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

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

3. Proof by Contradiction Proof by Contrapositive Proof by Contradiction

4. Example 2 AnalysisProof

5. Proof by Contradiction AnalysisProof by Contradiction of If-then Theorem Suppose the negation of the conclusion is true. Find a contradiction. State the conclusion.

6. Proof by Contradiction The method also work with statements other then If P then Q

7. Example 3 AnalysisProof

8. Proof by Cases

9. Example 4 AnalysisProof

10. Proof by Cases AnalysisProof by Cases of If-then Theorem Split the domain of interest into cases. Prove each case separately. State the conclusion. Note that the cases do not have to be mutually exclusive. They just have to cover all elements in the domain.

11. Existence Proofs

12. Example 5 AnalysisProof

13. Existence Proofs AnalysisExistence Proof Prove the statement by exhibiting an element in the domain of interest that satisfies the given conditions. State the conclusion.

14. MAT 2720 Discrete Mathematics Section 2.4 Mathematical Induction http://myhome.spu.edu/lauw

15. Preview Review Mathematical Induction Why? How?

16. The Needs… Theorems involve infinitely many, yet countable, number of statements.

17. Principle of Mathematical Induction (PMI) PMI: It suffices to show 1. P(1) is true. 2. If P(k) is true, then P(k+1) is also true, for all k.

18. Principle of Mathematical Induction (PMI) PMI: It suffices to show 1. P(1) is true. (Basic Step) 2. If P(k) is true, then P(k+1) is also true, for all k (Inductive Step)

19. Format of Solutions In this course, it is extremely important for you to follow the exact solution format of using mathematical induction. Do not skip steps.

20. Format of Solutions In this course, it is extremely important for you to follow the exact solution format of using mathematical induction. Do not skip steps.

21. Example 1 Use mathematical induction to prove that whenever n is a nonnegative integer.

22. Checklist A 0,0,

23. Checklist B

24. Checklist C

26. Checklist D

27. Proof by Mathematical Induction Declare P(n) and the domain of n. Basic Step Write down the statement of the first case. Do not do any simplifications or algebra on the statement of the first case. Explain why it is true. For A=B type, simplify and/ or manipulate each side and see that they are the same. Inductive Step Write down the k-th case. This is the inductive hypothesis. Write down the (k+1)-th case. This is what you need to prove to be true. For A=B type, we usually start form one side of the equation and show that it equals to the other side. In the process, you need to use the inductive hypothesis. Conclude that p(k+1) is true. Make the formal conclusion by quoting the PMI

28. Example 2

30. Example 3