Show that the choice axiom is equivalent to the statement that for any indexed family \(\ts{A_\alpha}_{\alpha \in J}\) of nonempty sets, with \(J \neq 0\), the cartesian product
\[ \prod_{\alpha \in J} A_\alpha \]
is not empty.
Solution
Assume the axiom of choice. Given a family \(\{A_\alpha\}_{\alpha\in J}\) of nonempty sets, choice supplies a function \(c:J\to\bigcup_\alpha A_\alpha\) with \[ c(\alpha)\in A_\alpha \] for every \(\alpha\). This is exactly an element \((c(\alpha))_{\alpha\in J}\) of the Cartesian product, so the product is nonempty.
Conversely, assume every product of a family of nonempty sets is nonempty. Given any family \(\{A_\alpha\}_{\alpha\in J}\) of nonempty sets, choose an element \[ (a_\alpha)_{\alpha\in J}\in\prod_{\alpha\in J}A_\alpha. \] Then \(c(\alpha)=a_\alpha\) is a choice function. Hence the product-nonemptiness statement is equivalent to the axiom of choice.