1. MAT 3749 Introduction to Analysis Section 0.1 Part I Methods of Proof I http://myhome.spu.edu/lauw

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3. Announcement Major applications and OMH 202

4. Preview of Reviews Set up common notations. Direct Proof Counterexamples Indirect Proofs - Contrapositive

5. Background: Common Symbols and Set Notations Integers This is an example of a Set : a collection of distinct unordered objects. Members of a set are called elements. 2,3 are elements of, we use the notations

6. Background: Common Symbols Implication Example

7. Goals We will look at how to prove or disprove Theorems of the following type: Direct Proofs Indirect proofs

8. Theorems Example:

9. Theorems Example: Underlying assumption:

10. Example 1 AnalysisProof

11. Note Keep your analysis for your HW Do not type or submit the analysis

12. Direct Proof Direct Proof of If-then Theorem Restate the hypothesis of the result. Restate the conclusion of the result. Unravel the definitions, working forward from the beginning of the proof and backward from the end of the proof. Figure out what you know and what you need. Try to forge a link between the two halves of your argument.

13. Example 2 Analysisproof

14. Counterexamples To disprove we simply need to find one number x in the domain of discourse that makes false. Such a value of x is called a counterexample

15. Example 3 AnalysisThe statement is false

16. Example 4

17. Indirect Proof: Contrapositive To prove we can prove the equivalent statement in contrapositive form : or

18. Rationale Why?

19. Background: Negation Statement: n is odd Negation of the statement: n is not odd Or: n is even

20. Background: Negation Notations Note:

21. Contrapositive The contrapositive form of is

22. Example 4 AnalysisProof: We prove the contrapositive:

23. Contrapositive AnalysisProof by Contrapositive of If-then Theorem Restate the statement in its equivalent contrapositive form. Use direct proof on the contrapositive form. State the origin statement as the conclusion.

24. Classwork Very fun to do. Keep your voices down…you do not want to spoil the fun for the other groups.