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Unformatted text preview: 2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p → q . Here are some strategies we have available to try. • Trivial Proof: If we know q is true then p → q is true regardless of the truth value of p . • Vacuous Proof: If p is a conjunction of other hypotheses and we know one or more of these hypotheses is false, then p is false and so p → q is vacuously true regardless of the truth value of q . • Direct Proof: Assume p , and then use the rules of inference, axioms, defi nitions, and logical equivalences to prove q . • Indirect Proof or Proof by Contradiction: Assume p and ¬ q and derive a contradiction r ∧ ¬ r . • Proof by Contrapositive: (Special case of Proof by Contradiction.) Give a direct proof of ¬ q → ¬ p . Assume ¬ q and then use the rules of inference, axioms, definitions, and logical equivalences to prove ¬ p .(Can be thought of as a proof by contradiction in which you assume p and ¬ q and arrive at the contradiction p ∧ ¬ p .) • Proof by Cases: If the hypothesis p can be separated into cases p 1 ∨ p 2 ∨ ··· ∨ p k , prove each of the propositions, p 1 → q , p 2 → q , ..., p k → q , separately. (You may use different methods of proof for different cases.) Discussion We are now getting to the heart of this course: methods you can use to write proofs. Let’s investigate the strategies given above in some detail. 2.2. Trivial Proof/Vacuous Proof. Example 2.2.1 . Prove the statement: If there are 100 students enrolled in this course this semester, then 6 2 = 36 . Proof. The assertion is trivially true, since the conclusion is true, independent of the hypothesis (which, may or may not be true depending on the enrollment). Example 2.2.2 . Prove the statement. If 6 is a prime number, then 6 2 = 30 . 2. METHODS OF PROOF 70 Proof. The hypothesis is false, therefore the statement is vacuously true (even though the conclusion is also false). Discussion The first two methods of proof, the “Trivial Proof” and the “Vacuous Proof” are certainly the easiest when they work. Notice that the form of the “Trivial Proof”, q → ( p → q ), is, in fact, a tautology. This follows from disjunction introduction, since p → q is equivalent to ¬ p ∨ q . Likewise, the “Vacuous Proof” is based on the tautology ¬ p → ( p → q ). Exercise 2.2.1 . Fill in the reasons for the following proof of the tautology ¬ p → ( p → q ) . [ ¬ p → ( p → q )] ⇔ [ p ∨ ( ¬ p ∨ q )] ⇔ [( p ∨ ¬ p ) ∨ q ] ⇔ T ∨ q ⇔ T Exercise 2.2.2 . Let A = { 1 , 2 , 3 } and R = { (2 , 3) , (2 , 1) } ( ⊆ A × A ) . Prove: if a,b,c ∈ A are such that ( a,b ) ∈ R and ( b,c ) ∈ R then ( a,c ) ∈ R ....
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 Fall '08
 PenelopeKirby
 Negative and nonnegative numbers, Natural number, Prime number

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