1. Lecture 2.3: Set Theory, and Functions*
CS 250, Discrete Structures, Fall 2011
Nitesh Saxena
*Adopted from previous lectures by Cinda Heeren

2. Lecture 2.3 -- Set Theory, and Functions
Course Admin
HW1 just due
We will provide the solution soon
We will start to grade it
Mid Term 1: Sep 27 (Thurs)
Review Sep 25 (Tues)
Covers Chapter 1 and Chapter 2
HW2 coming out: early next week
Due Oct 2 (Tues)
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Lecture Set Theory, and Functions

3. Lecture 2.3 -- Set Theory, and Functions
Outline
Sets: Inclusion/Exclusion Principle
Functions
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Lecture Set Theory, and Functions

4. A Proof (direct and indirect)
A  B = 
Pv that if (A - B) U (B - A) = (A U B) then
Suppose to the contrary, that A  B  , and that x  A  B.
A U B = 
A = B
A  B = 
A-B = B-A = 
Then x cannot be in A-B and x cannot be in B-A.
Then x is not in (A - B) U (B - A).
But x is in A U B since (A  B)  (A U B).
Thus, A  B = .
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Lecture Set Theory, and Functions

5. Set Theory - Inclusion/Exclusion
Example:
How many people are wearing a watch? a
How many people are wearing sneakers? b
How many people are wearing a watch OR sneakers? a + b
Wrong.
What’s wrong? A B
|A  B| = |A| + |B| - |A  B|
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Lecture Set Theory, and Functions

6. Set Theory - Inclusion/Exclusion
Example:
There are 217 cs majors.
157 are taking cs125.
145 are taking cs173.
98 are taking both.
How many are taking neither?
125
173
217 - ( ) = 13
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Lecture Set Theory, and Functions

7. Set Theory – Generalized Inclusion/Exclusion
Suppose we have: B A C
Now let’s do it for 4 sets!
And I want to know |A U B U C|
kidding.
|A U B U C| = |A| + |B| + |C|
- |A  B| - |A  C| - |B  C|
+ |A  B  C|
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Lecture Set Theory, and Functions

8. Lecture 2.3 -- Set Theory, and Functions
Suppose we have:
-50
-25
And I ask you to describe the yellow function.
What’s a function?
y = f(x) = -(1/2)x - 25
Notation: f: RR, f(x) = -(1/2)x - 25
co-domain
domain
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Lecture Set Theory, and Functions

9. Functions: Definitions
A function f : A B is given by a domain set A, a codomain set B, and a rule which for every element a of A, specifies a unique element f (a) in B
f (a) is called the image of a, while a is called the pre-image of f (a)
The range (or image) of f is defined by
f (A) = {f (a) | a  A }.
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Lecture Set Theory, and Functions

10. Lecture 2.3 -- Set Theory, and Functions
Function or not? A B B A
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Lecture Set Theory, and Functions

11. Lecture 2.3 -- Set Theory, and Functions
Functions: examples
Ex: Let f : Z  R be given by f (x ) = x 2
Q1: What are the domain and co-domain?
Q2: What’s the image of -3 ?
Q3: What are the pre-images of 3, 4?
Q4: What is the range f ?
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Lecture Set Theory, and Functions

12. Lecture 2.3 -- Set Theory, and Functions
Functions: examples
f : Z  R is given by f (x ) = x 2
A1: domain is Z, co-domain is R
A2: image of -3 = f (-3) = 9
A3: pre-images of 3: none as 3 isn’t an integer!
pre-images of 4: -2 and 2
A4: range is the set of perfect squares = {0,1,4,9,16,25,…}
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Lecture Set Theory, and Functions

13. Lecture 2.3 -- Set Theory, and Functions
Functions: examples
A = {Michael, Tito, Janet, Cindy, Bobby}
B = {Katherine Scruse, Carol Brady, Mother Teresa}
Let f: A  B be defined as f(a) = mother(a).
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa
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Lecture Set Theory, and Functions

14. Functions - image set For any set S  A, image(S) = {f(a) : a  S}
image(S) = f(S)
For any set S  A, image(S) = {f(a) : a  S}
So, image({Michael, Tito}) = {Katherine Scruse} image(A) = B - {Mother Teresa}
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa
image(A) is also called range
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Lecture Set Theory, and Functions

15. Functions – preimage set
For any S  B, preimage(S) = {a  A: f(a)  S}
So, preimage({Carol Brady}) = {Cindy, Bobby} preimage(B) = A
preimage(S) = f-1(S)
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa
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Lecture Set Theory, and Functions

16. Functions: images and preimages
Ex: f : Z  R with f (x ) = x 2
Q1: Calculate f –1(3)
Q2: Calculate f –1(4)
Q3: Calculate f ( {-9,-5,-3,0,1,2,3,4} )
Q4: Calculate
f –1({-9,-5,-3,0,0.25,1,2,2.25,3,4})
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Lecture Set Theory, and Functions

17. Functions: images and preimages
Ex: f : Z  R with f (x ) = x 2
A1: f –1(3) = 
A2: f –1(4) = {-2, 2}
A3: f ( {-9,-5,-3,0,1,2,3,4} )
= {81,25,9,0,1,4,16}
A4: f –1({-9,-5,-3,0,0.25,1,2,2.25,3,4})
= {0,-1,1,-2,2}
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Lecture Set Theory, and Functions

18. Every b  B has at most 1 preimage.
Functions - injection
A function f: A  B is one-to-one (injective, an injection) if a,b,c, (f(a) = b  f(c) = b)  a = c
Not one-to-one
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa
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Lecture Set Theory, and Functions

19. Functions - surjection
Every b  B has at least 1 preimage.
Functions - surjection
A function f: A  B is onto (surjective, a surjection) if b  B, a  A, f(a) = b
Not onto
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa
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Lecture Set Theory, and Functions

20. Functions - bijection
A function f: A  B is bijective if it is one-to-one and onto.
Every b  B has exactly 1 preimage.
A B C D A-
Alice Bob Tom Charles Eve
Isaak Bri Lynette Aidan Evan
Cinda Dee Deb Katrina Dawn
An important implication of this characteristic:
The preimage (f-1) is a function!
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Lecture Set Theory, and Functions

21. Lecture 2.3 -- Set Theory, and Functions
Functions - examples
Suppose f: R+  R+, f(x) = x2.
Is f one-to-one?
Is f onto?
Is f bijective?
yes
yes
yes
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Lecture Set Theory, and Functions

22. Lecture 2.3 -- Set Theory, and Functions
Functions - examples
Suppose f: R  R+, f(x) = x2.
Is f one-to-one?
Is f onto?
Is f bijective? no
yes no
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Lecture Set Theory, and Functions

23. Lecture 2.3 -- Set Theory, and Functions
Functions - examples
Suppose f: R  R, f(x) = x2.
Is f one-to-one?
Is f onto?
Is f bijective? no no no
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Lecture Set Theory, and Functions

24. Lecture 2.3 -- Set Theory, and Functions
Functions - examples
Q: Which of the following are 1-to-1, onto, a bijection? If f is invertible, what is its inverse?
f : Z  R is given by f (x ) = x 2
f : Z  R is given by f (x ) = 2x
f : R  R is given by f (x ) = x 3
f : Z  N is given by f (x ) = |x |
f : {people}  {people} is given by
f (x ) = the father of x.
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Lecture Set Theory, and Functions

25. Lecture 2.3 -- Set Theory, and Functions
Functions - examples
f : Z  R, f (x ) = x 2: none
f : Z  Z, f (x ) = 2x : 1-1
f : R  R, f (x ) = x 3: 1-1, onto, bijection, inverse is f (x ) = x (1/3)
f : Z  N, f (x ) = |x |: onto
f (x ) = the father of x : none
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Lecture Set Theory, and Functions

26. Lecture 2.3 -- Set Theory, and Functions
Today’s Reading
Rosen 2.3 and 2.4
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Lecture Set Theory, and Functions