Reflexive relation
From Freepedia
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:
- <math>\forall a \in X,\ a R a</math>
For example, "is greater than or equal to" is a reflexive relation but "is greater than" is irreflexive.
Other examples of reflexive relations include:
- "is equal to" (equality)
- "is a subset of" (set inclusion)
- "is less than or equal to" and "is greater than or equal to" (inequality)
- "divides" (divisibility)
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
- <math>\forall a \in X,\ \lnot a R a</math>.
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).
