Suppose Ai i I and Bi i I are indexed families of sets a P

Suppose {Ai | i I} and {Bi | i I} are indexed families of sets.

(a) Prove that iI (Ai × Bi) (iI Ai) × (iI Bi).

(b) For each (i, j) I × I let C(i, j) = Ai × B j, and let P = I × I. Prove that pPC p = (iI Ai) × (iI Bi).

Solution

(a) Let (x, y) be arbitrary. Suppose (x, y) iI (Ai X Bi).

Since (x, y) iI(Ai X Bi), there exists an iI with xAi and yBi.

So x {xliI(xAi)} and y{yliI(yBi)}

Therefore, x iI Ai and yiI Bi.

This is equivalent to (iI Ai) X (iI Bi).

Hence, iI (Ai X Bi)(UiI Ai) X (iI Bi).