1. MAT 2720 Discrete Mathematics Section 2.1 Mathematical Systems, Direct proofs, and Counterexamples http://myhome.spu.edu/lauw

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4. Preview Set up common notations. Direct Proof Counterexamples

5. Implication Example

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

7. Theorems Example:

8. Theorems Example: Underlying assumption:

9. Example 1 AnalysisProof

10. Direct Proof 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.

11. Example 2 Analysisproof

12. Example 3 AnalysisProof

13. 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

14. Example 4 AnalysisThe statement is false

15. MAT 2720 Discrete Mathematics Section 2.2 More Methods of Proof Part I http://myhome.spu.edu/lauw

16. Goals Indirect Proofs Contrapositive Contradiction Proof by Contrapositive is considered as a special case of proof by contradiction Proof by cases

17. Example 1

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

19. Rationale Why?

20. Background: Negation (1.2) Statement: n is odd Negation of the statement: n is not odd Or: n is even

21. Background: Negation (1.2) Notations Note:

22. Contrapositive (1.3) The contrapositive form of is

23. Example 1 AnalysisProof: We prove the contrapositive:

24. 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.