Contrapositive
Calculus & Advanced MathThe 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.
Example
Original: "If it rains, the ground is wet." Contrapositive: "If the ground is not wet, then it did not rain." This is just as true as the original.
Key Insight
The contrapositive is always logically equivalent to the original. Proving the contrapositive is a powerful strategy when the original form is hard to work with directly.
Definition
The contrapositive of $P \to Q$ is $\neg Q \to \neg P$, and it is logically equivalent to $P \to Q$ (same truth table). This makes proof by contrapositive a valid proof technique: to prove $P \to Q$, assume $\neg Q$ and prove $\neg P$.
Example
Prove: "If $n^2$ is even, then $n$ is even." Contrapositive: "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. QED.
Key Insight
Proof by contrapositive is especially useful in number theory and analysis, where the "not Q" assumption often has concrete algebraic consequences.
Definition
$P \to Q \equiv \neg Q \to \neg P$ by classical propositional logic (verified by truth table or by the equivalence $P \to Q \equiv \neg P \vee Q = \neg Q \to \neg P$). In intuitionistic logic, this equivalence does NOT hold generally: the contrapositive is weaker than the original.
Example
Proof by contrapositive of the irrationality of $\sqrt{2}$: equivalent to "if $\sqrt{2} = p/q$ in lowest terms, then $2 = p^2/q^2$ is a contradiction" proven by the well-known parity argument.
Key Insight
The classical equivalence of a statement and its contrapositive relies on the law of excluded middle. Constructive (intuitionistic) mathematics, which rejects excluded middle, treats these as distinct proof obligations.