Finding the equivalence class and quotient set of relation

Question

I have the relation: $ \forall x \in \mathbb{R}: xRy \Leftrightarrow |x - 3| = |y - 3| $

I need to find the equivalence class and quotient set of the relation.

I think the equivalence class is:

$ \forall x \in \mathbb{R}: Cl(x) = \left \{ x \right \} $

but I don't know if it it is a valid answer.

And also I don't know how to represent the quotient set since it's infinite, but it would be the set of all the classes for every x in R.

Answer 1

It is not correct.

$|x-3|=|y-3|\Leftrightarrow x-3=y-3 \, OR \, x-3=3-y \Leftrightarrow x=y \, OR \, x+y=6.$ E.g., the equivalence class of $4$ is $\{2,4\}$. Equivalence classes can be represented by numbers greater than or equal to $3$.

Answer 2

The equivalence class of $x$ is the two points of distance $\mid x-3\mid$ from $3$ (except in the case of $x=3$, when there is just $\{3\}$).

The quotient will then be, for each $d\gt0$, the set of two points at distance $d$ from $3$, $\{3-d,3+d\}$, together with the singleton $\{3\}$.