Choice function

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.

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. 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. \forall x \in X \, ( f(x) \in F(x) ) \,. 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. See Selection theorem. A survey of selectors was published by Claude Ambrose Rogers and John E. Jayne in 2002. ==Bourbaki tau function== Nicolas Bourbaki used epsilon calculus for their foundations that had a \tau symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if P(x) is a predicate, then \tau_{x}(P) is one particular object that satisfies P (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example P( \tau_{x}(P)) was equivalent to (\exists x)(P(x)) . However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice. Hilbert realized this when introducing epsilon calculus. ==See also== Axiom of countable choice Axiom of dependent choice Hausdorff paradox Hemicontinuity ==Notes== ==References==

Text derived from https://en.wikipedia.org/w/index.php?oldid=1375760115 under CC BY-SA 4.0; modifications have been made.