How Many Choice Functions Exist for Finite and Infinite Sets?

[honestrosewater]

If |X| = n > 0, there are 1C1^nC12C1^nC2...nC1^nCn many choice functions from (power set of X minus empty set) P(X)\{{}} to X?
Just curious. I'm jumping ahead a bit, but that makes sense to me. Is it correct? Is it much more difficult to understand when X is infinite?

 

[Hurkyl]

Sounds right.

Infinite sets make things more interesting! For instance, there might be 0 choice functions, if you deny the axiom of choice! Though, if there's at least one choice function on a set S, and S has an element with cardinality a, then there must be at least a choice functions.

By a "choice function on S", I mean a function f:S --> US such that f(x) is in x. Or, equivalently, an element in the cartesian product of all the elements of S.