Propositional Logic and Operators

Let talk a bit more about the statement $p \to q$ (= "$p$ implies $q$" = "if $p$, then $q$".)

There is a relationship between $p \to q$ and the statements $q \to p$, $\neg p \to \neg q$, and $\neg q \to \neg p$:

The statement:	is called the:
$q \to p$	converse of the statement $p \to q$.
$\neg p \to \neg q$	inverse of the statement $p \to q$.
$\neg q \to \neg p$	contrapositive of the statement $p \to q$.

We will soon see that $p \to q$ is equivalent to its contrapositive statement $\neg q \to \neg p$, but isn't necessarily equal to the converse statement $q \to p$. [Note that the converse: $q \to p$, however, is, equivalent to the inverse: $\neg p \to \neg q$.]

By saying 'equivalent', we mean that the two statements have the same truth value, no matter what the truth values of the individual statements $p$ and $q$ are. In other words, $\neg q \to \neg p$ is just another way to say $p \to q$ and vice versa.