Converse, Inverse, and Contrapositive

Let the implication

$P \to Q$

be a proposition with hypothesis $P$ and conclusion $Q$. Three related implications are easy to confuse.

Name	Form
Original implication	$P \to Q$
Converse	$Q \to P$
Inverse	$\neg P \to \neg Q$
Contrapositive	$\neg Q \to \neg P$

The converse and inverse are logically equivalent to each other, but neither is generally equivalent to the original implication. The contrapositive is logically equivalent to the original implication:

$$P \to Q \quad\Longleftrightarrow\quad \neg Q \to \neg P.$$

ExampleA counterexample to the converse

The statement “if an integer is divisible by $4$, then it is even” is true. Its converse, “if an integer is even, then it is divisible by $4$,” is false because $2$ is even but not divisible by $4$.

ExampleA contrapositive proof

To prove that an integer $n$ is even whenever $n^2$ is even, it is convenient to prove the contrapositive. Assume that $n$ is odd, so $n=2k+1$ for some integer $k$. Then

$$n^2=(2k+1)^2=2(2k^2+2k)+1,$$

which is odd. Therefore, if $n^2$ is even, $n$ cannot be odd and must be even.

The logical equivalence of an implication and its contrapositive is a transformation of propositions. It does not make the converse true, and it should not be confused with proof by contradiction, even though both methods often avoid a direct proof.