1. Discrete Mathematical

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

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

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

5. 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).
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa

6. Functions - image & preimage
What about the range?
image(S) = f(S)
For any set S  A, image(S) = {b : a  S, f(a) = b} So, image({Michael, Tito}) = {Katherine Scruse} image(A) = B - {Mother Teresa}
Some say it means codomain, others say, image. Since it’s ambiguous, we don’t use it at all.
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa

7. Functions - image & preimage
preimage(S) = f-1(S)
For any S  B, preimage(S) = {a: b  S, f(a) = b} So, preimage({Carol Brady}) = {Cindy, Bobby} preimage(B) = A
Michael Tito Janet Cindy Bobby
Katherine Scruse
Carol Brady
Mother Teresa

8. 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

9. 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

10. Functions - bijection
A function f: A  B is bijective if it is one-to-one and onto.
Every b  B has exactly 1 preimage.
Isaak Bri Lynette Aidan Evan
Cinda Dee Deb Katrina Dawn
Isaak Bri Lynette Aidan Evan
Cinda Dee Deb Katrina Dawn
An important implication of this characteristic:
The preimage (f-1) is a function!

11. Functions - examples
Suppose f: R+  R+, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective?
yes
yes
yes

12. Functions - examples
Suppose f: R  R+, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective? no
yes no

13. Functions - examples
Suppose f: R  R, f(x) = x2. Is f one-to-one? Is f onto? Is f bijective? no no no

14. 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))