Definition:Reflexive Relation

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

Definition 1

$\RR$ is reflexive if and only if:

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

Definition 2

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

$\Delta_S \subseteq \RR$


Arbitrary Reflexive Relation

Let $V_0 = \set {a, b, c}$.

A reflexive relation on $V_0$ must include the ordered pairs:

$\tuple {a, a}, \tuple {b, b}, \tuple {c, c}$

Reflexive Relation on Cartesian Plane

The subset of the Cartesian plane defined as:

$\RR := \set {\tuple {x, y} \in \R^2: x \le y \le x + 1}$

determines a relation on $\R^2$ which is reflexive, but neither symmetric nor transitive.

Distance Less than 1

Let $\sim$ be the relation on the set of real numbers $\R$ defined as:

$x \sim y \iff \size {x - y} < 1$

Then $\sim$ is reflexive and symmetric, but not transitive.

Also see

  • Results about reflexivity of relations can be found here.