Diagonal Relation is Equivalence/Examples/Integers

Examples of Use of Diagonal Relation 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.

Proof

This is an instance of Diagonal Relation is Equivalence.

The result follows from Equivalence Classes of Diagonal Relation.

$\blacksquare$

Sources

• 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: Exercises: $1.6 \ \text {a)}$