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There are two descriptions of axiom of choice that are considered equivalent.

The first one goes like this:

Let $({X_i})_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

Then there exists an indexed family $({x_i})_{i \mathop \in I}$ such that $\forall i \in I: x_i \in X_i$

The second one goes like this:

That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.

I don't see how the second one is equivalent to the first one. Can someone illustrate with an example that both the descriptions mean the same thing?

Lone Learner
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  • Do you know the definition of an arbitrary Cartesian product? – Rushabh Mehta Jun 19 '20 at 15:42
  • I could probably add a few more duplicates. I found all of them by search "axiom of choice cartesian products". In the future, it is worth to spend a couple of minutes searching for the obvious search term before starting to write your question. – Asaf Karagila Jun 19 '20 at 15:47

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