Reflexive relation
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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>
A relation that is not reflexive is irreflexive or aliorelative.
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.
The statement
- <math>\forall a \in X,\ a = a<math>
is called the axiom of equality in some systems.de:Reflexivität es:Relacin reflexiva pl:Relacja zwrotna zh:自反关系