Proof that an implication is equivalent to its contrapositive by a truth table

Proof

We compare the truth tables of $p \to q$ and $\neg q \to \neg p$.

1. Let $p$ and $q$ be propositions. In classical logic, each of them is true or false, so there are four rows: TT, TF, FT and FF.
2. By the definition of implication, $p \to q$ is false exactly when $p$ is true and $q$ is false, that is, only in the row TF.
3. Again by the definition of implication, $\neg q \to \neg p$ is false exactly when $\neg q$ is true and $\neg p$ is false.
4. By the definition of negation, $\neg q$ is true exactly when $q$ is false, and $\neg p$ is false exactly when $p$ is true.
5. By steps 3 and 4, $\neg q \to \neg p$ is false exactly when $p$ is true and $q$ is false, that is, only in the row TF.
6. The table is this.

7. By steps 2 and 5, the two implications have the same truth value in each row. So they are logically equivalent: $p \to q \equiv \neg q \to \neg p$.