Let Sα (α ∈ a ) be an indexed family of sets. We define the Cartesian product ×α ∈ A Sα to be the set of all functions f having domain a such that f (α ) ∈ Sα for all α ∈ a. (a) Show that if a = {1,...

Let Sα (α ∈ a ) be an indexed family of sets. We define the Cartesian product ×α ∈ A Sα to be the set of all functions f having domain a such that f (α ) ∈ Sα for all α ∈ a.

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

(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.