Let $\displaystyle S = \{(x,y)| x,y \in \mathbb{R} \} $. If $\displaystyle (a,b) $ and $\displaystyle (c,d) $ belong to $\displaystyle S $, define $\displaystyle (a,b)R(c,d) $ if $\displaystyle a^2+b^2 = c^2+d^2 $. Prove that $\displaystyle R $ is an equivalence relation on $\displaystyle S $. Give a geometrical description of the equivalence classes of $\displaystyle S $.

Proof.So $\displaystyle (a,b)R(a,b) $ (reflexivity). Also $\displaystyle (a,b)R(b,a) $ (symmetry). Also if $\displaystyle (a,b)R(c,d) $ and $\displaystyle (c,d)R(e,f) $, then $\displaystyle (a,b)R(e,f) $ (transitivity). $\displaystyle \square $

Consider $\displaystyle a \in S $ where $\displaystyle a = (a_1, a_2) $. Then $\displaystyle [a] = \{x \in S| x \sim a \} $. In this case, $\displaystyle a $ is an ordered pair. So it would be a circle?