1. MAT 3100 Introduction to Proof
03 Methods of Proof I

2. I cannot see some of the symbols in PPT and HW!
Download the trial version of MATH TYPE equation edition from design science

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

4. Recall: 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

5. Background: Common Symbols
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
Analysis
Proof

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

11. 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.

12. Example 2
Analysis
proof

13. Pause Here
Classwork

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
Analysis
The 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. Recall: Negation
Statement: n is odd Negation of the statement: n is not odd Or: n is even

20. Recall: Negation
Notations

21. Contrapositive
The contrapositive form of is

22. Example 4
Analysis
Proof: We prove the contrapositive:

23. Contrapositive Analysis Proof 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. Group Explorations 03 Very fun to do.
Keep your voices down…you do not want to spoil the fun for the other groups.