class2 sets

7/27/2019 Class2 Sets

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Instructor:Instructor: NirmalNirmal GuptaGupta

Instructor InInstructor In--Charge: Dr.Charge: Dr. MukeshMukesh RohilRohil


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TEXT BOOK:TEXT BOOK:

T1: Mott ,T1: Mott , KandelKandel, & Baker : Discrete Mathematics for Computer, & Baker : Discrete Mathematics for Computer, , , ., , , .

REFERENCE BOOKS:REFERENCE BOOKS:

: u: emen s o scre e a ema cs, c raw , e,: u: emen s o scre e a ema cs, c raw , e,19851985

R2: K H Rosen: Discrete Mathematics & its Applications, TMH, 6e,R2: K H Rosen: Discrete Mathematics & its Applications, TMH, 6e,..

CMT forCMT for all kinds of course information and alsoall kinds of course information and also lecturelecture slides inslides inorma sorma s, up a e, up a e a era er eaceac sess on.sess on.


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Test 1 (25Feb)Test 1 (25Feb) 40 Marks40 Marks

Quiz (4Feb)Quiz (4Feb) 40 Marks40 Marks

Project/AssignmentsProject/Assignments 40 Marks40 Marks

ComprehensiveComprehensive 80 Marks80 Marks


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(bits).(bits).

Therefore both a com uter sTherefore both a com uter s

structure (circuits)structure (circuits) andand

can be described by discrete math.can be described by discrete math.


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Introduction & OverviewIntroduction & Overview

MathematicalMathematical InductionInduction

GraphsGraphs

Boolean AlgebraBoolean Algebra

rouproup


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Set: Collection of objects ( elements )Set: Collection of objects ( elements )

aa AA a is an element of Aa is an element of Aa is a member of Aa is a member of A

aa AA a is not an element of Aa is not an element of A

= a= a11, a, a22, , a, , ann con a nscon a ns

Order of elements is meaninglessOrder of elements is meaningless

It does not matter how often the same elementIt does not matter how often the same elementis listed.is listed.


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Sets A and B are equal if and only if they containSets A and B are equal if and only if they contain..

Examples:Examples:

A = {9, 2, 7,A = {9, 2, 7, --3}, B = {7, 9,3}, B = {7, 9, --3, 2} :3, 2} : A = BA = B

= , , ,= , , ,B = {cat, horse, squirrel, dog} :B = {cat, horse, squirrel, dog} : AA BB

= og, ca , orse ,= og, ca , orse ,B = {cat, horse, dog, dog} :B = {cat, horse, dog, dog} : A = BA = B


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Standard Sets:Standard Sets:

Natural numbersNatural numbers NN = {0, 1, 2, 3, }= {0, 1, 2, 3, }

-- --,, ,, , , , ,, , , ,Positive IntegersPositive Integers ZZ++ = {1, 2, 3, 4, }= {1, 2, 3, 4, }

Real NumbersReal Numbers RR = {47.3,= {47.3, --12,12, , }, }

Rational NumbersRational Numbers QQ = {1.5, 2.6,= {1.5, 2.6, --3.8, 15, }3.8, 15, }(correct definition will follow)(correct definition will follow)


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A =A = empty set/null setempty set/null set

== ,,

A = {{b, c}, {c, x, d}}A = {{b, c}, {c, x, d}}

A = {{x, y}}A = {{x, y}}Note: {x, y}Note: {x, y} A, but {x, y}A, but {x, y} {{x, y}}{{x, y}}

A = {x | P(x)}A = {x | P(x)}set of all x such that P(x)set of all x such that P(x)

A = {x | xA = {x | x NN x > 7} = {8, 9, 10, }x > 7} = {8, 9, 10, }

set builder notationset builder notation


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We are now able to define the set of rationalWe are now able to define the set of rationalnumbers Q:numbers Q:

QQ = {a/b | a= {a/b | a ZZ bb ZZ++}}

ororQQ = {a/b | a= {a/b | a ZZ bb ZZ bb 0}0}

And how about the set of real numbers R?And how about the set of real numbers R?

RR = {r | r is a real number}= {r | r is a real number}

That is the best we can do.That is the best we can do.


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AA BB A is a subset of BA is a subset of BAA B if and only if every element of A is alsoB if and only if every element of A is also

an e ement o .an e ement o .

We can completely formalize this:We can completely formalize this:

x xx x xx

Exam les:Exam les:

A = {3, 9}, B = {5, 9, 1, 3}, AA = {3, 9}, B = {5, 9, 1, 3}, A B ?B ? truetrue

A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, AA = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A B ?B ?

falsefalse

truetrue

A = 1 2 3 B = 2 3 4 AA = 1 2 3 B = 2 3 4 A B ?B ?


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Useful rules:Useful rules:A = BA = B (A(A B)B) (B(B A)A)

(A(A B)B) (B(B C)C) AA CC (see Venn Diagram)(see Venn Diagram)

AABB

CC


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Useful rules:Useful rules:

AA A for any set AA for any set A

Proper subsets:Proper subsets:

AA BB x (x (xx AA xx BB)) x (x (xx BB xx AA))

oror

AA BB x (x (xx AA xx BB)) x (x (xx BB xx AA))


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If a set S contains n distinct elements, nIf a set S contains n distinct elements, n NN,,we call S awe call S a finite setfinite set withwith cardinalit ncardinalit n..

Examples:Examples:

A = {Mercedes, BMW, Porsche}, |A| = 3A = {Mercedes, BMW, Porsche}, |A| = 3

B = 1, 2, 3 , 4, 5 , 6B = 1, 2, 3 , 4, 5 , 6 B = 4B = 4

C =C = | C| = 0| C| = 0

== ==

E = { xE = { x NN | x| x 7000 }7000 } E is inf init e!E is inf init e!


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P(A)P(A) power set of Apower set of AP A = B BP A = B B AA contains all subsets of Acontains all subsets of A

Examples:Examples:

A = {x, y, z}A = {x, y, z}

==

A =A =

P(A) = {P(A) = { }}

Note: |A| = 0, |P(A)| = 1Note: |A| = 0, |P(A)| = 1


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Cardinality of power sets:Cardinality of power sets:

| P(A) | = 2| P(A) | = 2|A||A|

Imagine each element in A has anImagine each element in A has an onon//offoff

switch

switch

Each possible switch configuration in A correspondsEach possible switch configuration in A corresponds

to one element in 2to one element in 28877665544332211AA

yyyyyyyyyyyyyyyyyy

xxxxxxxxxxxxxxxxxx

zzzzzzzzzzzzzzzzzz

For 3 element s in A, t her e ar eFor 3 element s in A, t her e ar e= e emen s n= e emen s n


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TheThe ordered nordered n--tupletuple (a(a11, a, a

22, a, a

33, , a, , a

nn) is an) is an orderedordered

collectioncollection of objects.of objects.

Two ordered nTwo ordered n--tuples (atuples (a11, a, a22, a, a33, , a, , ann) and) and

(b(b11, b, b22, b, b33, , b, , bnn) are equal if and only if they) are equal if and only if they

contain exactly the same elementscontain exactly the same elements in the samein the sameorderorder, i.e. a, i.e. aii = b= bii for 1for 1 ii n.n.

TheThe Cartesian productCartesian product of two sets is defined as:of two sets is defined as:

== ,,Example:Example: A = {x, y}, B = {a, b, c}A = {x, y}, B = {a, b, c}

, , , , , , , , , , ,, , , , , , , , , , ,


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TheThe Cartesian productCartesian product of two sets is defined as:of two sets is defined as:AA B = {(a, b) | aB = {(a, b) | a AA bb B}B}

Example:Example:

= == =, , ,, , ,

== (good, student),(good, student), (good, prof),(good, prof), (bad, student),(bad, student), (bad, prof)(bad, prof)

, ,, , , ,, , , ,, , ,,==


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Note that:Note that:AA ==

A =A =

--|A|A B| = |A|B| = |A| |B||B|

The Cartesian product ofThe Cartesian product of two or more setstwo or more sets isis

defined as:defined as:

AA11 AA22 AAnn = {(a= {(a11, a, a22, , a, , ann) | a) | aii A for 1A for 1 ii n}n}

class2 sets

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