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.