CS 23022 Discrete Mathematical Structures

1. CS 23022Discrete Mathematical Structures Mehdi Ghayoumi MSB rm 132 mghayoum@kent.edu Ofchr: Thur, 9:30-11:30a

2. Announcements Homework 3 available. Due 06/19, 8a. Solutions are Available at 8.01 a Midterm 1: Monday,June-30, location 121 rm.

3. Functions

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

5. Functions Suppose we have: And I ask you to describe the yellow function. What’s a function? f(x) = x +20

6. Functions Suppose we have: And I ask you to describe the yellow function. What’s a function? f(x) = x -20

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

8. Functions Suppose we have: And I ask you to describe the yellow function. What’s a function? f(x) = -x+20

9. Functions Suppose we have: And I ask you to describe the yellow function. What’s a function? f(x) = -x-20

10. Michael Tito Janet Cindy Bobby Katherine Carol Brady Mother Teresa Functions A = {Michael, Tito, Janet, Cindy, Bobby} B = {Katherine, Carol Brady, Mother Teresa} Let f: A  B be defined as f(a) = mother(a).

11. Michael Tito Janet Cindy Bobby Katherine Carol Brady Mother Teresa Some say it means codomain, others say, image. Functions - image & preimage For any set S  A, image(S) = {b : a  S, f(a) = b} So, image({Michael, Tito}) = {Katherine } image(A) = B - {Mother Teresa} image(S) = f(S)

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

13. Michael Tito Janet Cindy Bobby Katherine Carol Brady Mother Teresa Suppose S is {Janet, Cindy} preimage(image(S)) = A Functions - image & pre image What is preimage(image(S))?

14. 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 one-to-one Every b  B has at most 1 preimage.

15. 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 one-to-one Every b  B has at most 1 preimage.

16. Functions – Not-injection Not one-to-one

17. .  Functions - surjective • A function f : A → B is said to surjective or onto if the range of f is equal to B(co-domain), i.e. every element of B has its pre-image in A under the given function f. • So ∀ b ∈ B ∃ a ∈ A, such that f(a) =b onto

18. Functions - injection not onto

19. Functions - bijection A function f: A  B is bijective if it is one-to-one and onto. Every b  B has exactly 1 preimage.

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

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

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

23. 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))

24. Functions - composition

25. Functions - composition

26. Functions - composition Let f = {(1,3), (2,1), (3,4), (4,6)} and let g = {(1,5), (2,3), (3,4), (4,1), (5,3), (6,2)}. Then g o f = {(1,4), (2,5), (3,1), (4,2)}. Given f(x) = 2x + 3 and g(x) = –x2 + 5…. find ( f o g)(x). (f o g)(x) = f (g(x))     = f (–x2 + 5)     = 2(             ) + 3     ... setting up to insert the input formula     = 2(–x2 + 5) + 3     = –2x2 + 10 + 3     = –2x2 + 13