Equivalence Relation/Examples/Even Sum

Example of Equivalence Relation
Let $\Z$ denote the set of integers.

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

Then $\mathcal R$ is an equivalence relation.

The equivalence classes are:
 * $\eqclass 0 {\mathcal R}$
 * $\eqclass 1 {\mathcal R}$

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
Let $x \in \Z$.

Then:
 * $x + x = 2 x$

and so $x + x$ is an even integer.

Thus:
 * $\forall x \in \Z: x \mathrel {\mathcal R} x$

and $\mathcal R$ is seen to be reflexive.

Symmetry
Thus $\mathcal R$ is seen to be symmetric.

Transitivity
Thus $\mathcal R$ is seen to be transitive.

$\mathcal R$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

We have that:
 * $x \mathrel {\mathcal R} 0 \iff x \text { is even}$
 * $x \mathrel {\mathcal R} 1 \iff x \text { is odd}$

and the equivalence classes of $\mathcal R$ are $\eqclass 0 {\mathcal R}$ and $\eqclass 1 {\mathcal R}$ from the Fundamental Theorem on Equivalence Relations.