After proving implication and equivalence between predicates, we now examine related conditional forms. Converse, inverse, and contrapositive of quantified implications are obtained by switching and negating the predicate parts of a universal implication. Only the contrapositive is always logically equivalent to the original implication. In this lesson, quantified implications will be compared through definitions, proof, and counterexamples.

A statement of the form $$\begin{equation*} \forall x[p(x)\to q(x)] \end{equation*}$$ is called a \textbf{quantified implication}. It says that for every element $x$ in the universe, if $p(x)$ is true, then $q(x)$ is true.