Applications: Mathematical Proofs and Automated Reasoning 225 The scheme of a proof by contradiction is of the following form Assume P1 , , Pn Assume ¬C (a sequence of valid inferences) Derive both A and ¬A for some proposition A Conclude that ¬C cannot be true We therefore conclude C Example 182 Here is an example of a proof by contradiction of the statement: If x and y are integers, x is odd and xy is even, then y is even Proof Suppose x, y are integers, x is odd, and xy is even Suppose also that y is not even Therefore, y is odd Then there is an integer k such that x = 2k + 1, since x is odd There is also an integer m such that y = 2m + 1, since y is odd Therefore, xy = (2k + 1)(2m + 1) = = 2(2km + k + m) + Therefore, xy is odd On the other hand, we have assumed that xy is even and, therefore, xy is not odd xy is therefore odd and xy is not odd 10 This is a contradiction, which completes the proof Example 183 Euclid’s proof by contradiction that √ is irrational Proof √ Suppose it is not the case that is irrational √ Therefore, is rational √ can therefore be represented as√an irreducible fraction, that is, there are relatively prime integers m and n such that = m n Therefore, = m2 , n2 hence 2n2 = m2