On the set $\mathbb{R}^2$, define $(x,y) R (a,b)$ if and only if $x^2-y =a^2-b$. Show that $R$ is an equivalence relation.
1) $\forall (x,y)\in\mathbb{R^2}$ , we have $(x,y) R (x,y)$ since $x^2-y = x^2-y$, which shows that $R$ is reflexive
2) If $(x,y) R (a, b)$, then $x^2-y=a^2-b$ , and so $a^2-b=x^2-y$ , which implies that $(a,b)R(x,y)$, and so $R$ is symmetric
3) If $(x,y)R(a,b)$ and $(a,b)R(c,d)$ , then $x^2-y=a^2-b$ and $a^2-b=c^2-d$ The transitive law for equality implies that $x^2-y=c^2-d$, and therefore $(x,y)R(c,d)$, so $R$ is transitive.
We conclude that $R$ is an equivalence relation.
Is correct my proof?