Proof by contrapositive
In logic, the contrapositive of a conditional statement is formed by negating both terms and reversing the direction of inference. More specifically, the contrapositive of the statement "if "A", then "B"" is "if not "B", then not "A"." A statement and its contrapositive are logically equivalent, in the sense that if the statement is true, then its contrapositive is true and vice versa. In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. In other words, the conclusion "if "A", then  B" is inferred by constructing a proof of the claim if not "B", then not "A"" instead. More often than not, this approach is preferred if the contrapositive is easier to prove than the original conditional statement itself. Logically, the validity of proof by contrapositive can be demonstrated by the use of the following truth table, where it is shown that p" → "q and formula_1q" → "formula_1p share the same truth values in all scenarios: Difference with proof by contradiction. Proof by contradiction: Assume formula_3 is true, formula_4 is false, thus formula_3 is false, and formula_6 is true. Proof  by contrapositive: To prove formula_7, prove its contrapositive statement, which is formula_8. Example. Let formula_9 be an integer. Although a direct proof can be given, we choose to prove this statement by contraposition. The contrapositive of the above statement is: This latter statement can be proven as follows: suppose that "x" is not even, then "x" is odd. The product of two odd numbers is odd, hence formula_14 is odd. Thus formula_10 is not even. Having proved the contrapositive, we can then infer that the original statement is true.