Definition:Choice Function

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

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

A choice function on $$S$$ is a mapping $$f: \mathbb S \to \bigcup \mathbb S$$ defined as:
 * $$\forall S \in \mathbb S: \exists x \in S: f \left({S}\right) = x$$

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

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

The codomain of $$f$$ is $$\bigcup \mathbb S$$, that is, the union of all the sets which comprise $$\mathbb S$$

Axiom of Choice
It is allowed that, given any set $$S$$ one can select an element from it.

However, what is debated is whether, given a set of sets $$\mathbb S$$, one can construct a mapping $$f$$ such that every set in $$\mathbb S$$ can be an element of the domain of $$f$$.

So it's not:
 * "Can I choose an element from any given set (in $$\mathbb S$$)?"

as much as:
 * "Can I construct a (mechanistic) procedure that will always return some element of any set (in $$\mathbb S$$)?"

The Axiom of Choice (AoC) is a philosophical position which states that a set of nonempty sets always has a choice function.

Non-Acceptance of Axiom of Choice
If one does not accept the AoC, then a choice function can be proved to exist for the following categories of $$\mathbb S$$:

A Choice Function Exists for All Finite Sets
If $$\mathbb S$$ is finite, we can construct a choice function on $$\mathbb S$$ by picking one element from each member of $$\mathbb S$$.

A Choice Function Exists for Set of Well-Ordered Sets
Thus, if every member of $$\mathbb S$$ is a well-ordered, then we can create a choice function $$f$$ defined as:
 * $$\forall S \in \mathbb S: f \left({S}\right) = \inf \left({S}\right)$$

A Choice Function Exists for Well-Orderable Union of Sets
If the union $$\bigcup \mathbb S$$ is well-orderable, we can create a choice function for $$\bigcup \mathbb S$$.

Also see

 * Well-Ordering Theorem


 * The Well-Ordering Theorem is Equivalent to the Axiom of Choice, which demonstrates the truth of the converse of the Well-Ordering Theorem.