Are f and g Injective and Surjective if g\circf is Injective or Surjective?

[Ka Yan]

Could anybody help me check whether my judgements ture or false? (MJ = My Judgement)

Suppose f maps A into B, and g maps B into C

1. If f and g are injective, then g$$\circ$$f is injective;
(MJ)but that when g$$\circ$$f is injective, the injectivity of f and g are unsure.

2. If f and g are surjective, then g$$\circ$$f is surjective;
(MJ)and that when g$$\circ$$f is surjective, f and g are both surjective.

Thx!

 

[HallsofIvy]

What reasons do you have for those judgements? Can you think up a simple example where f and g are not injective but g$\circ$f is? Try the case where A, B, and C have only a few members.