Let $∼$ be a relation on $\mathbb{R}$ and $x ∼ y \iff x=y$ or $x+y=6$.

Question

Let $∼$ be a relation on $\mathbb{R}$ and $x ∼ y \iff x=y$ or $x+y=6$. I proved that $∼$ is an equivalence relation. Now I have to find a complete set of representatives. I know that $[a]_∼ := \{b \in \mathbb{R} \mid b ∼ a\}$, but I don't know how to use this to find a CSR.

Answer 1

$[3,\infty)\subset \mathbb R$ should be a suitable set of representatives.

For distinct $x,y \ge 3$, both $x\ne y$ and $x+y > 6$, so $x \nsim y$.

For other $x < 3$, both $6-x > 3$ and $6-x\sim x$, so $x$ is equivalent to some element in $[3, \infty)$.