Q:

#Describe the smallest equivalence relation

Describe the smallest equivalence relation on the set of real numbers which contains the line $x - y = 1$ in the $(x, y)$-plane.
My Answer:
$R=\text{locus}\{x-y=n|n\in \mathbb{Z} \} $
(I changed $\mathbb{N}$ to$\mathbb{Z}$, thanks to Fred.)
Is this the smallest equivalence relation? Need help!!

A:

No. Let $R$ be the smallest equivalence relation. Then $R$ is reflexive, hence for all $x \in \mathbb R$: $xRx$. But there is no $ n \in \mathbb N$ such that $x-x=n$.
Furthermore: if $xRy$ then $yRx$, but $x-y=-(y-x)$.
Therfore try:
$R=\{(x,y) \in \mathbb R^2: \exists k \in \mathbb Z: x-y =k\}$.

