Definition:Choice Function

Definition
Let $\mathbb S$ be a set of sets such that:
 * $\forall S \in \mathbb S: S \ne \O$

that is, none of the sets in $\mathbb S$ may be empty.

A choice function on $\mathbb S$ is a mapping $f: \mathbb S \to \ds \bigcup \mathbb S$ satisfying:
 * $\forall S \in \mathbb S: \map f S \in S$

That is, for a given set in $\mathbb S$, a choice function selects an element from that set.

The domain of $f$ is $\mathbb S$.

Power Set
The concept of the choice function is often seen in the context of the power set of a given set $S$:

Also see

 * Zermelo's Well-Ordering Theorem


 * Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice


 * In some situations, AoC is not needed to get a choice function:
 * Principle of Finite Choice
 * Choice Function Exists for Set of Well-Ordered Sets
 * Choice Function Exists for Well-Orderable Union of Sets