What is a good way to prove that the function $\Bbb Z_n \rightarrow \Bbb Z_u \times\Bbb Z_v$ is well defined?

Question

Given a function of the type $\Bbb Z_n \rightarrow \Bbb Z_u \times\Bbb Z_v$ ($[x]_n\rightarrow([x]_u, [x]_v)$) where of course $n=u*v$ how can I prove that is a well defined function?

Answer 1

For $f:\mathbb{Z}_n\to\mathbb{Z}_u\times\mathbb{Z}_v$, $[x]_n\mapsto ([x]_u, [x]_v)$

It might go like this:

To prove that this function is well-defined, we have to check if it really does not matter which representative of the equivalence class we choose.

So let $[x]_n=[y]_n$. Then $x=nk+r$ and $y=nl+r$ with $0\leq r<n$.

Now $f([x]_n-[y]_n)=f([x-y]_n)=([x-y]_u, [x-y]_v)\stackrel{x-y=n(k-l)}{=}([n(k-l)]_u, [n(k-l)]_v)\stackrel{n=uv}{=}([0]_u,[0]_v)$