question about the axiom of choice

Question

We know axiom of choice states that:

Given any collection $\{ S_i : i \in I \} $ of nonempty sets, there exists a choice function $f: I \to \bigcup_{i \in I} S_i $ such that $f(i) \in S_i $ for all $i \in I$.

Question: I am bit confused because in one books I saw the axiom of choice as: For any collection $\{ S_i : i \in I \} $ of nonempty sets, $\prod_{i \in I} S_i $ is non empty.

Why are they equivalent?

I am 103% sure that this question has been asked before, at least twice. — Asaf Karagila, Apr 14 '14 at 11:45

score 0 · Accepted Answer · answered Apr 14 '14 at 01:14

0

The product of a collection $\{S_i\}$ is, by definition, the collection of all choice functions $I\to \bigcup_{i\in I}S_i$.

answered Apr 14 '14 at 01:14

Ittay Weiss

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question about the axiom of choice

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