choice function

A choice function on a set $S$ is a function $f$ with domain $S$ such that $f(x)\in x$ for all $x\in S$.

A choice function on $S$ simply picks one element from each member of $S$. So in order for $S$ to have a choice function, every member of $S$ must be a nonempty set. The Axiom of Choice (http://planetmath.org/AxiomOfChoice) (AC) states that every set of nonempty sets does have a choice function.

Without AC the situation is more complicated, but we can still show that some sets have a choice function. Here are some examples:

• If $S$ is a finite set of nonempty sets, then we can construct a choice function on $S$ by picking one element from each member of $S$. This requires only finitely many choices, so we don’t need to use AC.

• If every member of $S$ is a well-ordered nonempty set, then we can pick the least element of each member of $S$. In this case we may be making infinitely many choices, but we have a rule for making the choices, so AC is not needed. The distinction between “well-ordered” and “well-orderable” is important here: if the members of $S$ were merely well-orderable, we would first have to choose a well-ordering of each member, and this might require infinitely many arbitrary choices, and therefore AC.

• If every member of $S$ is a nonempty set, and the union $\cup S$ is well-orderable, then we can choose a well-ordering for this union, and this induces a well-ordering on every member of $S$, so we can now proceed as in the previous example. In this case we were able to well-order every member of $S$ by making just one choice, so AC wasn’t needed. (This example shows that the Well-Ordering Principle, which states that every set is well-orderable, implies AC. The converse is also true, but less trivial — see the proof (http://planetmath.org/ProofOfZermelosWellOrderingTheorem).)