1. MAT 3749 Introduction to Analysis Section 1.3 Part I Countable Sets 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) Countable and Uncountable Sets

3. References Section 1.3 Howland, Appendices A-C

4. 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

5. Is this a Function? (I)

6. Is this a Function? (II)

7. One-to-One Functions

8. Equivalent Criteria

9. Example 1 Determine if the given function is injective. Prove your answer.

10. Onto Functions

11. Equivalent Criteria

12. Example 2 Determine if the given function is surjective. Prove your answer.

13. Counting Problems…

15. Bijections

16. Inverse Functions

17. Equivalent Sets

18. Example 3 The set of odd integers (O) and even integers (E) are equivalent. Plan: 1. Define a function from O to E. 2. Show that the function is well defined. 3. Show that the function is bijective.

19. Countable Sets

20. Remark

21. Theorem

22. Analysis

23. Proof

26. Corollary (HW)

27. Theorem

28. Proof Outline