1. 1 Discrete Structures – CNS 2300 Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 1 The Foundations: Logic, Sets, and Functions

2. 2 Section 1.8 Functions

3. 3 What is a Function? Let A and B be sets. 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 the function f to the element a of A. If f is a function from A to B, we write

4. 4 Domain and Codomain If f is a function from A to B, we say that A is the domain of f and B is the codomain of f. 1234512345 abcdabcd f =f(2) DomainCodomain

5. 5 Domain and Codomain If f is a function from A to B, we say that A is the domain of f and B is the codomain of f. 1234512345 abcdabcd f =f(2) Domain Codomain 2 is the pre- image of b b is the image of 2

6. 6 Domain and Range DomainCodomain 12341234 1234567812345678 f(x) = 2x

7. 7 Domain and Range DomainCodomain 12341234 1234567812345678 f(x) = 2x Range

8. 8 Let f 1 and f 2 be functions from A to R. Then f 1 + f 2 and f 1 f 2 are also functions from A to R defined by (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) (f 1 f 2 )(x) = f 1 (x) f 2 (x)

9. 9 Let f be a function from the set A to the set B and let S be a subset of A. The image of S is the subset of B that consists of the images of the elements of S. We denote the image of S by f(s), so that

10. 10 One-to-one Functions A function f is said to be one-to-one, or injective, if and only if f(x) = f(y) implies that x = y for all x and y in the domain of f. A function is said to be an injection if it is one- to-one.

11. 11 One-to-One Functions a b c x y z

12. 12 NOT One-to-one functions a b c x y z

13. 13 One-to-one functions???? f(x) = 3x+5 f(x) = x 2 - 1 f(x) = x 3 f(x) = |x-1|

14. 14 Strictly Increasing Functions A function f whose domain and codomain are subsets of the set of real numbers is called strictly increasing if f (a) f (b) whenever a<b and a and b are in the domain of f.

15. 15 Strictly Increasing Function

16. 16 Strictly Decreasing Function

17. 17 Neither

18. 18 Onto Functions A function f from A to B is called onto, or surjective, if and only if for every element there is an element with f(a)=b. A function f is called a surjection if it is onto.

19. 19 Onto Function a b c x y z d Every element in the co- domain has something mapped to it.

20. 20 NOT an Onto Function a b c x y z d There exists an element in the co- domain that does not have anything mapped to it.

21. 21 Onto functions???? f(x) = 3x+5 f(x) = x 2 - 1 f(x) = x 3 f(x) = |x-1|

22. 22 One-to-One Correspondence The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto.

23. 23 Bijections???? f(x) = 3x+5 f(x) = x 2 - 1 f(x) = x 3 f(x) = |x-1|

24. 24 Inverse Functions Let f be a one-to-one correspondence (bijection) from the set A to the set B. The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f(a) = b. The inverse function of f is denoted by f -1. Hence f -1 (b) = a when f(a) = b.

25. 25 Inverse Functions 12341234 35793579 f(x) = 2x+1 f(1) f(2) f(3) f(4) f -1 (3) f -1 (5) f -1 (7) f -1 (9)

26. 26 Invertible Functions A one-to-one correspondence is called invertible since an inverse function can be defined for the function. f(x) = 2x + 1 f(x) = x 2 + 1 f(x) = x 3

27. 27 Composition of Functions Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The composition of the functions f and g, denoted by is defined by

28. 28 Composition of functions 2 f(x)=2x+1 5 g(x)=x 2 25 g(f(x)) = g(2x+1) = (2x+1) 2

29. 29 Some Important Functions Ceiling Function Floor Function

30. 30 Problems from the text Page 108-111 3, 5, 9, 15, 17, 23, 29, 35, 50, 53, 59 Homework will not be collected. However, you should do enough problems to feel comfortable with the concepts. For these sections the following problems are suggested.

31. 31 finished