Let A = A1 × A2 × … and B = B1 × B2 × ….(a) Show that if Bi ⊂ Ai- for all i, then B ⊂ A. (...

Let A = A1 × A2 × … and B = B1 × B2 × ….

(a) Show that if Bi ⊂ Ai- for all i, then B ⊂ A. (Strictly speaking, if we are given a function mapping the index set  into the union of the sets Bi, we must change its range before it can be considered as a function mapping  into the union of the sets Ai. We shall ignore this technicality when dealing with cartesian products).

(b) Show the converse of (a) holds if B is nonempty.

(c) Show that if A is nonempty, each Ai is nonempty. Does the converse hold? (We will return to this question in the exercises of § 19.)

(d) What is the relation between the set A ⋃ B and the cartesian product of the sets Ai ⋃ Bi? What is the relation between the set A ⋂ B and the cartesian product of the sets Ai ⋂ Bi?