1. ## A question about notation

Let S be a set and A a non-empty subset of S. A choice function f is defined as $f: A \to S$ such that $f(A) \in A$.

The last part is what is causing the problem. Does this mean that $f(A) \in A$ implies that f(A) is a single element of the subset A? If so I can work with this, but what little experience I have says f(A) should be a subset (not a single element) of S so I'm a little confused.

Thank you.
-Dan

2. A choice function f is usually defined on a collection of subsets of some set S, not on a fixed subset A of S. Then for every A in the collection, f(A) is in A.

In your scenario, strictly speaking, f cannot be applied to the whole A because f is defined on elements of A. You are right the the f(A) probably should be interpreted as {f(a) | a in A}, so, yes, f(A) is probably not an element of A or of S. Sometimes this application is denoted by f[A] to distinguish it from the regular application of a function to an element of the domain.

3. Some students have found this web page useful.

4. Thank you for the link. I've got it now.

-Dan