Choice function

{{Short description|Mathematical function}} {{for|the combinatorial choice function C(n, k)|Combination|Binomial coefficient}}

Let ''X'' be a set of sets none of which are empty. Then a '''choice function''' ('''selector''', '''selection''') on ''X'' is a mathematical function ''f'' that is defined on ''X'' such that ''f'' is a mapping that assigns each element of ''X'' to one of its elements.

== An example == 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''.

== History and importance == Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem,<ref name="Zermelo, 1904">{{cite journal| first=Ernst| last=Zermelo| year=1904| title=Beweis, dass jede Menge wohlgeordnet werden kann| journal=Mathematische Annalen| volume=59| issue=4| pages=514–16| doi=10.1007/BF01445300| url=http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526}}</ref> 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.

*If <math>X</math> is a finite set of nonempty sets, then one can construct a choice function for <math>X</math> by picking one element from each member of <math>X.</math> This requires only finitely many choices, so neither AC or ACω is needed. *If every member of <math>X</math> is a nonempty set, and the union <math>\bigcup X</math> is well-ordered, then one may choose the least element of each member of <math>X</math>. In this case, it was possible to simultaneously well-order every member of <math>X</math> by making just one choice of a well-order of the union, so neither AC nor ACω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)

== Choice function of a multivalued map == Given two sets <math>X</math> and <math>Y</math>, let <math>F</math> be a multivalued map from <math>X</math> to <math>Y</math> (equivalently, <math>F:X\rightarrow\mathcal{P}(Y)</math> is a function from <math>X</math> to the power set of <math>Y</math>).

A function <math>f: X \rightarrow Y</math> is said to be a '''selection''' of <math>F</math>, if:

<math display="block">\forall x \in X \, ( f(x) \in F(x) ) \,.</math>

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.<ref>{{cite book | last = Border | first = Kim C. | title = Fixed Point Theorems with Applications to Economics and Game Theory | year = 1989 | publisher = Cambridge University Press | isbn = 0-521-26564-9 }}</ref> See Selection theorem.

==Bourbaki tau function== Nicolas Bourbaki used epsilon calculus for their foundations that had a <math> \tau </math> symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if <math> P(x) </math> is a predicate, then <math>\tau_{x}(P)</math> is one particular object that satisfies <math>P</math> (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example <math> P( \tau_{x}(P))</math> was equivalent to <math> (\exists x)(P(x))</math>.<ref>{{cite book|last=Bourbaki|first=Nicolas|title=Elements of Mathematics: Theory of Sets|date=1968 |publisher=Hermann |isbn=0-201-00634-0}}</ref>

However, Bourbaki's choice operator is stronger than usual: it's a ''global'' choice operator. That is, it implies the axiom of global choice.<ref>John Harrison, "The Bourbaki View" [http://www.rbjones.com/rbjpub/logic/jrh0105.htm eprint].</ref> Hilbert realized this when introducing epsilon calculus.<ref>"Here, moreover, we come upon a very remarkable circumstance, namely, that all of these transfinite axioms are derivable from a single axiom, one that also contains the core of one of the most attacked axioms in the literature of mathematics, namely, the axiom of choice: <math>A(a)\to A(\varepsilon(A))</math>, where <math>\varepsilon</math> is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, ''From Frege to Gödel'', p. 382. From [http://ncatlab.org/nlab/show/choice+operator nCatLab].</ref>

==Notes== {{Reflist}}

==References== {{PlanetMath attribution|id=6419|title=Choice function}}

Category:Basic concepts in set theory Category:Axiom of choice