Hello! I am new to this forum and I hope I will soon familiarize with posting new threads. Now, during the winter break I started to study for myself the set theory and especially the axiom of choice. I found the next problem in a book:

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 of non-empty sets X there is a choice function.

3) For any non-empty set X, there exists a function f:P(X)\{∅}→X so that for any non-empty set A⊆X, f(A) ∈ A.

Now, I have already tried and also succeeded to prove 2 of the 6 possible implications between the statements. But I simply can't realize how to prove the next implications: 1=>2, 2=>3 and 3=>1.

Thank you in advance!

Best regards,

Alice