Definition:Non-reflexive Relation

Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

$\mathcal R$ is non-reflexive if and only if it is neither reflexive nor antireflexive.

Also known as

Some sources use the term irreflexive.

However, as irreflexive is also found in other sources to mean antireflexive, it is better to use the clumsier, but less ambiguous, non-reflexive.

Examples

Arbitrary Non-reflexive Relation 1

Let $V_1 = \set {y, z}$.

Let $S$ be the relation on $V_1$ defined as:

$S = \set {\tuple {y, y}, \tuple {y, z} }$

Then $S$ is neither:

a reflexive relation, as $\tuple {z, z} \notin S$

nor:

an antireflexive relation, as $\tuple {y, y} \in S$

Thus $S$ is a non-reflexive relation.

Arbitrary Non-reflexive Relation 2

Let $S = \set {a, b}$.

Let $\RR$ be the relation on $S$ defined as:

$\RR = \set {\paren {a, a} }$

Then $\RR$ is neither:

a reflexive relation, as $\tuple {b, b} \notin \RR$

nor:

an antireflexive relation, as $\tuple {a, a} \in \RR$

Thus $\RR$ is a non-reflexive relation.

Also see

Definition:Reflexivity

Results about reflexivity of relations can be found here.

Sources

1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 4.5$: Properties of Relations
1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations