u ⇀ u1) ∧ (u1 ⇀ u2) ∧ · · · ∧ (uk−1 ⇀ v)
Nodes u, v connected by a walk: u# v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
A tour is a walk from a node to itself: u# u
Discrete Mathematics I – p. 255/292
![Page 1156: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1156.jpg)
Graphs
G = (V, E)
A walk: sequence (u, u1, . . . , uk−1, v), such that
(u ⇀ u1) ∧ (u1 ⇀ u2) ∧ · · · ∧ (uk−1 ⇀ v)
Nodes u, v connected by a walk: u# v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
A tour is a walk from a node to itself: u# u
Discrete Mathematics I – p. 255/292
![Page 1157: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1157.jpg)
Graphs
G = (V, E)
A walk: sequence (u, u1, . . . , uk−1, v), such that
(u ⇀ u1) ∧ (u1 ⇀ u2) ∧ · · · ∧ (uk−1 ⇀ v)
Nodes u, v connected by a walk: u# v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
A tour is a walk from a node to itself: u# u
Discrete Mathematics I – p. 255/292
![Page 1158: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1158.jpg)
Graphs
�2
�
3
�
4
�10
�
0
�
5
�
1
�
6
�7
8
9
0# 5 : 0 ⇀ 3 ⇀ 1 ⇀ 4 ⇀ 6 ⇀ 3 ⇀ 0 ⇀ 2 ⇀ 5
A tour: 0 ⇀ 3 ⇀ 1 ⇀ 4 ⇀ 6 ⇀ 3 ⇀ 5 ⇀ 2 ⇀ 0
Discrete Mathematics I – p. 256/292
![Page 1159: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1159.jpg)
Graphs
�2
�
3
�
4
�10
�
0
�
5
�
1
�
6
�7
8
9
0# 5 : 0 ⇀ 3 ⇀ 1 ⇀ 4 ⇀ 6 ⇀ 3 ⇀ 0 ⇀ 2 ⇀ 5
A tour: 0 ⇀ 3 ⇀ 1 ⇀ 4 ⇀ 6 ⇀ 3 ⇀ 5 ⇀ 2 ⇀ 0
Discrete Mathematics I – p. 256/292
![Page 1160: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1160.jpg)
Graphs
�2
�
3
�
4
�10
�
0
�
5
�
1
�
6
�7
8
9
0# 5 : 0 ⇀ 3 ⇀ 1 ⇀ 4 ⇀ 6 ⇀ 3 ⇀ 0 ⇀ 2 ⇀ 5
A tour: 0 ⇀ 3 ⇀ 1 ⇀ 4 ⇀ 6 ⇀ 3 ⇀ 5 ⇀ 2 ⇀ 0
Discrete Mathematics I – p. 256/292
![Page 1161: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1161.jpg)
Graphs
For all v ∈ V , (v) is a walk v # v of length 0
For all u, v ∈ V , (u# v) ⇒ (v # u)
For all u, v, w ∈ V , (u# v) ∧ (v # w) ⇒ (u# w)
Therefore R# : V ↔ V is an equivalence relation
Classes of R# called connected components
A graph is connected, if every two nodes areconnected
Discrete Mathematics I – p. 257/292
![Page 1162: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1162.jpg)
Graphs
For all v ∈ V , (v) is a walk v # v of length 0
For all u, v ∈ V , (u# v) ⇒ (v # u)
For all u, v, w ∈ V , (u# v) ∧ (v # w) ⇒ (u# w)
Therefore R# : V ↔ V is an equivalence relation
Classes of R# called connected components
A graph is connected, if every two nodes areconnected
Discrete Mathematics I – p. 257/292
![Page 1163: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1163.jpg)
Graphs
For all v ∈ V , (v) is a walk v # v of length 0
For all u, v ∈ V , (u# v) ⇒ (v # u)
For all u, v, w ∈ V , (u# v) ∧ (v # w) ⇒ (u# w)
Therefore R# : V ↔ V is an equivalence relation
Classes of R# called connected components
A graph is connected, if every two nodes areconnected
Discrete Mathematics I – p. 257/292
![Page 1164: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1164.jpg)
Graphs
For all v ∈ V , (v) is a walk v # v of length 0
For all u, v ∈ V , (u# v) ⇒ (v # u)
For all u, v, w ∈ V , (u# v) ∧ (v # w) ⇒ (u# w)
Therefore R# : V ↔ V is an equivalence relation
Classes of R# called connected components
A graph is connected, if every two nodes areconnected
Discrete Mathematics I – p. 257/292
![Page 1165: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1165.jpg)
Graphs
For all v ∈ V , (v) is a walk v # v of length 0
For all u, v ∈ V , (u# v) ⇒ (v # u)
For all u, v, w ∈ V , (u# v) ∧ (v # w) ⇒ (u# w)
Therefore R# : V ↔ V is an equivalence relation
Classes of R# called connected components
A graph is connected, if every two nodes areconnected
Discrete Mathematics I – p. 257/292
![Page 1166: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1166.jpg)
Graphs
For all v ∈ V , (v) is a walk v # v of length 0
For all u, v ∈ V , (u# v) ⇒ (v # u)
For all u, v, w ∈ V , (u# v) ∧ (v # w) ⇒ (u# w)
Therefore R# : V ↔ V is an equivalence relation
Classes of R# called connected components
A graph is connected, if every two nodes areconnected
Discrete Mathematics I – p. 257/292
![Page 1167: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1167.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1168: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1168.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1169: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1169.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1170: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1170.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1171: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1171.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1172: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1172.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1173: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1173.jpg)
Graphs
G = (V, E)
A path is a walk where all nodes are distinct
Nodes u, v connected by a path: u v
u = u0 ⇀ u1 ⇀ u2 ⇀ . . . ⇀ uk−1 ⇀ uk = v
∀i, j ∈ Nk+1 : i 6= j ⇒ ui 6= uj
A cycle is a tour of length ≥ 3 where all nodes exceptthe final are distinct: u v ⇀ u
A graph without cycles called acyclic
Discrete Mathematics I – p. 258/292
![Page 1174: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1174.jpg)
Graphs
�2
�
3
�
4
�10
�
0
�
5
�
1
�
6
�7
8
9
0 5 : 0 ⇀ 2 ⇀ 7 ⇀ 10 ⇀ 8 ⇀ 3 ⇀ 5
A cycle: 3 ⇀ 8 ⇀ 10 ⇀ 9 ⇀ 4 ⇀ 6 ⇀ 3
Discrete Mathematics I – p. 259/292
![Page 1175: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1175.jpg)
Graphs
�2
�
3
�
4
�10
�
0
�
5
�
1
�
6
�7
8
9
0 5 : 0 ⇀ 2 ⇀ 7 ⇀ 10 ⇀ 8 ⇀ 3 ⇀ 5
A cycle: 3 ⇀ 8 ⇀ 10 ⇀ 9 ⇀ 4 ⇀ 6 ⇀ 3
Discrete Mathematics I – p. 259/292
![Page 1176: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1176.jpg)
Graphs
�2
�
3
�
4
�10
�
0
�
5
�
1
�
6
�7
8
9
0 5 : 0 ⇀ 2 ⇀ 7 ⇀ 10 ⇀ 8 ⇀ 3 ⇀ 5
A cycle: 3 ⇀ 8 ⇀ 10 ⇀ 9 ⇀ 4 ⇀ 6 ⇀ 3
Discrete Mathematics I – p. 259/292
![Page 1177: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1177.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof. (u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1178: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1178.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof. (u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1179: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1179.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof. (u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1180: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1180.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof. (u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1181: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1181.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof.
(u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1182: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1182.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof. (u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1183: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1183.jpg)
Graphs
G = (V, E)
R : V ↔ V — equivalence relation?
Prove: For all u, v ∈ V , there is a path u v, iffthere is a walk u# v.
R , R# : V ↔ V R = R#
Proof. (u v) ⇒ (u# v): trivial
(u# v) ⇒ (u v): induction on walk length
Discrete Mathematics I – p. 260/292
![Page 1184: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1184.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1185: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1185.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1186: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1186.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1187: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1187.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1188: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1188.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1189: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1189.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1190: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1190.jpg)
Graphs
Induction base: (u) is both u# u and u u
Inductive step: consider walk u# w ⇀ v
Assume statement holds for u# w: path u w
Case 1: path u w does not visit v
Then u w ⇀ v is a path
Case 2: path u w visits v: u v w ⇀ v
Take initial u v
Discrete Mathematics I – p. 261/292
![Page 1191: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1191.jpg)
Graphs
G = (V, E) v ∈ V
The degree of node v is the number of nodes adjacentto v
deg(v) = |{u ∈ V | v ⇀ u}|
a
b c
d
ef
deg(a) = deg(d) = 2
deg(b) = deg(c) = 4
deg(e) = deg(f) = 4
Discrete Mathematics I – p. 262/292
![Page 1192: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1192.jpg)
Graphs
G = (V, E) v ∈ V
The degree of node v is the number of nodes adjacentto v
deg(v) = |{u ∈ V | v ⇀ u}|
a
b c
d
ef
deg(a) = deg(d) = 2
deg(b) = deg(c) = 4
deg(e) = deg(f) = 4
Discrete Mathematics I – p. 262/292
![Page 1193: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1193.jpg)
Graphs
G = (V, E) v ∈ V
The degree of node v is the number of nodes adjacentto v
deg(v) = |{u ∈ V | v ⇀ u}|
a
b c
d
ef
deg(a) = deg(d) = 2
deg(b) = deg(c) = 4
deg(e) = deg(f) = 4
Discrete Mathematics I – p. 262/292
![Page 1194: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1194.jpg)
Graphs
G = (V, E) v ∈ V
The degree of node v is the number of nodes adjacentto v
deg(v) = |{u ∈ V | v ⇀ u}|
�a
�
b
�
c
�
d
�
e
�
f
deg(a) = deg(d) = 2
deg(b) = deg(c) = 4
deg(e) = deg(f) = 4
Discrete Mathematics I – p. 262/292
![Page 1195: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1195.jpg)
Graphs
G = (V, E) v ∈ V
The degree of node v is the number of nodes adjacentto v
deg(v) = |{u ∈ V | v ⇀ u}|
�a
�
b
�
c
�
d
�
e
�
f
deg(a) = deg(d) = 2
deg(b) = deg(c) = 4
deg(e) = deg(f) = 4
Discrete Mathematics I – p. 262/292
![Page 1196: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1196.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
a
b c
d
ef
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1197: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1197.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1198: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1198.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a
⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1199: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1199.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b
⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1200: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1200.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c
⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1201: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1201.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f
⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1202: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1202.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e
⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1203: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1203.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d
⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1204: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1204.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c
⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1205: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1205.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e
⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1206: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1206.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1207: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1207.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f
⇀ a
Discrete Mathematics I – p. 263/292
![Page 1208: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1208.jpg)
Graphs
An Euler tour of graph G is a tour which visits everyedge in E exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ c ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b ⇀ f ⇀ a
Discrete Mathematics I – p. 263/292
![Page 1209: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1209.jpg)
Graphs
G = (V, E)
Theorem: Graph G has an Euler tour iff
• G is connected
• every node in V has even degree
Gives an efficient test for Euler tour existence
Discrete Mathematics I – p. 264/292
![Page 1210: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1210.jpg)
Graphs
G = (V, E)
Theorem: Graph G has an Euler tour iff
• G is connected
• every node in V has even degree
Gives an efficient test for Euler tour existence
Discrete Mathematics I – p. 264/292
![Page 1211: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1211.jpg)
Graphs
G = (V, E)
Theorem: Graph G has an Euler tour iff
• G is connected
• every node in V has even degree
Gives an efficient test for Euler tour existence
Discrete Mathematics I – p. 264/292
![Page 1212: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1212.jpg)
Graphs
G = (V, E)
Theorem: Graph G has an Euler tour iff
• G is connected
• every node in V has even degree
Gives an efficient test for Euler tour existence
Discrete Mathematics I – p. 264/292
![Page 1213: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1213.jpg)
Graphs
G = (V, E)
Theorem: Graph G has an Euler tour iff
• G is connected
• every node in V has even degree
Gives an efficient test for Euler tour existence
Discrete Mathematics I – p. 264/292
![Page 1214: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1214.jpg)
Graphs
Proof.
G has Euler tour ⇒ G connected: trivial
G has Euler tour ⇒ every node has even degree
Consider v ∈ V
Suppose v visited k times by Euler tour
Every visit uses 2 new edges (in and out)
Hence deg(v) = 2k
Discrete Mathematics I – p. 265/292
![Page 1215: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1215.jpg)
Graphs
Proof.
G has Euler tour ⇒ G connected: trivial
G has Euler tour ⇒ every node has even degree
Consider v ∈ V
Suppose v visited k times by Euler tour
Every visit uses 2 new edges (in and out)
Hence deg(v) = 2k
Discrete Mathematics I – p. 265/292
![Page 1216: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1216.jpg)
Graphs
Proof.
G has Euler tour ⇒ G connected: trivial
G has Euler tour ⇒ every node has even degree
Consider v ∈ V
Suppose v visited k times by Euler tour
Every visit uses 2 new edges (in and out)
Hence deg(v) = 2k
Discrete Mathematics I – p. 265/292
![Page 1217: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1217.jpg)
Graphs
Proof.
G has Euler tour ⇒ G connected: trivial
G has Euler tour ⇒ every node has even degree
Consider v ∈ V
Suppose v visited k times by Euler tour
Every visit uses 2 new edges (in and out)
Hence deg(v) = 2k
Discrete Mathematics I – p. 265/292
![Page 1218: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1218.jpg)
Graphs
Proof.
G has Euler tour ⇒ G connected: trivial
G has Euler tour ⇒ every node has even degree
Consider v ∈ V
Suppose v visited k times by Euler tour
Every visit uses 2 new edges (in and out)
Hence deg(v) = 2k
Discrete Mathematics I – p. 265/292
![Page 1219: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1219.jpg)
Graphs
Proof.
G has Euler tour ⇒ G connected: trivial
G has Euler tour ⇒ every node has even degree
Consider v ∈ V
Suppose v visited k times by Euler tour
Every visit uses 2 new edges (in and out)
Hence deg(v) = 2k
Discrete Mathematics I – p. 265/292
![Page 1220: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1220.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1221: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1221.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1222: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1222.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1223: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1223.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1224: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1224.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1225: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1225.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1226: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1226.jpg)
Graphs
G connected ∧ every node has even degree⇒ G has an Euler tour
Take any v0 ∈ V
Consider any walk v0 ⇀ v1 ⇀ . . . ⇀ vk 6= v0
Node vk has an odd number of visited edges
deg(vk) is even ⇒ vk has an unvisited edge
Extend walk: v0 ⇀ v1 ⇀ . . . ⇀ vk ⇀ vk+1
Repeat until v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Discrete Mathematics I – p. 266/292
![Page 1227: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1227.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1228: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1228.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1229: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1229.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1230: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1230.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1231: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1231.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1232: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1232.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1233: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1233.jpg)
Graphs
Consider tour v0 ⇀ v1 ⇀ . . . ⇀ vm = v0
Suppose some vi has unvisited edge to vm+1
By symmetry, let vi = v0
Extend walk: v0 ⇀ . . . ⇀ vk ⇀ vm = v0 ⇀ vm+1
Repeat until every vi has no unvisited edges
G connected =⇒ all edges in E visited
Therefore, G has an Euler tour
Discrete Mathematics I – p. 267/292
![Page 1234: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1234.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1235: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1235.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a
⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1236: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1236.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b
⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1237: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1237.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c
⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1238: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1238.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f
⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1239: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1239.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1240: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1240.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b
⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1241: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1241.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f
⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1242: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1242.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e
⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1243: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1243.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d
⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1244: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1244.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c
⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1245: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1245.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e
⇀ b
Discrete Mathematics I – p. 268/292
![Page 1246: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1246.jpg)
Graphs
Example:
�a
�
b
�
c
�d
�e
�
f
a ⇀ b ⇀ c ⇀ f ⇀ a
b ⇀ c ⇀ f ⇀ a ⇀ b ⇀ f ⇀ e ⇀ d ⇀ c ⇀ e ⇀ b
Discrete Mathematics I – p. 268/292
![Page 1247: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1247.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
a
b c
d
ef
a ⇀ b ⇀ e ⇀ d ⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1248: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1248.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ e ⇀ d ⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1249: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1249.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a
⇀ b ⇀ e ⇀ d ⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1250: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1250.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b
⇀ e ⇀ d ⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1251: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1251.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ e
⇀ d ⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1252: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1252.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ e ⇀ d
⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1253: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1253.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ e ⇀ d ⇀ c
⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1254: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1254.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ e ⇀ d ⇀ c ⇀ f
⇀ a
Discrete Mathematics I – p. 269/292
![Page 1255: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1255.jpg)
Graphs
A Hamiltonian cycle of graph G is a cycle whichvisits every node in V exactly once
�a
�
b
�
c
�
d
�
e�
f
a ⇀ b ⇀ e ⇀ d ⇀ c ⇀ f ⇀ a
Discrete Mathematics I – p. 269/292
![Page 1256: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1256.jpg)
Graphs
Exercise: find an efficient test for existence ofHamiltonian cycle. . .
. . . and claim your $1 000 000!
See www.claymath.org for details
Discrete Mathematics I – p. 270/292
![Page 1257: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1257.jpg)
Graphs
Exercise: find an efficient test for existence ofHamiltonian cycle. . .
. . . and claim your $1 000 000!
See www.claymath.org for details
Discrete Mathematics I – p. 270/292
![Page 1258: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1258.jpg)
Graphs
Exercise: find an efficient test for existence ofHamiltonian cycle. . .
. . . and claim your $1 000 000!
See www.claymath.org for details
Discrete Mathematics I – p. 270/292
![Page 1259: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1259.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a subgraph of G, if V ′ ⊆ V , E ′ ⊆ E
0
1
2
3
4G 0
1
2
3
G′
G′ ⊆ G
Discrete Mathematics I – p. 271/292
![Page 1260: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1260.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a subgraph of G, if V ′ ⊆ V , E ′ ⊆ E
0
1
2
3
4G 0
1
2
3
G′
G′ ⊆ G
Discrete Mathematics I – p. 271/292
![Page 1261: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1261.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a subgraph of G, if V ′ ⊆ V , E ′ ⊆ E
�
0
�1
�
2
� 3
�4G
�
0
�1
�2
� 3
G′
G′ ⊆ G
Discrete Mathematics I – p. 271/292
![Page 1262: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1262.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a subgraph of G, if V ′ ⊆ V , E ′ ⊆ E
�
0
�1
�
2
� 3
�4G
�
0
�1
�2
� 3
G′
G′ ⊆ G
Discrete Mathematics I – p. 271/292
![Page 1263: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1263.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a spanning subgraph of G, if V ′ = V , E ′ ⊆ E
0
1
2
3
4G 0
1
2
3
4G′
G′ v G
Discrete Mathematics I – p. 272/292
![Page 1264: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1264.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a spanning subgraph of G, if V ′ = V , E ′ ⊆ E
0
1
2
3
4G 0
1
2
3
4G′
G′ v G
Discrete Mathematics I – p. 272/292
![Page 1265: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1265.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a spanning subgraph of G, if V ′ = V , E ′ ⊆ E
�
0
�1
�
2
� 3
�4G
�
0
�1
�2
� 3
4G′
G′ v G
Discrete Mathematics I – p. 272/292
![Page 1266: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1266.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
G′ is a spanning subgraph of G, if V ′ = V , E ′ ⊆ E
�
0
�1
�
2
� 3
�4G
�
0
�1
�2
� 3
4G′
G′ v G
Discrete Mathematics I – p. 272/292
![Page 1267: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1267.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1268: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1268.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1269: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1269.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1270: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1270.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1271: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1271.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1272: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1272.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1273: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1273.jpg)
Graphs
V — a finite set G(V ) — set of all graphs on V
R⊆ : G(V ) ↔ G(V )
∀G : G ⊆ G G v G
∀G, G′ : (G′ ⊆ G) ∧ (G ⊆ G′) ⇒ G = G′
∀G, G′, G′′ : (G′′ ⊆ G′) ∧ (G′ ⊆ G) ⇒ G′′ ⊆ G
Therefore, R⊆ is a partial order
Similarly, Rv is a partial order
Discrete Mathematics I – p. 273/292
![Page 1274: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1274.jpg)
Graphs
Recall: a graph is
• connected, if every two nodes connected
• acyclic, if there is no cycle
A connected acyclic graph is called a tree
Discrete Mathematics I – p. 274/292
![Page 1275: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1275.jpg)
Graphs
Recall: a graph is
• connected, if every two nodes connected
• acyclic, if there is no cycle
A connected acyclic graph is called a tree
Discrete Mathematics I – p. 274/292
![Page 1276: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1276.jpg)
Graphs
Recall: a graph is
• connected, if every two nodes connected
• acyclic, if there is no cycle
A connected acyclic graph is called a tree
Discrete Mathematics I – p. 274/292
![Page 1277: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1277.jpg)
Graphs
Recall: a graph is
• connected, if every two nodes connected
• acyclic, if there is no cycle
A connected acyclic graph is called a tree
Discrete Mathematics I – p. 274/292
![Page 1278: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1278.jpg)
Graphs
Recall: a graph is
• connected, if every two nodes connected
• acyclic, if there is no cycle
A connected acyclic graph is called a tree
� ��
�
��
��
�
�
Discrete Mathematics I – p. 274/292
![Page 1279: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1279.jpg)
Graphs
G = (V, E) — a tree
Prove: |V | = |E|+ 1.
Proof. Induction base: V = {v}, E = ∅|E| = 0 |V | = 1 = |E|+ 1
Discrete Mathematics I – p. 275/292
![Page 1280: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1280.jpg)
Graphs
G = (V, E) — a tree
Prove: |V | = |E|+ 1.
Proof. Induction base: V = {v}, E = ∅|E| = 0 |V | = 1 = |E|+ 1
Discrete Mathematics I – p. 275/292
![Page 1281: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1281.jpg)
Graphs
G = (V, E) — a tree
Prove: |V | = |E|+ 1.
Proof. Induction base: V = {v}, E = ∅
|E| = 0 |V | = 1 = |E|+ 1
Discrete Mathematics I – p. 275/292
![Page 1282: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1282.jpg)
Graphs
G = (V, E) — a tree
Prove: |V | = |E|+ 1.
Proof. Induction base: V = {v}, E = ∅|E| = 0 |V | = 1 = |E|+ 1
Discrete Mathematics I – p. 275/292
![Page 1283: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1283.jpg)
Graphs
Inductive step: assume statement holds for all propersubgraphs of G
Take any edge (u, v) ∈ E.
Let G′ = (V, E \ {(u, v), (v, u)}).
G′
u v
Consider R : V ↔ V in G′
Discrete Mathematics I – p. 276/292
![Page 1284: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1284.jpg)
Graphs
Inductive step: assume statement holds for all propersubgraphs of G
Take any edge (u, v) ∈ E.
Let G′ = (V, E \ {(u, v), (v, u)}).
G′
u v
Consider R : V ↔ V in G′
Discrete Mathematics I – p. 276/292
![Page 1285: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1285.jpg)
Graphs
Inductive step: assume statement holds for all propersubgraphs of G
Take any edge (u, v) ∈ E.
Let G′ = (V, E \ {(u, v), (v, u)}).
G′
u v
Consider R : V ↔ V in G′
Discrete Mathematics I – p. 276/292
![Page 1286: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1286.jpg)
Graphs
Inductive step: assume statement holds for all propersubgraphs of G
Take any edge (u, v) ∈ E.
Let G′ = (V, E \ {(u, v), (v, u)}).
G′
�u � v
Consider R : V ↔ V in G′
Discrete Mathematics I – p. 276/292
![Page 1287: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1287.jpg)
Graphs
Inductive step: assume statement holds for all propersubgraphs of G
Take any edge (u, v) ∈ E.
Let G′ = (V, E \ {(u, v), (v, u)}).
G′
�u � v
Consider R : V ↔ V in G′
Discrete Mathematics I – p. 276/292
![Page 1288: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1288.jpg)
Graphs
Suppose u v in G′
G′u v
Then u v ⇀ u a cycle in G — contradiction
Therefore u 6 v in G′
Discrete Mathematics I – p. 277/292
![Page 1289: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1289.jpg)
Graphs
Suppose u v in G′
G′
�u � v
Then u v ⇀ u a cycle in G — contradiction
Therefore u 6 v in G′
Discrete Mathematics I – p. 277/292
![Page 1290: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1290.jpg)
Graphs
Suppose u v in G′
G′
�u � v
Then u v ⇀ u a cycle in G — contradiction
Therefore u 6 v in G′
Discrete Mathematics I – p. 277/292
![Page 1291: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1291.jpg)
Graphs
Suppose u v in G′
G′
�u � v
Then u v ⇀ u a cycle in G — contradiction
Therefore u 6 v in G′
Discrete Mathematics I – p. 277/292
![Page 1292: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1292.jpg)
Graphs
Vu = [u] Vv = [v] (in G′)
Eu = {(x, y) | x, y ∈ Vu} Ev = {(x, y) | x, y ∈ Vv}Gu = (Vu, Eu) Gv = (Vv, Ev)
Gu Gvu v
G connected =⇒ Gu, Gv connected
G acyclic =⇒ Gu, Gv acyclic
Discrete Mathematics I – p. 278/292
![Page 1293: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1293.jpg)
Graphs
Vu = [u] Vv = [v] (in G′)
Eu = {(x, y) | x, y ∈ Vu} Ev = {(x, y) | x, y ∈ Vv}
Gu = (Vu, Eu) Gv = (Vv, Ev)
Gu Gvu v
G connected =⇒ Gu, Gv connected
G acyclic =⇒ Gu, Gv acyclic
Discrete Mathematics I – p. 278/292
![Page 1294: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1294.jpg)
Graphs
Vu = [u] Vv = [v] (in G′)
Eu = {(x, y) | x, y ∈ Vu} Ev = {(x, y) | x, y ∈ Vv}Gu = (Vu, Eu) Gv = (Vv, Ev)
Gu Gvu v
G connected =⇒ Gu, Gv connected
G acyclic =⇒ Gu, Gv acyclic
Discrete Mathematics I – p. 278/292
![Page 1295: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1295.jpg)
Graphs
Vu = [u] Vv = [v] (in G′)
Eu = {(x, y) | x, y ∈ Vu} Ev = {(x, y) | x, y ∈ Vv}Gu = (Vu, Eu) Gv = (Vv, Ev)
Gu Gv
�u � v
G connected =⇒ Gu, Gv connected
G acyclic =⇒ Gu, Gv acyclic
Discrete Mathematics I – p. 278/292
![Page 1296: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1296.jpg)
Graphs
Vu = [u] Vv = [v] (in G′)
Eu = {(x, y) | x, y ∈ Vu} Ev = {(x, y) | x, y ∈ Vv}Gu = (Vu, Eu) Gv = (Vv, Ev)
Gu Gv
�u � v
G connected =⇒ Gu, Gv connected
G acyclic =⇒ Gu, Gv acyclic
Discrete Mathematics I – p. 278/292
![Page 1297: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1297.jpg)
Graphs
Vu = [u] Vv = [v] (in G′)
Eu = {(x, y) | x, y ∈ Vu} Ev = {(x, y) | x, y ∈ Vv}Gu = (Vu, Eu) Gv = (Vv, Ev)
Gu Gv
�u � v
G connected =⇒ Gu, Gv connected
G acyclic =⇒ Gu, Gv acyclic
Discrete Mathematics I – p. 278/292
![Page 1298: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1298.jpg)
Graphs
By induction hypothesis:|Vu| = |Eu|+ 1 |Vv| = |Ev|+ 1
|V | = |Vu|+ |Vv| = (|Eu|+ 1) + (|Ev|+ 1) =
(|Eu|+ |Ev|+ 1) + 1 = |E|+ 1
Discrete Mathematics I – p. 279/292
![Page 1299: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1299.jpg)
Graphs
By induction hypothesis:|Vu| = |Eu|+ 1 |Vv| = |Ev|+ 1
|V | = |Vu|+ |Vv| = (|Eu|+ 1) + (|Ev|+ 1) =
(|Eu|+ |Ev|+ 1) + 1 = |E|+ 1
Discrete Mathematics I – p. 279/292
![Page 1300: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1300.jpg)
Graphs
By induction hypothesis:|Vu| = |Eu|+ 1 |Vv| = |Ev|+ 1
|V | = |Vu|+ |Vv| = (|Eu|+ 1) + (|Ev|+ 1) =
(|Eu|+ |Ev|+ 1) + 1 = |E|+ 1
Discrete Mathematics I – p. 279/292
![Page 1301: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1301.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1302: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1302.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1303: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1303.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1304: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1304.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2.
|E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1305: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1305.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |
But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1306: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1306.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1307: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1307.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1308: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1308.jpg)
Graphs
G = (V, E) — a tree
Corollary: G has a node of degree 1.
Proof. G connected ⇒ no nodes of degree 0.
Suppose all degrees ≥ 2. |E| ≥ 2 · |V |/2 = |V |But |E| = |V | − 1. Hence assumption false.
Therefore G has a node of degree 1.
A node of degree 1 in a tree is called a leaf.
Discrete Mathematics I – p. 280/292
![Page 1309: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1309.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
Recall: G′ is a spanning subgraph of G, if V ′ = V ,E ′ ⊆ E
Rv : G(V ) ↔ G(V )
Consider restricting Rv to the set of all
• connected graphs in G(V )
• acyclic graphs in G(V )
Discrete Mathematics I – p. 281/292
![Page 1310: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1310.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
Recall: G′ is a spanning subgraph of G, if V ′ = V ,E ′ ⊆ E
Rv : G(V ) ↔ G(V )
Consider restricting Rv to the set of all
• connected graphs in G(V )
• acyclic graphs in G(V )
Discrete Mathematics I – p. 281/292
![Page 1311: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1311.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
Recall: G′ is a spanning subgraph of G, if V ′ = V ,E ′ ⊆ E
Rv : G(V ) ↔ G(V )
Consider restricting Rv to the set of all
• connected graphs in G(V )
• acyclic graphs in G(V )
Discrete Mathematics I – p. 281/292
![Page 1312: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1312.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
Recall: G′ is a spanning subgraph of G, if V ′ = V ,E ′ ⊆ E
Rv : G(V ) ↔ G(V )
Consider restricting Rv to the set of all
• connected graphs in G(V )
• acyclic graphs in G(V )
Discrete Mathematics I – p. 281/292
![Page 1313: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1313.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
Recall: G′ is a spanning subgraph of G, if V ′ = V ,E ′ ⊆ E
Rv : G(V ) ↔ G(V )
Consider restricting Rv to the set of all
• connected graphs in G(V )
• acyclic graphs in G(V )
Discrete Mathematics I – p. 281/292
![Page 1314: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1314.jpg)
Graphs
G = (V, E) G′ = (V ′, E ′)
Recall: G′ is a spanning subgraph of G, if V ′ = V ,E ′ ⊆ E
Rv : G(V ) ↔ G(V )
Consider restricting Rv to the set of all
• connected graphs in G(V )
• acyclic graphs in G(V )
Discrete Mathematics I – p. 281/292
![Page 1315: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1315.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-minimal in the set of allconnected graphs on V
Proof. G connected
Need to prove: G acyclic iff G v-minimal
Equivalent to: G has a cycle iff G not v-minimal
Discrete Mathematics I – p. 282/292
![Page 1316: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1316.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-minimal in the set of allconnected graphs on V
Proof. G connected
Need to prove: G acyclic iff G v-minimal
Equivalent to: G has a cycle iff G not v-minimal
Discrete Mathematics I – p. 282/292
![Page 1317: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1317.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-minimal in the set of allconnected graphs on V
Proof. G connected
Need to prove: G acyclic iff G v-minimal
Equivalent to: G has a cycle iff G not v-minimal
Discrete Mathematics I – p. 282/292
![Page 1318: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1318.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-minimal in the set of allconnected graphs on V
Proof. G connected
Need to prove: G acyclic iff G v-minimal
Equivalent to: G has a cycle iff G not v-minimal
Discrete Mathematics I – p. 282/292
![Page 1319: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1319.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-minimal in the set of allconnected graphs on V
Proof. G connected
Need to prove: G acyclic iff G v-minimal
Equivalent to: G has a cycle iff G not v-minimal
Discrete Mathematics I – p. 282/292
![Page 1320: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1320.jpg)
Graphs
G has a cycle ⇒ G not v-minimal
Suppose G has a cycle
Remove any edge from cycle
Remaining graph connected
Hence G not v-minimal
Discrete Mathematics I – p. 283/292
![Page 1321: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1321.jpg)
Graphs
G has a cycle ⇒ G not v-minimal
Suppose G has a cycle
Remove any edge from cycle
Remaining graph connected
Hence G not v-minimal
Discrete Mathematics I – p. 283/292
![Page 1322: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1322.jpg)
Graphs
G has a cycle ⇒ G not v-minimal
Suppose G has a cycle
Remove any edge from cycle
Remaining graph connected
Hence G not v-minimal
Discrete Mathematics I – p. 283/292
![Page 1323: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1323.jpg)
Graphs
G has a cycle ⇒ G not v-minimal
Suppose G has a cycle
Remove any edge from cycle
Remaining graph connected
Hence G not v-minimal
Discrete Mathematics I – p. 283/292
![Page 1324: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1324.jpg)
Graphs
G has a cycle ⇒ G not v-minimal
Suppose G has a cycle
Remove any edge from cycle
Remaining graph connected
Hence G not v-minimal
Discrete Mathematics I – p. 283/292
![Page 1325: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1325.jpg)
Graphs
G not v-minimal ⇒ G has a cycle
Suppose G not v-minimal
For some u, v ∈ V , removing edge u ⇀ v does notdisconnect the graph
Therefore there is another path u v
Hence G has a cycle
Discrete Mathematics I – p. 284/292
![Page 1326: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1326.jpg)
Graphs
G not v-minimal ⇒ G has a cycle
Suppose G not v-minimal
For some u, v ∈ V , removing edge u ⇀ v does notdisconnect the graph
Therefore there is another path u v
Hence G has a cycle
Discrete Mathematics I – p. 284/292
![Page 1327: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1327.jpg)
Graphs
G not v-minimal ⇒ G has a cycle
Suppose G not v-minimal
For some u, v ∈ V , removing edge u ⇀ v does notdisconnect the graph
Therefore there is another path u v
Hence G has a cycle
Discrete Mathematics I – p. 284/292
![Page 1328: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1328.jpg)
Graphs
G not v-minimal ⇒ G has a cycle
Suppose G not v-minimal
For some u, v ∈ V , removing edge u ⇀ v does notdisconnect the graph
Therefore there is another path u v
Hence G has a cycle
Discrete Mathematics I – p. 284/292
![Page 1329: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1329.jpg)
Graphs
G not v-minimal ⇒ G has a cycle
Suppose G not v-minimal
For some u, v ∈ V , removing edge u ⇀ v does notdisconnect the graph
Therefore there is another path u v
Hence G has a cycle
Discrete Mathematics I – p. 284/292
![Page 1330: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1330.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-maximal in the set of allacyclic graphs on V
Proof. G acyclic
Need to prove: G connected iff G v-maximal
Equivalent to:G disconnected iff G not v-maximal
Discrete Mathematics I – p. 285/292
![Page 1331: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1331.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-maximal in the set of allacyclic graphs on V
Proof. G acyclic
Need to prove: G connected iff G v-maximal
Equivalent to:G disconnected iff G not v-maximal
Discrete Mathematics I – p. 285/292
![Page 1332: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1332.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-maximal in the set of allacyclic graphs on V
Proof. G acyclic
Need to prove: G connected iff G v-maximal
Equivalent to:G disconnected iff G not v-maximal
Discrete Mathematics I – p. 285/292
![Page 1333: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1333.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-maximal in the set of allacyclic graphs on V
Proof. G acyclic
Need to prove: G connected iff G v-maximal
Equivalent to:G disconnected iff G not v-maximal
Discrete Mathematics I – p. 285/292
![Page 1334: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1334.jpg)
Graphs
G = (V, E)
Prove: G is a tree iff G is v-maximal in the set of allacyclic graphs on V
Proof. G acyclic
Need to prove: G connected iff G v-maximal
Equivalent to:G disconnected iff G not v-maximal
Discrete Mathematics I – p. 285/292
![Page 1335: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1335.jpg)
Graphs
G disconnected ⇒ G not v-maximal
Suppose G disconnected
Add any edge between two connected components
Resulting graph acyclic
Hence G not v-maximal
Discrete Mathematics I – p. 286/292
![Page 1336: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1336.jpg)
Graphs
G disconnected ⇒ G not v-maximal
Suppose G disconnected
Add any edge between two connected components
Resulting graph acyclic
Hence G not v-maximal
Discrete Mathematics I – p. 286/292
![Page 1337: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1337.jpg)
Graphs
G disconnected ⇒ G not v-maximal
Suppose G disconnected
Add any edge between two connected components
Resulting graph acyclic
Hence G not v-maximal
Discrete Mathematics I – p. 286/292
![Page 1338: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1338.jpg)
Graphs
G disconnected ⇒ G not v-maximal
Suppose G disconnected
Add any edge between two connected components
Resulting graph acyclic
Hence G not v-maximal
Discrete Mathematics I – p. 286/292
![Page 1339: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1339.jpg)
Graphs
G disconnected ⇒ G not v-maximal
Suppose G disconnected
Add any edge between two connected components
Resulting graph acyclic
Hence G not v-maximal
Discrete Mathematics I – p. 286/292
![Page 1340: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1340.jpg)
Graphs
G not v-maximal ⇒ G disconnected
Suppose G not v-maximal
For some u, v ∈ V , adding edge u ⇀ v does notcreate cycle
Therefore u, v are in different connected components
Hence G disconnected
Discrete Mathematics I – p. 287/292
![Page 1341: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1341.jpg)
Graphs
G not v-maximal ⇒ G disconnected
Suppose G not v-maximal
For some u, v ∈ V , adding edge u ⇀ v does notcreate cycle
Therefore u, v are in different connected components
Hence G disconnected
Discrete Mathematics I – p. 287/292
![Page 1342: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1342.jpg)
Graphs
G not v-maximal ⇒ G disconnected
Suppose G not v-maximal
For some u, v ∈ V , adding edge u ⇀ v does notcreate cycle
Therefore u, v are in different connected components
Hence G disconnected
Discrete Mathematics I – p. 287/292
![Page 1343: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1343.jpg)
Graphs
G not v-maximal ⇒ G disconnected
Suppose G not v-maximal
For some u, v ∈ V , adding edge u ⇀ v does notcreate cycle
Therefore u, v are in different connected components
Hence G disconnected
Discrete Mathematics I – p. 287/292
![Page 1344: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1344.jpg)
Graphs
G not v-maximal ⇒ G disconnected
Suppose G not v-maximal
For some u, v ∈ V , adding edge u ⇀ v does notcreate cycle
Therefore u, v are in different connected components
Hence G disconnected
Discrete Mathematics I – p. 287/292
![Page 1345: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1345.jpg)
Graphs
A graph is called planar, if it can be drawn on theplane without edge crossings.
Examples: any tree, any cycle
Complete graphs:K(1), K(2), K(3), K(4). Not K(5).
Complete bipartite graphs: K(2, 3). Not K(3, 3).
Discrete Mathematics I – p. 288/292
![Page 1346: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1346.jpg)
Graphs
A graph is called planar, if it can be drawn on theplane without edge crossings.
Examples: any tree
, any cycle
Complete graphs:K(1), K(2), K(3), K(4). Not K(5).
Complete bipartite graphs: K(2, 3). Not K(3, 3).
Discrete Mathematics I – p. 288/292
![Page 1347: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1347.jpg)
Graphs
A graph is called planar, if it can be drawn on theplane without edge crossings.
Examples: any tree, any cycle
Complete graphs:K(1), K(2), K(3), K(4). Not K(5).
Complete bipartite graphs: K(2, 3). Not K(3, 3).
Discrete Mathematics I – p. 288/292
![Page 1348: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1348.jpg)
Graphs
A graph is called planar, if it can be drawn on theplane without edge crossings.
Examples: any tree, any cycle
Complete graphs:K(1), K(2), K(3), K(4).
Not K(5).
Complete bipartite graphs: K(2, 3). Not K(3, 3).
Discrete Mathematics I – p. 288/292
![Page 1349: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1349.jpg)
Graphs
A graph is called planar, if it can be drawn on theplane without edge crossings.
Examples: any tree, any cycle
Complete graphs:K(1), K(2), K(3), K(4). Not K(5).
Complete bipartite graphs: K(2, 3). Not K(3, 3).
Discrete Mathematics I – p. 288/292
![Page 1350: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1350.jpg)
Graphs
A graph is called planar, if it can be drawn on theplane without edge crossings.
Examples: any tree, any cycle
Complete graphs:K(1), K(2), K(3), K(4). Not K(5).
Complete bipartite graphs: K(2, 3).
Not K(3, 3).
Discrete Mathematics I – p. 288/292
![Page 1351: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1351.jpg)
Graphs
G = (V, E) How to test if G is planar?
Subdivision: Let u ⇀ v. Add new node x
Replace u ⇀ v by u ⇀ x ⇀ v
G non-planar ⇒ new graph non-planar
Discrete Mathematics I – p. 289/292
![Page 1352: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1352.jpg)
Graphs
G = (V, E) How to test if G is planar?
Subdivision: Let u ⇀ v.
Add new node x
Replace u ⇀ v by u ⇀ x ⇀ v
G non-planar ⇒ new graph non-planar
Discrete Mathematics I – p. 289/292
![Page 1353: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1353.jpg)
Graphs
G = (V, E) How to test if G is planar?
Subdivision: Let u ⇀ v. Add new node x
Replace u ⇀ v by u ⇀ x ⇀ v
G non-planar ⇒ new graph non-planar
Discrete Mathematics I – p. 289/292
![Page 1354: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1354.jpg)
Graphs
G = (V, E) How to test if G is planar?
Subdivision: Let u ⇀ v. Add new node x
Replace u ⇀ v by u ⇀ x ⇀ v
G non-planar ⇒ new graph non-planar
Discrete Mathematics I – p. 289/292
![Page 1355: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1355.jpg)
Graphs
G = (V, E) How to test if G is planar?
Subdivision: Let u ⇀ v. Add new node x
Replace u ⇀ v by u ⇀ x ⇀ v
G non-planar ⇒ new graph non-planar
Discrete Mathematics I – p. 289/292
![Page 1356: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1356.jpg)
Graphs
Only K(5) and K(3, 3) are “really” non-planar.
Theorem (Kuratowski). A graph is planar iff it has nosubgraph obtained from K(5) or K(3, 3) bysubdivisions.
Proof: difficult.
Discrete Mathematics I – p. 290/292
![Page 1357: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1357.jpg)
Graphs
Only K(5) and K(3, 3) are “really” non-planar.
Theorem (Kuratowski). A graph is planar iff it has nosubgraph obtained from K(5) or K(3, 3) bysubdivisions.
Proof: difficult.
Discrete Mathematics I – p. 290/292
![Page 1358: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1358.jpg)
Graphs
Only K(5) and K(3, 3) are “really” non-planar.
Theorem (Kuratowski). A graph is planar iff it has nosubgraph obtained from K(5) or K(3, 3) bysubdivisions.
Proof: difficult.
Discrete Mathematics I – p. 290/292
![Page 1359: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1359.jpg)
Graphs
Recall: if G = (V, E) a tree, then |V | = |E|+ 1.
Generalisation: G = (V, E) — planar
Drawing of G partitions the plane into faces
Let F be the set of all faces
Discrete Mathematics I – p. 291/292
![Page 1360: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1360.jpg)
Graphs
Recall: if G = (V, E) a tree, then |V | = |E|+ 1.
Generalisation: G = (V, E) — planar
Drawing of G partitions the plane into faces
Let F be the set of all faces
Discrete Mathematics I – p. 291/292
![Page 1361: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1361.jpg)
Graphs
Recall: if G = (V, E) a tree, then |V | = |E|+ 1.
Generalisation: G = (V, E) — planar
Drawing of G partitions the plane into faces
Let F be the set of all faces
Discrete Mathematics I – p. 291/292
![Page 1362: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1362.jpg)
Graphs
Recall: if G = (V, E) a tree, then |V | = |E|+ 1.
Generalisation: G = (V, E) — planar
Drawing of G partitions the plane into faces
Let F be the set of all faces
Discrete Mathematics I – p. 291/292
![Page 1363: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1363.jpg)
Graphs
Examples:
G is a tree: |V | = |E|+ 1 |F | = 1
G has one cycle: |V | = |E| |F | = 2
Theorem (Euler). For any planar graph G,|V | − |E|+ |F | = 2.
Proof: induction.
Discrete Mathematics I – p. 292/292
![Page 1364: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1364.jpg)
Graphs
Examples:
G is a tree: |V | = |E|+ 1 |F | = 1
G has one cycle: |V | = |E| |F | = 2
Theorem (Euler). For any planar graph G,|V | − |E|+ |F | = 2.
Proof: induction.
Discrete Mathematics I – p. 292/292
![Page 1365: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1365.jpg)
Graphs
Examples:
G is a tree: |V | = |E|+ 1 |F | = 1
G has one cycle: |V | = |E| |F | = 2
Theorem (Euler). For any planar graph G,|V | − |E|+ |F | = 2.
Proof: induction.
Discrete Mathematics I – p. 292/292
![Page 1366: Discrete Mathematics I - Computer Science Departmenttiskin/teach/dm1/o.pdf · Discrete maths in depth, highly recommended! ... Discrete Mathematics and its Applications Rosen (McGraw-Hill,](https://reader033.vdocuments.net/reader033/viewer/2022052712/5ae8a8e37f8b9a2904906ed6/html5/thumbnails/1366.jpg)
Graphs
Examples:
G is a tree: |V | = |E|+ 1 |F | = 1
G has one cycle: |V | = |E| |F | = 2
Theorem (Euler). For any planar graph G,|V | − |E|+ |F | = 2.
Proof: induction.
Discrete Mathematics I – p. 292/292