Proofs:Contraposition

Let $P$ and $Q$ be propositions such that $P\implies Q$. Then, the contrapositive of the conditional statement "$P\implies Q$" (read as "if P, then Q" or "P implies Q") is "$\neg Q\implies \neg P$ (read as "if not Q, then not P" or "not Q implies not P"). The contrapositive is logically equivalent to the original conditional; that is, a conditional and its contrapositive always have the same truth values. For example, the contrapositive of "if $x>0$, then $x$ is positive" is "if $x$ is NOT positive, then $x\leq 0$". These two statements are logically equivalent, and are both true. Sometimes, proving the contrapositive of a conditional is easier than proving the conditional itself.

For example: Let x be an integer. If $x^{2}$ is even, then x is even. While we can attempt to prove this conditional statement directly, it is easier to show that if x is not even, then $x^{2}>$ is not even, given that an odd number times an odd number must be odd (that is, not even).

Resources

• Proof by Contrapositive, Wikipedia
• Proof by Contrapositive, Dartmouth University
• Proof by Contrapositive, Fresno State University