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MAT 3749 Introduction to Analysis

MAT 3749 Introduction to Analysis. Section 0.1 Part I Methods of Proof I. http://myhome.spu.edu/lauw. I cannot see some of the symbols in HW!. Download the trial version of MATH TYPE equation edition from design science http://www.dessci.com/en/products/mathtype/trial.asp. Announcement.

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MAT 3749 Introduction to Analysis

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

  2. I cannot see some of the symbols in HW! • Download the trial version of MATH TYPE equation edition from design science • http://www.dessci.com/en/products/mathtype/trial.asp

  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

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

  12. Direct Proof

  13. Example 2

  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

  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

  23. Contrapositive

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

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