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::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is an [[axiom]] of [[Zermelo-Fraenkel]] set theory holding that:<ref>Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is an [[axiom]] of [[Zermelo-Fraenkel]] set theory holding that:<ref>Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

:given any collection of sets, however large, we can pick one element from each set in the collection.

::given any collection of sets, however large, we can pick one element from each set in the collection.

In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set."  More precisely, the Axiom of Choice states that:

In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set."  More precisely, the Axiom of Choice states that: