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:

<math>\forall a \in X,\ a R a<math>

A relation that is not reflexive is irreflexive.

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.