A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns some element of each set S in that collection to S by f(S); f(S) maps S to some element of S. In other words, f is a choice function for X if and only if it belongs to the direct product of X.
Let X = { {1,4,7}, {9}, {2,7} }. Then the function f defined by f({1, 4, 7}) = 7, f({9}) = 9 and f({2, 7}) = 2 is a choice function on X.
Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem,[1] which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, some sets can still be shown to have a choice function.
Given two sets X and Y, let F be a multivalued map from X to Y (equivalently, is a function from X to the power set of Y).
A function is said to be a selection of F, if:
The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential inclusions, optimal control, and mathematical economics.[2] See Selection theorem.
Nicolas Bourbaki used epsilon calculus for their foundations that had a symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if is a predicate, then is one particular object that satisfies (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example was equivalent to .[3]
However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice.[4] Hilbert realized this when introducing epsilon calculus.[5]
This article incorporates material from Choice function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.