The Axiom of Choice and the Cartesian Product.

Question

I understand that the axiom of choice, given the axioms of ZF set theory, is equivalent to the statement that "the Cartesian product of any family of nonempty sets is nonempty." I've been unable to find this proof. Could someone sketch it for me? Or provide me with a source at least?

Answer 1

Suppose $X=\{X_i\mid i\in I\}$ is a family of nonempty sets.

If there exists a choice function, then $\langle f(i)\mid i\in I\rangle$ is an element of the product $\prod_{i\in I}X_i$.

If $\prod_{i\in I}X_i$ is nonempty then there is $f=\langle x_i\mid i\in I\rangle$ in this product, which is a sequence of $x_i$ such that $x_i\in X_i$. The function $f(i)=x_i$ is a choice function.

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Indeed as Nate comments, it is most common to define the product $\prod_{i\in I}X_i$ as the set of functions $f:I\to\bigcup\{X_i\mid i\in I\}$ such that $f(i)\in X_i$ for all $i\in I$.

One can easily observe that under this definition the product is exactly the set of choice functions, therefore the product is nonempty if and only if there exists a choice function.