An implication

$$P\Rightarrow Q$$

asserts that whenever $P$ is true, $Q$ must also be true. $P$ is a sufficient condition for $Q$, while $Q$ is a necessary condition for $P$.

Converse and contrapositive

The converse is

$$Q\Rightarrow P,$$

and does not follow automatically from the original implication.

The contrapositive is

$$\neg Q\Rightarrow\neg P.$$

An implication and its contrapositive are logically equivalent.

Equivalence

Two statements are equivalent when each implies the other:

$$P\Leftrightarrow Q.$$

A proof of equivalence therefore normally contains two directions.

Theorem structure

Many mathematical results have the form

assumptions  →  conclusion

Reading the assumptions and conclusion separately prevents common errors such as using a theorem backwards without proving its converse.