Diagonal Relation is Equivalence/Examples/Numbers

Examples of Use of Diagonal Relation 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.

Proof

This is an instance of Diagonal Relation is Equivalence.

The result follows from Equivalence Classes of Diagonal Relation.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equivalence relation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equivalence relation