1.43 THEOREM. Let X and Y be sets and f : XY. (i) If {Ea}aA is a collection of subsets of X, then (U f ( U Ea) = U

(a) and ( ^ E f \aA Eanf(Ea). (ii) If B and C are subsets of X, then f(CB) 2 f(C) \ f(B). (iii) If {Ea}aEA is a collection of subsets of Y, then f-1 (UE)

a) = U(E) and f-1 (iv) If B and C are subsets of Y, then (n) = 0. | E = (Ea). EA f(C\B) = f(C) \ (B). (v) If EC f(X), then f(f(E)) = E, but if EC X, then f(f(E)) 2 E. PROOF. (i) By definition, y = f(Uae AEa) if and only if y = f(x) for some x Ea and a A. This is equivalent to y Uae Af (Ea). Similarly, y f(naEA E) if and