This winter I started to study much harder the set theory and especially the axiom of choice. Unfortunately, I have problems with solving the next exercise:
Prove that the 3 statements of the axiom of choice are equivalent :
1) For any non-empty collection $X$ of pairwise disjoint non-empty sets, there exists a choice set.
2) For any non-empty collection $X$ there is a choice function.
3) For any non-empty set $X$, there exists a function $f:P(X)\setminus\{\varnothing\}\to X$ so that for any non-empty set $A\subseteq X$, $f(A) \in A$.