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 (strictly) positive integers.

Let $x \divides y$ denote that $x$ is a divisor of $y$

Then $\divides$ is not an equivalence relation.

Let $\Z$ denote the set of integers.

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

Then $\mathcal R$ is not an equivalence relation.

Proof
Let $x = 1$.

Then:
 * $x + x = 2$

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

Thus $\mathcal R$ is seen to be non-reflexive.

Hence by definition $\mathcal R$ is not an equivalence relation.