Definition:Reflexivity

From ProofWiki
Jump to navigation Jump to search

Definition

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


Reflexive

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

$\forall x \in S: \tuple {x, x} \in \mathcal R$


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

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

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


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.