Skip to main content

Section 3.1 Using the Contrapositive

One of the most useful logical equivalencies to prove a conditional statement is that a conditional statement \(P \to Q\) is logically equivalent to its contrapositive, \(\neg Q \to \neg P\text{.}\) This means that if we prove the contrapositive of the conditional statement, then we have proven the conditional statement. The following are some important points to remember.

  • A conditional statement is logically equivalent to its contrapositive.

  • Use a direct proof to prove that \(\neg Q \to \neg P\) is true.

  • Caution: One difficulty with this type of proof is in the formation of correct negations. (We need to be very careful doing this.)

  • We might consider using a proof by contrapositive when the statements \(P\) and \(Q\) are stated as negations.

Proof.

We will prove this result by proving the contrapositive of the statement, which is

For each integer \(n\text{,}\) if \(n\) is an odd integer, then \(n^2\) is an odd integer.
So we assume that \(n\) is an odd integer and prove that \(n^2\) is an odd integer. Since \(n\) is odd, there exists an integer \(k\) such that \(n = 2k + 1\text{.}\) Hence,

\begin{align*} n^2 \amp = (2k + 1)^2\\ \amp = 4k^2 + 4k + 1\\ \amp = 2(2k^2 + 2k) + 1 \end{align*}

Since the integers are closed under addition and multiplication, \((2k^2 + 2k)\) is an integer and so the last equation proves that \(n^2\) is an odd integer. This proves that for all integers \(n\text{,}\) if \(n\) is an odd integer, then \(n^2\) is an odd integer. Since this is the contrapositive of the proposition, we have completed a proof of the proposition.