Elementary Foundations: An introduction to topics in discrete mathematics

We simply compare the truth tables.

As the two “output” columns are identical, we conclude that the statements are equivalent.

The inverse of the conditional $p\to q$ is $\mathrm{\neg }p\to \mathrm{\neg }q\text{.}$ The contrapositive of this new conditional is $\mathrm{\neg }\mathrm{\neg }q\to \mathrm{\neg }\mathrm{\neg }p\text{,}$ which is equivalent to $q\to p$ by double negation.