How to check if a function is injective and surjective

Question

I'm currentlly doing a course in abstract algebra and I often have to prove a map is surjective or injective. It's always done the same way, we take $f(a)=f(b)$ and deduce $a=b$, or we show that for every $y$ in the range there is an element x in the domain such that $f(x)=y$. I was wondering if there are alternative ways we can use to prove a map is injective/surjective?

Answer 1

Suppose that $X,Y,Z$ are sets and $f: \ X \to Y$ and $g : \ Y \to Z$ are functions. We have the following implications: if $g \circ f$ is a surjection, then $g$ is surjection. If $g \circ f$ is an injection, then $f$ is an injection. I encourage you to prove this lemma yourself if you haven't seen it before.

Answer 2

If the map is an arbitrary function, there is no better way.

However, if the map is a homomorphism, you can prove it is injective by showing its kernel is trivial.