# Definition:Reflexivity

## Definition

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

### Reflexive

$\RR$ is reflexive if and only if:

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

### Coreflexive

$\mathcal R$ is coreflexive if and only if:

$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$

### Antireflexive

$\RR$ is antireflexive if and only if:

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

### Non-reflexive

$\mathcal R$ is non-reflexive if and only if it is neither reflexive nor antireflexive.

## Also see

• Results about reflexivity of relations can be found here.