CS 173: Discrete Mathematical Structures

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# CS 173: Discrete Mathematical Structures - PowerPoint PPT Presentation

CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: M 11a-12p CS 173 Announcements Homework 3 returned in section this week. Homework 4 available. Due 09/24, 8a. f(x) = -(1/2)x - 25 domain co-domain CS 173 Functions

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### CS 173:Discrete Mathematical Structures

Cinda Heeren

heeren@cs.uiuc.edu

Rm 2213 Siebel Center

Office Hours: M 11a-12p

CS 173 Announcements
• Homework 3 returned in section this week.
• Homework 4 available. Due 09/24, 8a.

Cs173 - Spring 2004

f(x) = -(1/2)x - 25

domain

co-domain

CS 173 Functions

Suppose we have:

And I ask you to describe the yellow function.

What’s a function?

Notation: f: RR, f(x) = -(1/2)x - 25

Cs173 - Spring 2004

CS 173 Functions

Definition: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f.

Cs173 - Spring 2004

B

A

A point!

A collection of points!

CS 173 Functions

Definition: a function f : A  B is a subset of AxB where  a  A, ! b  B and <a,b>  f.

B

A

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

CS 173 Functions

A = {Michael, Tito, Janet, Cindy, Bobby}

B = {Katherine Scruse, Carol Brady, Mother Teresa}

Let f: A  B be defined as f(a) = mother(a).

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

Some say it means codomain, others say, image. Since it’s ambiguous, we don’t use it at all.

f(S) = image(S)

CS 173 Functions - image & preimage

For any set S  A, image(S) = {b : a  S, f(a) = b}

So, image({Michael, Tito}) = {Katherine Scruse} image(A) = B - {Mother Teresa}

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

preimage(S) = f-1(S)

CS 173 Functions - image & preimage

For any S  B, preimage(S) = {a: b  S, f(a) = b}

So, preimage({Carol Brady}) = {Cindy, Bobby} preimage(B) = A

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

 S

CS 173 Functions - image & preimage

What is image(preimage(S))?

• S
• { }
• subset of S
• superset of S
• who knows?

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

Suppose S is {Janet, Cindy}

preimage(image(S)) = A

CS 173 Functions - image & preimage

What is preimage(image(S))?

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

CS 173 Functions - misc. properties
• f() = 
• f({a}) = {f(a)} (this is a definition, actually)
• f(A U B) = f(A) U f(B)
• f(A  B)  f(A)  f(B)

Cs173 - Spring 2004

CS 173 Functions - misc. properties

f(A  B)  f(A)  f(B)?

Choose an arbitrary c  f(A  B), and show that it must also be an element of f(A)  f(B).

f(A  B) = {x : a  (A  B), f(a) = x}

So, a (A  B) such that f(a) = c.

Since a  A, f(a) = c  f(A).

Since a  B, f(a) = c  f(B).

c  f(A), and c  f(B), so c  f(A)  f(B).

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

CS 173 Functions - misc. properties
• f-1() = 
• f-1(A U B) = f-1(A) U f-1(B)
• f-1(A  B) = f-1(A)  f-1(B)

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

Not one-to-one

Every b  B has at most 1 preimage.

CS 173 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

Cs173 - Spring 2004

Michael Tito Janet Cindy Bobby

Katherine Scruse

Mother Teresa

Not onto

Every b  B has at least 1 preimage.

CS 173 Functions - surjection

A function f: A  B is onto (surjective, a surjection) if b  B, a  A f(a) = b

Cs173 - Spring 2004

Isaak Bri Lynette Aidan Evan

Isaak Bri Lynette Aidan Evan

Cinda Dee Deb Katrina Dawn

Cinda Dee Deb Katrina Dawn

Every b  B has exactly 1 preimage.

An important implication of this characteristic:

The preimage (f-1) is a function!

CS 173 Functions - bijection

A function f: A  B is bijective if it is one-to-one and onto.

Cs173 - Spring 2004

yes

yes

yes

CS 173 Functions - examples

Suppose f: R+  R+, f(x) = x2.

Is f one-to-one?

Is f onto?

Is f bijective?

Cs173 - Spring 2004

no

yes

no

CS 173 Functions - examples

Suppose f: R  R+, f(x) = x2.

Is f one-to-one?

Is f onto?

Is f bijective?

Cs173 - Spring 2004

no

no

no

CS 173 Functions - examples

Suppose f: R  R, f(x) = x2.

Is f one-to-one?

Is f onto?

Is f bijective?

Cs173 - Spring 2004

CS 173 Functions - composition

Let f:AB, and g:BC be functions. Then the composition of f and g is:

(g o f)(x) = g(f(x))

Cs173 - Spring 2004

CS 173 Functions - a little problem

Let f:AB, and g:BC be functions.

Prove that if f and g are one to one, then g o f :AC is one to one.

Recall defn of one to one: f:A->B is 1to1 if f(a)=b and f(c)=b --> a=c.

Suppose g(f(x)) = y and g(f(w)) = y. Show that x=w.

f(x) = f(w) since g is 1 to 1.

Then x = w since f is 1 to 1.

Cs173 - Spring 2004

CS 173 Functions - another

Let f:AB, and g:BC be functions.

Prove that if f and g are onto, then g o f :AC is onto.

Cs173 - Spring 2004