Discrete math help, please?

Question

Let
f: A-->B
g: B-->C

Prove: If g o f is onto, then g is onto.

I appreciate any help! Thanks!

Answer 1

Let c ∈ C. Then there exists an a ∈ A such that
g o f(a) = c, since g o f is onto. But then f(a) ∈ B and g(f(a)) = c, so there is a b ∈ B such that g(b) = c. So g is onto.

We can also prove that if g o f is 1-1, then f is 1-1: Let x, y ∈ A be such that f(x) = f(y). Then g(f(x)) = g(f(y)), so g o f (x) = g o f (y); since g o f is 1-1, this implies x = y. So f(x) = f(y) implies x = y, i.e. f is 1-1.