Definition:Quasi-Reflexive Relation

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

Definition 1

$\RR$ is quasi-reflexive if and only if:

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

Definition 2

$\RR$ is quasi-reflexive if and only if:

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

where $\Field \RR$ denotes the field of $\RR$.

Definition 3

$\RR$ is quasi-reflexive if and only if $\RR$ is both left quasi-reflexive and right quasi-reflexive.

Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe

Let $\RR \subseteq V \times V$ be a relation.

$\RR$ is quasi-reflexive if and only if:

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

where $\Field \RR$ denotes the field of $\RR$.

Also see

• Equivalence of Definitions of Quasi-Reflexive Relation

• Definition:Left Quasi-Reflexive Relation
• Definition:Right Quasi-Reflexive Relation

• Results about quasi-reflexive relations can be found here.