Equivalence Relation/Examples/Even Sum

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

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

Then $\RR$ is an equivalence relation.

The equivalence classes are:
 * $\eqclass 0 \RR$
 * $\eqclass 1 \RR$

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:
 * $\forall x \in \Z: x \mathrel \RR x$

and $\RR$ is seen to be reflexive.

Symmetry
Thus $\RR$ is seen to be symmetric.

Transitivity
Thus $\RR$ is seen to be transitive.

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

Hence by definition it is an equivalence relation.

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

and the equivalence classes of $\RR$ are $\eqclass 0 \RR$ and $\eqclass 1 \RR$ from the Fundamental Theorem on Equivalence Relations.