Choice function
From Wikipedia, the free encyclopedia
A choice function is a mathematical function f whose domain X is a collection of nonempty sets such that for every S in X, f(S) is in S. In other words f chooses exactly one element from each set in X.
The axiom of choice (AC) states that every set of nonempty sets has a choice function. A weaker form of axiom of Choice, the axiom of countable choice (CC) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or CC, some sets can still be shown to have a choice function.
- If X is a finite set of nonempty sets, then one can construct a choice function for X by picking one element from each member of X. This requires only finitely many choices, so neither AC or CC is needed.
- If every member of X is a well-ordered nonempty set, then it is possible to pick the least element of each member of X. In this case infinitely many choices may be required, but there is a rule for making the choices, so again neither AC or CC is needed. The distinction between "well-ordered" and "well-orderable" is important here: if the members of X were merely well-orderable, it would be necessary to choose a well-ordering of each member, and this might require infinitely many arbitrary choices, and therefore AC (or CC, if X were countably infinite).
- If every member of X is a nonempty set, and the union
is well-orderable, then it is possible to choose a well-ordering for this union, and this induces a well-ordering on every member of X, so a choice function will exist as in the previous example. In this case it was possible to well-order every member of X by making just one choice, so neither AC nor CC was needed. (This example shows that the well-ordering theorem, which states that every set can be well-ordered, implies AC. The converse is also true, but less trivial.)
[edit] See also
This article incorporates material from Choice function on PlanetMath, which is licensed under the GFDL.
