1. MAT 2720 Discrete Mathematics Section 3.1 Functions http://myhome.spu.edu/lauw

2. Goals Review and Renew the concept of functions How to show that a function is an One-to- one function (Injection) How to show that a function is an Onto function (Surjection)

3. You Know a Lot About Functions You are supposed to know a lot… Domain, Range, Codomain Inverse Functions One-to-one, Onto Functions Composite Functions

4. Notations

5. From Continuous to Discrete

6. Is this a Function? (I)

7. Is this a Function? (II)

8. One-to-One Functions

9. Equivalent Criteria

10. Example 1 Determine if the given function is 1-1. Prove your answer. Proof:Analysis

11. Example 2 Determine if the given function is 1-1. Prove your answer. Proof:Analysis

12. Onto Functions

13. Equivalent Criteria

14. Example 3 Determine if the given function is onto. Prove your answer. Proof:Analysis

15. Example 4 Determine if the given function is onto. Prove your answer. Proof:Analysis

16. Counting Problems…

18. Bijection

19. Inverse Functions

20. MAT 2720 Discrete Mathematics Section 3.3 Relations Part I http://myhome.spu.edu/lauw

21. Goals Relations Generalize the concept of functions. Digraphs representations. A special type of relations will be studied in section 3.4.

22. This is NOT a Function (3.1) Nevertheless, it gives a “ relation ” between set X and Y.

23. Background: Cartesian Product (1.1)

24. Background: Cartesian Product (1.1) e.g.: The plane is the Cartesian Product of the lines.

25. Background: Cartesian Product (1.1) e.g.: The plane is the Cartesian Product of the lines. Another example:

26. Recall: Subset (1.1) X is a subset of Y if every element of X is also contained in Y. X is a subset of Y if X is a sub-collection of elements in Y. Notation:

27. Example

28. This is NOT a Function (3.1) It make sense to represent this “ relation ” between set X and Y by which is a subset of

29. Definition and Notations A relation from X to Y is a subset Sometimes, we write

30. This is NOT a Function (3.1) Notations:

31. Definition and Notations A relation from X to Y is a subset Sometimes, we write Domain of R = all possible value of x Range of R = all possible value of y

32. This is NOT a Function (3.1) Notations:

33. Example 1 X=Set of all U.S. cities, Y=Set of 50 states

34. Example 2

35. Example 3 X=Set of all U.S. cities

36. Definition and Notations A relation from X to X is called a relation on X

37. Example 4

38. Representation by a Digraph