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MAT 3749 Introduction to Analysis Section 1.3 Part I Countable Sets

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Presentation on theme: "MAT 3749 Introduction to Analysis Section 1.3 Part I Countable Sets"— Presentation transcript:

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…

14

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

24

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26 Corollary (HW)

27 Theorem

28 Proof Outline

29

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