Contrapositive
The contrapositive of "If P, then Q" is "If not Q, then not P," which is logically equivalent to the original statement.
Formula
\text{Contrapositive of } P \to Q \text{ is } \neg Q \to \neg P
Definition
The contrapositive of "If P, then Q" is "If not Q, then not P": you negate both parts and flip the order. Formally, the contrapositive of $P \to Q$ is $\neg Q \to \neg P$, and it is logically equivalent to $P \to Q$ (same truth table by classical propositional logic), making proof by contrapositive a valid technique: to prove $P \to Q$, assume $\neg Q$ and prove $\neg P$. In intuitionistic logic, however, this equivalence does not hold generally, the contrapositive is weaker than the original, since the classical equivalence relies on the law of excluded middle, which constructive mathematics rejects.
Example
"If it rains, the ground is wet" has the equally true contrapositive "If the ground is not wet, then it did not rain." Proving "If $n^2$ is even, then $n$ is even" by contrapositive means proving "If $n$ is odd, then $n^2$ is odd": if $n = 2k+1$, then $n^2 = 4k^2+4k+1 = 2(2k^2+2k)+1$, which is odd. The classic proof that $\sqrt{2}$ is irrational uses this technique: assuming $\sqrt{2} = p/q$ in lowest terms leads to a contradiction via a parity argument on $2 = p^2/q^2$.
Key Insight
The contrapositive is always logically equivalent to the original, so proving the contrapositive is a powerful strategy when the original form is hard to work with directly; it is especially useful in number theory and analysis, where the "not Q" assumption often has concrete algebraic consequences. Constructive (intuitionistic) mathematics, which rejects the law of excluded middle, treats the original and its contrapositive as distinct proof obligations.