Discrete Math: Proving Injectivity/Surjectivity of g°f

[Lolligirl]

1. Show by example that it is possible for g°f(x) to be surjective while f(x) is not
I am confused by the general pattern of injectivity (one-to-one) and surjectivity (onto). I know the following by looking through my book:

If f and g are surjective, then g°f is surjective.
If f is surjective and g is surjective, then g°f is surjective.
If f is surjective and g is injective, then g°f is injective.
If f is surjective, then f°f is surjective.
If g°f is surjective, then g is surjective.

But I don't know the answer to the rest that I haven't included, such as if f is an injection and g is a surjection, but g°f is not an injection.

What I'm asking is this: how do you prove something is surjective or injective? We have just started this topic so I'm not entirely sure if this is correct, but I think it would have something to do with, for surjectivity, proving that for every b ∈ B, there is a corresponding a ∈ A that maps to it. For injectivity, for every a ∈ A, there is only one b ∈ B that maps to it. How is this done?

 

[LCKurtz]

What I'm asking is this: how do you prove something is surjective or injective? We have just started this topic so I'm not entirely sure if this is correct, but I think it would have something to do with, for surjectivity, proving that for every b ∈ B, there is a corresponding a ∈ A that maps to it. For injectivity, for every a ∈ A, there is only one b ∈ B that maps to it. How is this done?

That is exactly right for surjective. For injective, assuming we are talking about maps from A to B you should say "for each b ∈ range(f) there is only one a ∈ A that maps to it".

The answer as to how you prove a map is injective or surjective, you demonstrate the above statements. The mechanics of doing so depends on the particular problem. And, of course, to show a map isn't injective or surjective you just demonstrate a counterexample to the defining statement.