Properties of relations

When a relation links a set to itself ($R\colon A\to A$) it may enjoy some notable properties, which classify it into important families.

Property — Properties of a relation

A relation $R$ on a set $A$ can be:

• Reflexive: for every $a\in A$, $a\mathbin{R} a$ (every element is related to itself).
• Symmetric: if $a\mathbin{R} b$ then $b\mathbin{R} a$ (every arrow has a corresponding opposite arrow).
• Antisymmetric: if $a\mathbin{R} b$ and $b\mathbin{R} a$ then $a=b$ (arrows there and back only on the loop).
• Transitive: if $a\mathbin{R} b$ and $b\mathbin{R} c$ then $a\mathbin{R} c$.

A reflexive, symmetric and transitive relation is an equivalence relation; a reflexive, antisymmetric and transitive one is an order relation.

The two combinations give rise to the two great families of relations we shall study: equivalences, which group together similar elements, and orders, which line them up.

Links

Topics: Set theory
Concepts: Relation · Order relation · Equivalence relation