Definition:Reflexive Relation/Class Theory

Definition

Let $V$ be a basic universe.

Let $A$ be a class, by definition a subclass of $V$.

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

$\RR$ is reflexive on $A$ if and only if:

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

Also defined as

Some sources define a reflexive relation on a basic universe $V$ as:

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

which is technically speaking a quasi-reflexive relation, and not a reflexive relation as such.

Also see

• Results about reflexive relations can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering