Equivalence Relation/Examples/Non-Equivalence/Sum of Integers is Divisible by 3

Example of Relation which is not Equivalence
Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:
 * $\forall x, y \in \Z: x \mathrel \RR y \iff x + y \text { is divisible by } 3$

Then $\RR$ is not an equivalence relation.

Proof
Let $x = 1$.

Then:
 * $x + x = 2$

and so $x + x$ is not divisible by $3$.

Thus $\RR$ is seen to be non-reflexive.

Hence by definition $\RR$ is not an equivalence relation.