Diagonal Relation is Equivalence

Theorem

The diagonal relation $\Delta_S$ on a set $S$ is always an equivalence in $S$.

Proof

Checking in turn each of the criteria for equivalence:

Reflexive

$\ds \forall x \in S: \, $

$\ds x$

$=$

$\ds x$

Definition of Equals

$\ds \leadsto \ \ $

$\ds \tuple {x, x}$

$\in$

$\ds \Delta_S$

Definition of Diagonal Relation

So $\Delta_S$ is reflexive.

$\Box$

Symmetric

$\ds \forall x, y \in S: \, $

$\ds \tuple {x, y}$

$\in$

$\ds \Delta_S$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds y$

Definition of Diagonal Relation

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds x$

Equality is Symmetric

$\ds \leadsto \ \ $

$\ds \tuple {y, x}$

$\in$

$\ds \Delta_S$

Definition of Diagonal Relation

So $\Delta_S$ is symmetric.

$\Box$

Transitive

$\ds \forall x, y, z \in S: \, $

$\ds \tuple {x, y}$

$\in$

$\ds \Delta_S \land \tuple {y, z} \in \Delta_S$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds y \land y = z$

Definition of Diagonal Relation

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds z$

Equality is Transitive

$\ds \leadsto \ \ $

$\ds \tuple {x, z}$

$\in$

$\ds \Delta_S$

Definition of Diagonal Relation

So $\Delta_S$ is transitive.

$\blacksquare$

Examples

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.

Sources

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.2$. Equivalence relations: Example $29$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations
1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Example $6.7$
1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations: $(1)$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 16$: Equivalence relations
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations
2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Example $2.3.3$