By Geoffrey C. Grabowski, Richard E. Dansky

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**Example text**

Let P be a formula of predicate logic which contains one or more predicate variables. P is said to be a tautology if it is true whenever all the predicate variables that it contains are replaced by actual predicates. Two formulas P and Q are said to be logically equivalent if P ↔ Q is a tautology, that is if P and Q always have the same truth value when the predicate variables they contain are replaced by actual predicates. The notation P ≡ Q asserts that P is logically equivalent to Q. Exercises 1.

Today, it is Sunday and it’s not raining. Therefore, it must be snowing. b) If there are anchovies on the pizza, Jack won’t eat it. If Jack doesn’t eat pizza, he gets angry. Jack is angry. Therefore, there were anchovies on the pizza. c) At 8:00, Jane studies in the library or works at home. It’s 8:00 and Jane is not studying in the library. So she must be working at home. 6 Proof Mathematics is unique in that it claims a certainty that is beyond all possible doubt or argument. A mathematical proof shows how some result follows by logic alone from a given set of assumptions, and once the result has been proven, it is as solid as the foundations of logic themselves.

We will prove that for all integers n, if n is even then n2 is even. (Definition: an integer n is even iff n = 2k for some integer k. ) Proof. ” We need to show that P (n) → Q(n) is true for all values of n. 5, we need to show that for any n, P (n) logically implies Q(n); or, equivalently, that Q(n) can be logically deduced from P (n); or, equivalently, that n is even ∴ n2 is even 48 CHAPTER 1. LOGIC AND PROOF is a valid argument. Here is a formal proof that n is even ∴ n2 is even is in fact a valid argument for any value of n: Let n be an arbitrary integer.