Proof By Contrapositive

Proof By Contrapositive

In logic, the contrapositive of a conditional statement of the form "if A then B" is formed by negating both terms and reversing the direction of inference. Thus, the contrapositive of the statement "if A, then B" is "if not B, then not A." A statement and its contrapositive are logically equivalent: if the statement is true, then its contrapositive is true, and vice versa.

In logic, proof by contrapositive is a form of proof that establishes the truth or validity of a proposition by demonstrating the truth or validity of the converse of its negated parts.

In other words, to prove by contraposition that, prove that .

Any proof by contrapositive can also be trivially formulated in terms of a Proof by contradiction: To prove the proposition, we consider the opposite, . Since we have a proof that, we have which arrives at the contradiction we want. So proof by contrapositive is in some sense "at least as hard to formulate" as proof by contradiction.