321 section, week 3 natalie linnell. functions a function from a to b is an assignment of exactly...
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![Page 1: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/1.jpg)
321 Section, Week 3
Natalie Linnell
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Functions• A function from A to B is an assignment of
exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by f to the element a of A. If f is a function from A to B, we write f: A→B. We say that f maps A to B
• Domain: A• Codomain: B• If f(a) = b, b is the image of a, a is the preimage of b
• Range of f is the set of all images of elements of A
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Identify the domain, codomain, range, image of a, preimage of 2
for the following A B
a
b
c
d
e
1
2
3
4
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What does it mean for a function to be one-to-one (aka injective)?
• Use a picture
F(a) = f(a’)->a=a’
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What does it mean for a function to be onto (aka surjective)?
Every b has an a f(a)=b Mention that a bijection is both
![Page 6: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/6.jpg)
Is this -a function?-one-to-one?-onto?
a
b
c
d
e
1
2
3
4
![Page 7: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/7.jpg)
Is this -a function?-one-to-one?-onto?
a
b
c
d
e
1
2
3
4
5
6
![Page 8: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/8.jpg)
What’s an inverse?
F(a) = b -> f-1(b) = a
![Page 9: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/9.jpg)
What’s the inverse of
• x
• x2
![Page 10: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/10.jpg)
What kinds of functions have inverses?
bijections
![Page 11: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/11.jpg)
What’s the composition of two functions?
• Draw a picture
Fog(a) = f(g(a))
![Page 12: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/12.jpg)
A few more things that you probably already know
• Increasing
• Strictly increasing
• Decreasing
• Strictly decreasing
• Product
• Sum
• Ceiling function
• Floor function
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Homework 1
• Associativity (parentheses matter!)
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HW1
• Proof style• Only if
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Fallacies
• Affirming the conclusion
• Denying the hypothesis
P->q, q, therefore p;;p->q, -p, therefore -q
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Machine representation of sets
• Store the set somewhere in a given order
• Represent a subset by a sequence of zeros and ones that express subset membership
• U = {1,2,3,4,5,6,7,8,9,10}
• Subset {1,2,3} = 1110000000
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Represent subsets of {1,2,3,4,5,6,7,8,9,10}
• Odd numbers
• Even numbers
• How do you find the complement of a subset given its binary representation?
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Union and intersection
• How would you compute the union of
1010101010 and 1111100000?
How would you compute the intersection of
1010101010 and 1111100000
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Use a direct proof to prove that the product of two rational numbers is
rational
![Page 20: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/20.jpg)
(AUB) (AUBUC)
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(B-A)U(C-A)=(BUC)-A
![Page 22: 321 Section, Week 3 Natalie Linnell. Functions A function from A to B is an assignment of exactly one element of B to each element of A. We write f(a)](https://reader035.vdocuments.net/reader035/viewer/2022070401/56649f155503460f94c2b3f7/html5/thumbnails/22.jpg)
Use a direct proof to show that every odd integer is the difference
of two squares
K+1, k; 1.6 #7