Proof by Contraposition

Proof Technique
Proof by contraposition is a rule of inference used in proofs.

This rule infers a conditional statement from its contrapositive.

It is based on the Rule of Transposition, which says that a conditional statement and its contrapositive have the same truth value:


 * $p \implies q \dashv \vdash \neg q \implies \neg p$

In other words, the conclusion "if A, then B" is drawn from the single premise "if not B, then not A."

Explanation
Proof by Contraposition can be expressed in natural language as follows:

If we know that by making an assumption


 * $\neg q$

we can deduce


 * $\neg p$

then it must be the case that


 * $p \implies q$.

Thus it provides a means of proving a logical implication.