226 Logic as a Tool m2 is therefore even m must therefore be even since, if it is odd, then m2 would be odd (see the example of a direct proof) There is therefore an integer a such that m = 2a Hence, m2 = 4a2 Therefore 2n2 = 4a2 , hence n2 = 2a2 10 n2 is therefore even 11 n must therefore be even, by the same argument as above 12 There is therefore an integer b such that n = 2b = 22ab is therefore not irreducible 13 The fraction m n 14 This contradicts the assumption that m is irreducible n 15 This contradiction completes the proof A variation of proof by contradiction is proof by contraposition This strategy is usually applied when the statement to be proved is of the type “if P , then C ,” that is, there is only one premise1 This proof strategy is based on the fact that the implication “if P , then C ” is logically equivalent to its contrapositive “if ¬C , then ¬P ” If we prove the latter, we therefore have a proof of the former Schematically, the proof by contraposition of “if P , then C ” has the following form Assume ¬C (a sequence of valid inferences) Derive ¬P We can also think of this strategy as obtained from the proof by contradiction strategy, by assuming P together with ¬C at the beginning and then proving ¬P to produce the desired contradiction We usually apply the proof by contraposition when it is easier to prove the contrapositive of the original implication Example 184 Here is a proof by contraposition of the statement if n is an integer such that n2 is even, then n is even Proof Suppose n is not even Therefore, n is odd Then (see the example of a direct proof) n2 is odd Therefore, n2 is not even This completes the proof It can in fact be applied in the general case, as we can always replace all premises by their conjunction