choice function

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

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 ChoiceMathworldPlanetmath ( (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 setMathworldPlanetmath 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 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 converseMathworldPlanetmath is also true, but less trivial — see the proof (

Title choice function
Canonical name ChoiceFunction
Date of creation 2013-03-22 14:46:26
Last modified on 2013-03-22 14:46:26
Owner yark (2760)
Last modified by yark (2760)
Numerical id 11
Author yark (2760)
Entry type Definition
Classification msc 03E25
Related topic AxiomOfChoice
Related topic AxiomOfCountableChoice
Related topic HausdorffParadox
Related topic ProofOfHausdorffParadox
Related topic OneToOneFunctionFromOntoFunction