Is $R$ an equivalence relation and if so what is the equivalence class?

Question

$R$ on $R^2$ by $uRv$ if $|v−u|∈Z$ where $|v−u|$ is the modulus of vector v-u

Is the following an equivalence relation? I cannot determine if it is reflective and transitive of not.

Answer 1

It is reflexive ( $0 \in \mathbb Z$) and symmetric ($|u-v|=|v-u|$) but it is not transitive: Take $v=(1,0), u=(0,0)$ and $w=(0,1)$. Then $|v-u|$ and $|u-w|$ are integers but $|v-w|=\sqrt 2$ is not.