Let : → , : → be total functions. We define: ℎ: × → × by ℎ(, ) = ((), ())
Prove: ℎ is onto iff , are onto
The logic is clear, if h is onto, (), () has to be able to generate every value in C and D, hence f and g are onto as well. And vice versa. But I have no idea on how to write it formally and rigorously...
Any help would be appreciated!
Saying $h$ is onto is equivalent to saying that for each $(c,d)\in C\times D$, there is at least one $(a,b)\in A\times B$ such that $h(a,b)=(c,d)$. But this is equivalent to saying that there is $(a,b)\in A\times B$ such that $f(a)=c$ and $g(b)=d$, which itself is equivalent to saying $f$ and $g$ are onto.
It is probably simpler to work direct and reciprocal implications separately.