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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 $A\subseteq X$, $f(A) \in A$.

  • 1
    Which of six possible implications between the statements can you prove by yourself?2012-01-07
  • 1
    In (2), you need that $X$ is a collection of non-empty sets, not a non-empty collection of sets.2012-01-07

2 Answers 2