Definition:Reflexive Relation

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Definition

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


Definition 1

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

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


Definition 2

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

$\Delta_S \subseteq \mathcal R$


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.