Notation 0.3. For n a positive integer we denote the set {0, 1, 2, ...n - 1} by Zn. Thus Z2 = {0, 1} We have a map E: F(X, Z2) - P(X), f - f (1). Given A C X we define XA E F(X, Z2) by XA(x) = 1 for x E A and XA(x) = 0 for re A. The function XA is called the characteristic function of A. 7. Problem Define the map E: P(X) - F(X, Z2) by E(A) = XA (a) Show that E and E are inverse to each other. (b) Use this and problem 5 to show that for X finite |P(X) | = 2 . It is because of this equality that an alternate notation for P(X ) is 2X