|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{for|the combinatorial choice function C(n, k)|Combination|Binomial coefficient}}
| | Marvella is what you can contact her but it's not the most female name out there. Years ago we moved to Puerto Rico and my family loves it. In her expert lifestyle she is a payroll clerk but she's always wanted her personal company. To gather coins is what his family and him enjoy.<br><br>Here is my web page; [http://bcadaptive.burstcreativegroup.com/content/ideal-way-battle-candida at home std test] |
| | |
| A '''choice function''' ('''selector''', '''selection''') is a [[mathematical function]] ''f'' that is defined on some collection ''X'' of nonempty [[Set (mathematics)|sets]] and assigns to each set ''S'' in that collection some element ''f''(''S'') of ''S''. In other words, ''f'' is a choice function for ''X'' if and only if it belongs to the [[direct product]] of ''X''.
| |
| | |
| == An Example ==
| |
| Let ''X'' = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} 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 ordering|well-ordered]]. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the [[axiom of countable choice]] (AC<sub>ω</sub>) states that every [[countable set]] of nonempty sets has a choice function. However, in the absence of either AC or AC<sub>ω</sub>, some sets can still be shown to have a choice function.
| |
| | |
| *If <math>X</math> is a [[finite set|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 AC<sub>ω</sub> is needed.
| |
| | |
| *If every member of <math>X</math> is a nonempty set, and the [[union (set theory)|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 AC<sub>ω</sub> was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)
| |
| | |
| == Refinement of the notion of choice function ==
| |
| A function <math>f: A \rarr B</math> is said to be a selection of a [[multivalued function|multivalued map]] φ:''A'' → ''B'' (that is, a function <math>\varphi:A\rarr\mathcal{P}(B)</math> from ''A'' to the [[power set]] <math>\mathcal{P}(B)</math>), if
| |
| :<math>\forall a \in A \, ( f(a) \in \varphi(a) ) \,.</math>
| |
| | |
| The existence of more regular choice functions, namely continuous or measurable selections (see:
| |
| <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>
| |
| ) is important in the theory of [[differential inclusion]]s, [[optimal control]], and mathematical economics.
| |
| | |
| ===Bourbaki tau function===
| |
| [[Nicolas Bourbaki]] used [[epsilon calculus]] for their foundations that had a <math> \tau </math> symbol which could be interpreted as choosing a object (if one existed) which satisfies a given proposition. So if <math> P(x) </math> is a predicate, then <math>\tau_{x}(P)</math> is the object which 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|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>
| |
| | |
| ==See also==
| |
| * [[Axiom of countable choice]]
| |
| * [[Hausdorff paradox]]
| |
| * [[Hemicontinuity]]
| |
| | |
| ==Notes==
| |
| {{reflist|1}}
| |
| | |
| ==References==
| |
| {{Reflist|2}}
| |
| | |
| {{PlanetMath attribution|id=6419|title=Choice function}}
| |
| | |
| [[Category:Basic concepts in set theory]]
| |
| [[Category:Axiom of choice]]
| |
Marvella is what you can contact her but it's not the most female name out there. Years ago we moved to Puerto Rico and my family loves it. In her expert lifestyle she is a payroll clerk but she's always wanted her personal company. To gather coins is what his family and him enjoy.
Here is my web page; at home std test