MAT 3100 Introduction to Proof

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Presentation transcript:

MAT 3100 Introduction to Proof 03 Methods of Proof I http://myhome.spu.edu/lauw

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

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

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

Background: Common Symbols Implication Example

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

Theorems Example:

Theorems Example: Underlying assumption:

Example 1 Analysis Proof

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

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.

Example 2 Analysis proof

Pause Here Classwork

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

Example 3 Analysis The statement is false

Example 4

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

Rationale Why?

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

Recall: Negation Notations

Contrapositive The contrapositive form of is

Example 4 Analysis Proof: We prove the contrapositive:

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.

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