Diagonal Relation is Equivalence/Examples

Examples of Use of Diagonal Relation is Equivalence

Equality of Integers is Equivalence

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

$\forall x, y \in \Z: x \mathrel \RR y \iff x = y$

Then $\RR$ is an equivalence relation such that the equivalence classes are singletons.

Equality of Numbers is Equivalence

Let $\SS$ denote the set of numbers.

Let $\RR$ denote the relation on $S$ defined as:

$\forall x, y \in S: x \mathrel \RR y \iff x = y$

Then $\RR$ is an equivalence relation such that the equivalence classes are singletons.