Let $p(x)=ax^2+bx+c, q(x)=dx^2+ex+f$ and $n\in\mathbb{z}$. Okay, I need to define the following opertations on $\mathbb{Z_n}$.
(1) $[r]_n\bigoplus[s]_n=[p(r)+q(s)]_n$
(2) $[r]_n\bigodot[s]_n=[p(r)q(s)]_n$
And determine whether or not $\bigoplus$ and $\bigodot$ are well defined, and prove your answer.
---- To be honest, I have no idea what I am supposed to do here. Do I have to prove like, $[r]_n\equiv r+kn$ or $[r]_n\equiv r\mod {n}$?
If $r=2,s=3$, How do I make these operation work? --$[2]_n\bigoplus[3]_n=[(4a+2b-c)+(9d+3x+f)]_n$? How this operation work?
An element $[r]_n \in \mathbb{Z}_n$ is actually an equivalence class. For example $[2] \in \mathbb{Z}_5$ stands for the equivalence class $\{\ldots,-8,-3,2,7,12,\ldots\}$. The operations $\oplus$ and $\odot$ require you to do arithmetic with elements of $\mathbb{Z}_n$. In order to do so we pick a representative from the equivalence class. The operations are called well defined if the outcome does not depend on which representative you pick. For example, we want to have $[2]_5 \oplus [1]_5 = [-8]_5 \oplus [11]_5$, so that the operation $\oplus$ makes sense on the equivalence classes as a whole.