Definition:Reflexive Relation/Class Theory
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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