Equivalence relations and If Then statements

# Equivalence relations and If Then statements - Math 100...

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Math 100 Logic, Equivalence Classes Martin H. Weissman 1. If...Then... statements Sometimes, you are asked to prove whether a sentence of the form “If A then B” is true or false. In mathematics, this has a precise logical meaning. The truth of the sentence “If A then B” depends on whether A and B are true or false – the dependence is illustrated by the following truth table: A B If A then B. True True True True False False False True True False False True So, if you are asked to prove that “if A then B”, then you have two ways of approaching this proof: You can take A as a hypothesis in your proof, and try to prove B from it. You can assume that A is true, and assume that B is false, and try to deduce a contradiction. For example, consider the structure of the following proof: Theorem 1. If n is a natural number, and n 2 is even, then n is even. Proof. We use proof by contradiction. Assume that n is a natural number, n 2 is even, and n is odd. Since n is odd, n 2 is odd. This contradicts the assumption that n 2 is even. / 2. Trivial Satisfaction Perhaps the least intuitive property of “If...then...” sentences is the following:

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