Question

The composition of a surjection with a surjection is a surjection.

Answer 1

We want to prove composition of two surjective functions is surjective.

Let f:X to Y and g:Y to Z are surjective.then g_of:X to Z.to prove g_of is surjective we need to prove that for all z in Z there exist a x in X such that (g_of)(x)=z.

Let z in Z be arbitrary.since g:Y to Z is surjective there exist a y in Y such that g(y)=z

Since f:X to Y is surjective there exist a x in X such that f(x)=y.

So (g_of)(x)=g(f(x))=g(y)=z

So g_of is surjective.

Hence composition of a surjection(or surjective) with a surjection is a surjection.hence proved.