Definition:Choice Function/Power Set

Definition
Let $S$ be a set.

Let $\mathbb S = \powerset S \setminus \set \O$ be the power set of $S$ excluding the empty set $\O$.

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

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

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