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.
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How to Use Python's choice Function

GTO2106: Impossible of Nonparadoxical Social Choice Functions

5.5.1  Selections
Transcription
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 wellordering theorem,^{[1]} which states that every set can be wellordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (AC_{ω}) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or AC_{ω}, some sets can still be shown to have a choice function.
 If is a finite set of nonempty sets, then one can construct a choice function for by picking one element from each member of This requires only finitely many choices, so neither AC or AC_{ω} is needed.
 If every member of is a nonempty set, and the union is wellordered, then one may choose the least element of each member of . In this case, it was possible to simultaneously wellorder every member of by making just one choice of a wellorder of the union, so neither AC nor AC_{ω} was needed. (This example shows that the wellordering theorem implies AC. The converse is also true, but less trivial.)
Choice function of a multivalued map
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.
Bourbaki tau function
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]}
See also
Notes
 ^ Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann". Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300.
 ^ Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0521265649.
 ^ Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. ISBN 0201006340.
 ^ John Harrison, "The Bourbaki View" eprint.
 ^ "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: , where is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, From Frege to Gödel, p. 382. From nCatLab.
References
This article incorporates material from Choice function on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.