After learning open statements and universe of discourse, we now ask how to turn an open statement into a complete statement without substituting a particular value. Existential and universal quantifiers do this by saying that a predicate is true for at least one element or for every element of the universe. These quantifiers are central to definitions, theorems, programming specifications, and mathematical proofs. In this lesson, existential and universal quantifiers will be read, interpreted, and tested through examples.

The symbol $\exists$ is called the \textbf{existential quantifier}. The statement $$\begin{equation*} \exists x\,p(x) \end{equation*}$$ is read as “there exists an $x$ such that $p(x)$.” It also means “for some $x$,” “for at least one $x$,” or “there is an $x$ such that $p(x)$.”