y=x^2 +3-1 1 2 xFigure B.5(e) f-^1 ((-oo,3] n (2,4)) = f-^1 (2,3] = {O}, and
r^1 (-oo, 3] n f-^1 (2, 4) = {O} n (-1, 1) = {O}.(f) f-^1 ((-oo,3] - (2, 4)) = f-^1 (-00,2] = 0, and
B.2 Functions 623f-^1 (-00,3]-f-^1 (2,4) = {O}-(-1, 1) = 0. D
The following theorem generalizes Theorem B.2.11 to families of sets, even
infinitely many sets.
Theorem B.2.13 (Functions and Collections of Sets) Suppose
f : A -t B is a function. Then
(a) If {C>.: >.EA} is a family of subsets of A, then{1) f ( LJ C>.) = LJ f(C>.) and
>.EA >.EA(2) f ( n c>-) ~ n f(C>.)·
>.EA >.EA(b) If {D>.: A EA} is a family of subsets of B, then{1) f-^1 ( U D>.) = U f-^1 (D>.) and
>.EA >.EAProof. Exercises 10-13. •