Prove that omega is onto.?

Question

Let alpha: A->B and omega: B->C be functions s.t. (omega)(alpha) is onto. Prove that omega is onto.

Answer 1

I guess you mean the composition omega o alpha is onto. Since this composition is onto, it's range is the whole C. So, every c in C is the image under omega o alpha of some a in A. So, c = omega(alpha(a)) for some a in A.

But alpha(a) = b is an element of B. Therefore, c = omega(b) for some b in B, which shows every c in C is the image under omega of an element of B. It follows omega is onto,