Definition:Antireflexive Relation
Definition
Let $\RR \subseteq S \times S$ be a relation in $S$.
$\RR$ is antireflexive if and only if:
- $\forall x \in S: \tuple {x, x} \notin \RR$
Also known as
Some sources use the term irreflexive.
However, as irreflexive is also found in other sources to mean non-reflexive, it is better to use the clumsier, but less ambiguous, antireflexive.
The term aliorelative can sometimes be found, but this is rare.
Examples
Non-Equality
The relation $\ne$ on the set of natural numbers $\N$ is antireflexive.
Strict Ordering
The relation $<$ on one of the standard number systems $\N$, $\Z$, $\Q$ and $\R$ is antireflexive.
Also see
- Results about reflexivity of relations can be found here.
Linguistic Note
The earliest use of the word aliorelative is found in the works of Charles Sanders Peirce, who probably coined it.
The word derives from the Latin alius, meaning other, together with relative, hence meaning a relation whose terms are related only to other terms.
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 4.5$: Properties of Relations
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Definition $19.2$
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: aliorelative
- 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