The Axiom of Choice

There are several equivalent formulations:

• The Cartesian product of nonempty sets is nonempty, even if the product is of an infinite family of sets.
• Given any set S of mutually disjoint nonempty sets, there is a set C containing a single member from each element of S. C can thus be thought of as the result of ``choosing" a representative from each set in S. Hence the name.

 
• Relevance of the Axiom of Choice