# Reflexive relation

In logic and mathematics, a binary relation R over a set X is reflexive if for all a in X, a is related to itself.

In mathematical notation, this is:

$\displaystyle \forall a \in X,\ a R a$

For example, "is greater than or equal to" is a reflexive relation but "is greater than" is irreflexive.

Other examples of reflexive relations include:

A reflexive relation that is also transitive is a preorder. A preorder that is antisymmetric is a partial order. A preorder that is symmetric is an equivalence relation.

An irreflexive (or aliorelative) relation is one that no element bears to itself, i.e., a binary relation R that satisfies

$\displaystyle \forall a \in X,\ \lnot a R a$ .

This is a stronger condition than failure of reflexivity, so a binary relation may be reflexive, irreflexive, or neither. "Less than" and "greater than" are irreflexive relations (no element is less than or greater than itself).