contrapositive

The contrapositive of a conditional statement is a logically equivalent form obtained by reversing and negating the components. If p → q is a statement, its contrapositive is ¬q → ¬p (If not q, then not p). In classical logic, p → q and ¬q → ¬p always share the same truth value, so they are logically equivalent.

This contrasts with the inverse and the converse. The inverse of p → q is ¬p → ¬q, and the converse is q → p. Neither the inverse nor the converse generally preserves the truth of the original statement, though the contrapositive does.

Examples help illustrate the idea. Consider: If it rains, the ground gets wet (p → q). The contrapositive is: If the ground is not wet, then it did not rain (¬q → ¬p). Another example: If a number is even, then it is divisible by 2 (p → q). The contrapositive is: If a number is not divisible by 2, then it is not even (¬q → ¬p).

In proofs, proving p → q by contrapositive can be advantageous. If it is easier to show ¬q → ¬p than to prove p → q directly, one may establish the contrapositive and thereby prove the original statement. A common instance is: to show that odd n implies odd n^2, prove the contrapositive: if n^2 is even, then n is even.

Overall, the contrapositive is a core tool in logic and mathematics, enabling alternative routes to establishing implications.