# 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.