Definition:Reflexivity

Definition

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

Reflexive

$\mathcal R$ is reflexive iff:

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

Coreflexive

$\mathcal R$ is coreflexive (pronounced co-reflexive, not core-flexive) iff:

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

Antireflexive

$\mathcal R$ is antireflexive iff:

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

Non-reflexive

$\mathcal R$ is non-reflexive iff it is neither reflexive nor antireflexive.

Also see

• Symmetry of Relations
• Transitivity of Relations

• Results about reflexivity of relations can be found here.