# Axiom of choice and cartesian product

I learned that Axiom of Choice states that the cartesian product of a family of non-empty sets $$X_i$$ indexed by a non-empty set $$I$$ is non-empty.

I think I can accept this axiom.

But I don’t understand how it guarantees that cartesian product can have more than one elements! It’s because ‘non-empty’ sounds to me ‘having at least one element’ here. So I think AC just implies that there exists at least one element in the product.

How do we insure that there exist all the possible functions from $$I$$ to $$\cup X_i$$?

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