Problem 8 (10 points). Let f : X Y be a function. If V c X, we define the image of V under f as f(V) = {f(x) 1 x E V). Prove that f is injective if and only if f(A-B) = f(A)-(B) for all A, BCX Hints: When showing f is injective implies f(A-B) = f(A)-f(B), show, that y E f(A-B) implies y E f(A) and conclude that f(A- B) C f(A) f(B). Then use the fact that f is injective to show that f(A) _ f(B) f(A-B). To show that f(A-B) = f(A)-f(B) implies f is injective, let zi,T2 E X be arbitrary and assume that A = {xi} and B = {x2} (for simplicity), which implies that f(A) = {f(x) and f(B) = {f(x2)). conclude something about f(A-B) and show that this implies that x-x2