Diagonal Relation is Equivalence/Examples/Integers
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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: $1.6 \ \text {a)}$