Definition reminders: A function f from A to B, written f: A + B is a relation f C Ax B such that, if a fb and a fb, then b=W. We write f(a) for the unique b such that a fb. Composition of functions is identical to composition of relations: if f: A + B and g:B C, then gof: A C where (gof)(a) = g(f(a)). A function f : A + B is injective if f(a) = f(a') implies a = a'. A function f : A + B is surjective if, for all be B, there is some a A such that f(a) = b. The identity on A is the function 1A: A + A such that 14(a) = a. A function f : A + B is left-cancelable if there is a function g: B + A such that go f = 1 A. A function f : A + B is right-cancelable if there is a function g: B + A such that fog=18. Let f: A + B and g: B+C. 1. (16 points) If f and g are injective, is gof injective? Justify your answer. 2. (16 points) If f and g are surjective, is gof surjective? Justify your answer. 3. (16 points) If g of is injective, are f and g both necessarily injective? Justify your answer. 4. (16 points) If g of is surjective, are f and g both necessarily surjective? Justify your answer. 5. (17 points) Prove (informally) that if f is left-cancelable, then f is injective. 6. (17 points) Prove (informally) that if f is right-cancelable, then f is surjective