Suppose that f X rightarrow Y and g Y rightarrow Z are surje

Suppose that f: X rightarrow Y and g: Y rightarrow Z are surjections. Prove that the composite g f: X rightarrow Z is a surjection.

Solution

Let z be in Z

Since g is surjection there is a y in Y so that

g(y)=z

Since f is a surjection so there is x in X so that f(x)=y

Hence, g(f(x))=g(y)=z

ie (gof)(x)=z

But z is arbitrary element in Z

So for any z in Z there exists x in X so that

(gof)(x)=z

Hence, gof is a surjecxtion