Axiom of Choice - Statement

Statement

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

For any set X of nonempty sets, there exists a choice function f defined on X.

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

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 same 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:

Given any family of nonempty sets, their Cartesian product is a nonempty set.

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