After learning how quantified statements become true or false, we now compare predicates over a common universe. Logical equivalence and implication of predicates extend familiar propositional ideas to open statements. Two predicates are equivalent when they agree for every element; one predicate implies another when the first being true always forces the second. In this lesson, logical equivalence and implication of predicates will be proved through universal statements.

Let $p(x)$ and $q(x)$ be open statements over the same universe. The predicates $p(x)$ and $q(x)$ are called \textbf{logically equivalent} if $$\begin{equation*} \forall x[p(x)\leftrightarrow q(x)] \end{equation*}$$ is true. In this case, $p(x)$ and $q(x)$ have the same truth value for every element of the universe.