Problem 8 (10 points). Let f : X ? Y be a function. If V-X, we define the image of V under f as f(v) = {f(z) | zen. Prove that f is injective if and only if f(A- B)-f(A)f(B) or all A, B CX Hints: When showing f is injective implies f(A B) f(A) f(B), show that y Ef(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) Sf(A - B). To show that f(A - B) f(A) - f (B) implies f is injective, let z1,r2 E X be arbitrary and assume that A-i) and B-2 (for simplicity), which implies that f(A)-If(xi)) and f(B)-{f(??)). Conclude something about f(A-B) and show that this implies that zi-Z2