Choice function

A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty 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,^[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 (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 $X$ is a finite set of nonempty sets, then one can construct a choice function for $X$ by picking one element from each member of $X.$ This requires only finitely many choices, so neither AC or AC_ω is needed.

If every member of $X$ is a nonempty set, and the union $\bigcup X$ is well-ordered, then one may choose the least element of each member of $X$ . In this case, it was possible to simultaneously well-order every member of $X$ by making just one choice of a well-order of the union, so neither AC nor AC_ω 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 $f:A\rightarrow B$ is said to be a selection of a multivalued map φ:A → B ( that is, a function $\varphi :A\rightarrow {\mathcal {P}}(B)$ from A to the power set ${\mathcal {P}}(B)$ ), if

\forall a\in A\,(f(a)\in \varphi (a))\,.

The existence of more regular choice functions, namely continuous or measurable selections (see: ^[2] ) is important in the theory of differential inclusions, optimal control, and mathematical economics.

Bourbaki tau function

Nicholas Bourbaki used a formalism for set theory that had a $\tau$ symbol which could be interpreted as choosing a set (if one existed) which satisfies a given proposition. So if $P(x)$ is a proposition $P(\tau (x)(P(x)))$ was equivalent to $(\exists x)(P(x))$ .^[3]

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.

References

Choice function at PlanetMath.

[Zermelo,_1904-1] Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann". Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300.

[2] Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0521265649.

[3] Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. ISBN 0201006340.

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