Let T be the statement: The sum of any two rational numbers is rational. Then T is true, but the following "proof is incorrect. Find the mistake. "Proof by contradiction: Suppose not. That is, suppose that the sum of any two rational numbers is not rational. This means that no matter what two rational numbers are chosen their sum is not rational. Now both 1 and 3 are 1/1 and 3 3/1, and so both are ratios of integers with rational because 1 a nonzero denominator. Hence, by a supposition, the sum of 1 and 3, which is 4, is not rational. But 4 is rational because 4 4/1 , which is a ratio of integers with a nonzero denominator. Hence 4 is both rational and not rational, which is a contradiction. This contradiction shows that the supposition is false and hence statement T is true.