structural:summarization
Statement

A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f (A) is an element of A. With this concept, the axiom can be stated:

Axiom — For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an element of that set.

Formally, this may be expressed as follows:

Thus, the negation of the axiom may be expressed as the existence of a collection of nonempty sets which has no choice function. Formally, this may be derived making use of the logical equivalence of ¬ ∀ X {\displaystyle \neg \forall X\left} to ∃ X {\displaystyle \exists X\left}.

Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X. This is not the most general situation of a Cartesian product of a family of sets, where a given set can occur more than once as a factor; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family. The axiom of choice asserts the existence of such elements; it is therefore equivalent to: