Exercise 2.7. Assume in each case that f is a function from X to Y. (1) If g is a function from Y to X such that gf is the identity on X, then f is injective and g is surjective. (2) A necessary and sufficient condition that f(An B) = f(A) f(B) for all subsets A and B of X is that f is injective. (3) A necessary and sufficient condition that f(x\A) Y\f(A) for all subsets A of X is that f is injective. (4) A necessary and sufficient condition that Y\f(A) = f(x\A) for all subsets A of X is that fis surjective.