# Equivalence relation $\varsigma$ on ${\mathbb{R}}^2$

How to show that there is exist equivalence relation $\varsigma$ on ${\mathbb{R}}^2$ such that the following conditions hold:

1. Exist only $7$ equivalence classes by $\varsigma$.
2. For every $x,y \in {\mathbb{R}}^2$ if the distance between $x$ and $y$ is $1$, then $x,y$ are in different equivalence classes.
-