Definition:Antireflexive Relation

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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 for antireflexive.

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.


Distinctness

Let $S$ be a set.

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

$\forall x, y \in S: x \mathrel \RR y$ if and only if $x$ is distinct from $y$

Then $\RR$ is antireflexive.


Also see

  • Results about antireflexive 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

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