1. ## proving functions onto

Hey I'm in need of a push in the right direction.
My problem is:
Assume A,B,C are arbitrary sets show if f:A -> B is onto and g: B->C is onto then g(f): A->C is onto. (EDIT - meant to say g(f))

What do I need to show to prove A-C is onto?

2. Every point is C has a pre-image in B and that pre-image has itself a pre-image in A.

If $\left( {\forall t \in C} \right)\left( {\exists x_t \in B} \right)\left[ {g(x_t ) = t} \right]\;\& \,\left( {\exists w_{x_t }\in A } \right)\left[ {f\left( {w_{x_t } } \right) = x_t } \right]$

So $g\left( {f\left( {w_{x_t } } \right)} \right) = g\left( {x_t } \right) = t$.

3. Do you mean, "Show g*f is onto"?

Suppose y in C.
Since g is onto, there is an x in B such that g(x) = y.
Since x in B, and since f is onto, there is a z in A such that f(z) = x.
So y = g(f(z)).

4. Thank you both. And yes typed the question poorly.