Let S∝ (∝∈.) be an indexed family of sets. We define the Cartesian product X. ∝∈.S∝ to be...

Let S∝ (∝∈.) be an indexed family of sets. We define the Cartesian product X. ∝∈.S∝ to be the set of all functions f having domain  such that f(∝)∈S∝ for all ∝∈'.

(a) Show that if . ' = {1, 2}, this definition gives a set that corresponds to the usual Cartesian product of two sets S1 × S2 in a natural way.

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(b) Show that the axiom of choice is equivalent to the following statement: The Cartesian product of a nonempty family of nonempty sets is nonempty.