After studying negation of quantified statements, we now allow more than one quantified variable. Multiple and nested quantifiers appear whenever a relation connects two or more objects, such as divisibility, order, equations, or array entries. The order of quantifiers matters when universal and existential quantifiers are mixed. In this lesson, multiple and nested quantifiers will be read, expanded over finite universes, and tested through examples.

A statement has \textbf{nested quantifiers} if one quantifier occurs within the scope of another quantifier. Examples include $$\begin{equation*} \forall x\exists y\,p(x,y) \end{equation*}$$ and $$\begin{equation*} \exists y\forall x\,p(x,y). \end{equation*}$$ The order of quantifiers is important when universal and existential quantifiers are mixed.