Definition:Reflexive Relation

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Definition

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:

$\Delta_S \subseteq \RR$


Class Theory

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

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:

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


Examples

Arbitrary Reflexive Relation

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

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

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


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.


Divisibility on Natural Numbers

Let $\sim$ be the relation on the set of natural numbers $\N$ defined as:

$x \sim y \iff x \divides y$

where $\divides$ denotes divisibility.

Then $\sim$ is reflexive.


Also see

  • Results about reflexive relations can be found here.


Sources

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