ABSTRACT

Note that only one quantifier suffices, since (∀x)[. . . ] is logically equivalent to¬(∃x)¬[. . . ].

A typical use of quantifiers occurs among the axioms for group theory. Given a set G with a binary operation · on G (i.e., a function from G × G to G),

(∃e)(∀x)[x · e = e · x = x] is the axiom that states that there is an element of G that functions as an identity, while

(∀x)(∃x∗)[x · x∗ = x∗ · x = e] is the axiom that states that every element of G has an inverse in G. Here, the quantifiers range over the universe (group) G.