\begin{definition}[Definition:Choice Function]
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 $f: \mathbb S \to \bigcup \mathbb S$ satisfying:
:$\forall S \in \mathbb S: \map f S \in S$.
That is, for any set in $\mathbb S$, a '''choice function''' selects an element from that set.
 
The domain of $f$ is $\mathbb S$.
\end{definition}