# 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:

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

### Reflexive Relation on Cartesian Plane

The subset of the Cartesian plane defined as:

$\RR := \set {\tuple {x, y} \in \R^2: x \le y \le x + 1}$

determines a relation on $\R^2$ which is reflexive, but neither symmetric nor transitive.

## Also see

• Results about reflexivity of relations can be found here.