Equivalence relation on $\mathbb{R}^2 \times \mathbb{R}^2$?

Question

I have already gotten around that ${\mathbb R}^2$ = ${\mathbb R}$ $\times$ ${\mathbb R}$.

I have the relation $C$ on ${\mathbb R}^2\times {\mathbb R}^2$: $((a_1, b_1), (a_2, b_2)) \in C$

iff

$a^2_1 + b^2_1 = a^2_2 + b^2_2$.

I am trying to wrap my head around if $C$ defines an equivalence relation on ${\mathbb R}^2\times{\mathbb R}^2$? If so, how do I determine the equivalence classes of ${\mathbb R}$?

Answer 1

Any condition of the form $f(x)=f(y)$ with $x,\,y\in A$ defines an equivalence relation on $A\times A$ (just check the axioms), with the equivalence classes being sets of the form $\{x\in A|f(x)=k\}$. In this case, the solution set of $a^2+b^2=k$ is a radius-$\sqrt{k}$ circle for $k>0$, or the point $a=b=0$ if $k=0$, of $\emptyset$ if $k<0$.