Let f : A - B be an injective (one-to-one) function. Define g : 2^A - 2^B as: g(C) = {f(x) | x in C}, for all subsets ...

Question

Let f : A -> B be an injective (one-to-one) function. Define g : 2^A -> 2^B as: g(C) = {f(x) | x in C}, for all subsets C of A. Define h : 2^B -> 2^A as: h(D) = {x | x in A, f(x) in D}, for all subsets D of B. Which of the following statements is always true? A. g(h(D)) subseteq D  B. g(h(D)) supseteq D  C. g(h(D)) intersect D = phi  D. g(h(D)) intersect (B - D) != phi

Accepted Answer

g(h(D)) subseteq D always holds, so the answer is Option A. Here g is the 'image' map (push a subset of A forward through f) and h is the 'preimage' map (pull a subset of B back through f). Starting from a subset D of B, we first take its preimage h(D) = all points of A that f sends into D, then push that forward with g. Every element we get back was, by construction, a value f(x) with x already chosen because f(x) is in D, so the result can never leave D. Hence g(h(D)) sits inside D.