Introduction to Proofs

1. Introduction to Proofs

2. The use of Reasoning and Logic in proofs • Inductive Reasoning- “reasoning from detailed facts to general principles” • Specific examples and assumes it generally • Deductive Reasoning- “reasoning from the general to the particular” • Generally prove topic, examples come after • Logic- “a system or mode of reasoning; the formal, guiding principles of a discipline.” • If p then q, p if and only if q, etc. • Definitions from www.thefreedictionary.com

3. Some Types/Strategies of Proofs • Direct Proof • Proof by Contradiction • Proof by Contrapositive • If, and Only If • Mathematical Induction • Proof by Exhaustion • Proof by Enumeration • The Pigeon hole Principle • Counterexamples

4. Two-column Proof

5. Direct Proof • If P, then Q. • This is a paragraph proof version of the two-column proof. • Direct proofs start with the hypothesis (assume it’s true), of a statement and deduce that the conclusion is true through a series of direct steps. -Those steps use definitions, theorems, axioms, etc.

6. Proof by Contradiction • This type of proof is meant to prove a proposition is either true or false and not both true and false (that is where the contradiction lies). In other words, assume P and not Q then prove the contradiction. • Need to mention somewhere in the proof that there is a contradiction with what was assumed.

7. If and only If Proofs • Need to prove: P if and only if Q. In other words, prove if P then Q and if Q then P. • Assume the hypothesis is true and prove the conclusion, then assume the conclusion is true and prove the hypothesis is true.

8. Proof by Contrapositive • P implies Q and Not Q implies not P • Example: “If it is my toothbrush, then it is purple.” is equivalent to “If it is not purple, then it is not my toothbrush.” • Assume the conclusion is not true, and then prove the hypothesis is not true also.

9. Proof by Induction • Base Case • Assume P(K) then prove P(k+1). • The stair analysis.

10. Bertrand Russell said… Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing... It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

11. References • http://www.tumblr.com/tagged/proof-that-girls-are-evil • A=B proof picture • http://zimmer.csufresno.edu/~larryc/proofs/proofs.html • www.thefreedictionary.com • Two-column proof image • http://www.personal.kent.edu • http://www-cs-students.stanford.edu • Logic Table Image • Image of Proof by Contradiction • Image for Induction proof • http://www.purplemath.com/modules/inductn3.htm