Let $\sim$ be the equivalence relation on $\mathbb{R}$ defined by: for all x,y in $\mathbb{R}$:
$ x\sim y$ iff $x - y \in \mathbb{Q}$
I also have an equivalence relation at $\mathbb{R^2}$ defined by: ($x_0$, $x_1$) $\sim_2$ ($y_0$, $y_1$) if and only if $x_0 - y_0 \in \mathbb{Q}$ and $x_1 - y_1 \in \mathbb{Q}$
Proof that the structure ($\mathbb{R}$,$\sim$) is isomorphic with ($\mathbb{R^2}$,$\sim_2$)
I already know in both structures I have countable many elements per equivalence class, and there are uncountably many, the same cardinality as $\mathbb{R}$, of that classes. So I wanted to make such an isomorphism that sends classes to other classes, but I don't know how to do it.
Is it sufficient to use the axiom of choice uncountable many times to choose a new class every time?