In this section, you'll learn the formal definition of converse, inverse, and contrapositive. You'll learn how to take a conditional statement in English and write its converse, inverse, and contrapositive in English. You'll also use truth tables to establish equivalence or nonequivalence between the conditional, converse, inverse, and contrapositive.

The **inverse** of a statement \(P \longrightarrow Q\) is the statement \(\neg P \longrightarrow \neg Q\).

The **converse** of a statement \(P \longrightarrow Q\) is the statement \(Q \longrightarrow P\).

The **contrapositive** of a statement \(P \longrightarrow Q\) is the statement \(\neg Q \longrightarrow \neg P\).

Prove equivalence or nonequivalence between conditional, inverse, converse, and contrapositive. For an equivalence, use a truth table. For a nonequivalence, use a truth table and proof by counterexample. Try to come up with two counterexamples for each nonequivalence proof, one of an everyday scenario, and one mathematical.