# 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