# Definition:Reflexive Relation

## Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

### Definition 1

$\RR$ is reflexive if and only if:

$\forall x \in S: \tuple {x, x} \in \RR$

### Definition 2

$\RR$ is reflexive if and only if it is a superset of the diagonal relation:

$\Delta_S \subseteq \RR$

## Examples

### Arbitrary Reflexive Relation

Let $V_0 = \set {a, b, c}$.

A reflexive relation on $V_0$ must include the ordered pairs:

$\tuple {a, a}, \tuple {b, b}, \tuple {c, c}$

## Also see

• Results about reflexivity of relations can be found here.