Reflexive/irreflexive relation Glossary
Definition
A binary relation is reflexive (on a domain of discourse) iff it relates everything (in that domain) to itself. That is, a reflexive relation R satisfies the condition
ALLx Rxx
R is irreflexive iff it does not relate anything to itself. That is, iff it satisfies
ALLx NOTRxx
Comments
Note that 'irreflexive' does not mean 'not reflexive'. It is a much stronger condition. Most relations, in fact, are neither reflexive nor irreflexive.
Where R is a relation between sets of things and single things, as in the case of the relation of logical consequence, it may satisfy a more general condition also called reflexivity (or extended reflexivity), meaning that R(s,a) holds whenever a is a member of s.
Examples
- The relation of weak order "less than or equal to" in the standard number systems, written '≤', is reflexive.
- The strict ordering relation, written '<', is irreflexive, as no number is properly less than itself.