Reflexive Relation

Definition

Given a set $A$, a relation $R:A\to A\to \mathrm{Prop}$ is reflexive iff every element is related to itself: $\forall x:A.R(x,x).$

Irreflexivity

A relation $R$ is irreflexive iff no element is related to itself: $\forall x:A.\neg R(x,x).$

Irreflexivity is strictly stronger than the negation of reflexivity: a relation may be neither reflexive nor irreflexive.

Examples

• The equality relation $=$ on any set is reflexive.
• The strict less-than relation $<$ on $\mathbb{N}$ is irreflexive.
• The relation $\le$ on $\mathbb{N}$ is reflexive but not irreflexive.

Related Concepts

• Transitive Relation
• Symmetric Relation
• Anti-Symmetric Relation
• Equivalence Relation
• Partial Order
• Predicate

References

• munkres2000-topology