Definition:Non-Reflexive Relation
Definition
Let $\RR \subseteq S \times S$ be a relation in $S$.
$\RR$ is non-reflexive if and only if it is neither reflexive nor antireflexive.
Also known as
Some sources do not hyphenate: nonreflexive relation.
Some sources use the term irreflexive relation when non-reflexive is the sense intended.
However, as irreflexive is also found in other sources to mean antireflexive, it is better to use the clumsier, but less ambiguous, non-reflexive.
Some sources avoid giving a name to a relation which is neither reflexive nor antireflexive, merely referring to such a relation by its properties.
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
- Results about non-reflexive relations can be found here.
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions