Equivalence Relation/Examples/Non-Equivalence/Sum is Integer

Example of Relation which is not Equivalence
Let $\R$ denote the set of real numbers.

Let $\sim$ denote the relation defined on $\R$ as:
 * $\forall \tuple {x, y} \in \R \times \R: x \sim y \iff x + y \in \Z$

Then $\sim$ is not an equivalence relation.

Proof
We have that $\sim$ symmetric:
 * $x + y \in \Z \implies y + x \in \Z$

from Real Addition is Commutative.

But $\sim$ is non-reflexive:
 * $2 \cdotp 3 + 2 \cdotp 3 = 4 \cdotp 6 \notin \Z$

for example.

Also, $\sim$ is non-transitive:

for example.

So $\sim$ is not an equivalence relation.