Given a function I and a subset A of its domain, let I(A) represent the range of lover the set A; that is, I(A) = {I(x) : x E A}. 12 Chapter 1. The Real Numbers (a) Let f(x) = x2. If A = [0,2] (the closed interval {x E R : ° ~ x ~ 2}) and B = [1,4]' find f(A) and f(B). Does f(A n B) = f(A) n f(B) in this case? Does f(A U B) = f(A) U f(B)? (b) Find two sets A and B for which f(A n B) =1= f(A) n f(B). (c) Show that, for an arbitrary function g : R ---+ R, it is always true that g(A n B) ~ g(A) n g(B) for all sets A, B ~ R. (d) Form and prove a conjecture about the relationship between g(A U B) and g(A) U g(B) for an arbitrary function g.