Let A, B and C be sets. Let f : A -> B and g : B -> C be functions.

1. Prove that if f and g are injective, then g o f is injective.

2. Suppose that g o f is bijective. Does it follow that g is injective? Prove your claim. (Hint:

If your answer is yes, show that g is indeed injective; if your answer is no, present a pair of

functions f and g as a counterexample.)

3. Suppose that g o f is bijective. Does it follow that f is injective? Prove your claim.

4. Suppose that g o f is bijective. Does it follow that g is surjective? Prove your claim.

5. Suppose that g o f is bijective. Does it follow that f is surjective? Prove your claim.