Proof by Contraposition

In a $\text{proof by contraposition}$, a statement of the form $\text{‘}p\to q\text{’}$ is proved by showing $\text{‘}\mathrm{\neg }q\to \mathrm{\neg }p\text{’}$. This is correct, because the statements are logically equivalent

Leeds University/Computer Science/Compulsory Modules/Fundamental Math Concepts/Proof Techniques/Definitions/Theorem 4.3

If $n$ is an integer and $3n+2$ is odd, then $n$ is odd.

Proof

Let $n$ be an arbitrary integer. We proceed by contraposition.
The first step in a proof by contraposition is to assume that the conclusion of the conditional statement $\text{‘}$If $3n+2$ is odd, then $n$ is odd$\text{’}$ is false, namely assume that $n$ is even.
Our goal is to how that the premise is false as well, i.e. we want to show that $3n+2$ is even. Since $n$ is even, by definition of an even integer, $n=2k$ for some integer $k$.
Substituting $2k$ for $n$, we find that

Hence $3n+2$ is even, because $3n+2=2\ell$ for some integer $\ell$, namely for $\ell =(3k+1)$. Thus we have reached our goal and the proof is complete.
$\text{Note: Instead of proving }\text{‘}p\to q\text{’}\text{, we proved }\text{‘}\mathrm{\neg }q\to \mathrm{\neg }p\text{’}\text{.}$
$\text{The proof of }\text{‘}\mathrm{\neg }q\to \mathrm{\neg }p\text{’}\text{ is a}$ direct proof