I know that what I am going to ask is pretty basic, borderline stupid, nevertheless it is bugging me. By definition I know that given a set $A$ and a equivalence relation $\rho$, then the items $[\alpha]_{\rho}$ and $A/\rho$ are defined as follows:
$[\alpha]_{\rho} = \{x \in A : \alpha\;\rho\; x\} \\A/\rho = \{[\alpha]_{\rho}:\alpha \in A\}$
let's consider the relation $\sim$ so defined in $J=\{0,1,2,3,4,5,6,7,8,9\}$:
$\begin{align} a\sim b \Leftrightarrow a^2-1 \equiv b^2-1 \mod 3\end{align}$
then given that if I had to determine $[5]_{\sim}$ and $[9]_{\sim}$, I would write
$[5]_{\sim} = \{1,2,4,5,7,8\} \\ [9]_{\sim} = \{0,3,9\}$
is it correct stating:
$J/\sim=\{[0], [1]\} = \{[2],[3]\}=\{[6],[8]\} =\; ...$
and so basically $J/\sim$ isn't univocally defined?