Proof by contrapositive

In logic, the contrapositive of a conditional statement is formed by negating both terms and reversing the direction of inference. Explicitly, 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.^[1]

In mathematics, proof by contraposition is a rule of inference used in proofs. This rule infers a conditional statement from its contrapositive.^[2] In other words, the conclusion "if A, then B" is drawn from the single premise "if not B, then not A."

Example

Let x be an integer.

To prove: If x² is even, then x is even.

Although a direct proof can be given, we choose to prove this statement by contraposition. The contrapositive of the above statement is:

If x is not even, then x² is not even.

This latter statement can be proven as follows. Suppose x is not even. Then x is odd. The product of two odd numbers is odd, hence x² = x·x is odd. Thus x² is not even.

Having proved the contrapositive, we infer the original statement.^[3]

Relation to proof by contradiction

Any proof by contrapositive can also be trivially formulated in terms of a proof by contradiction: To prove the proposition $P\Rightarrow Q$, we consider the opposite, $\lnot (P\Rightarrow Q)\equiv \lnot (\lnot P\vee Q)\equiv P\wedge \lnot Q$. Since we have a proof that $\lnot Q\Rightarrow \lnot P$, we have $P\wedge \lnot Q\Rightarrow P\wedge \lnot P\equiv \bot$ 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.

See also

• Contraposition
• Modus tollens
• Reductio ad absurdum

References

1. Regents Exam Prep, contrapositive definition
2. Larry Cusick's (CSU-Fresno) How to write proofs tutorial
3. Franklin, J.; A. Daoud (2011). Proof in Mathematics: An Introduction. Sydney: Kew Books. ISBN 0-646-54509-4. (p. 50).