Types of Relations

Relations defined on a single set $A$ (subsets of $A \times A$) can have special structural properties. Three properties are especially important.

Let $R$ be a relation on a set $A$.

Equivalence relation. A relation that is reflexive, symmetric and transitive all at once is called an equivalence relation. Equivalence relations formalise the everyday idea of "being alike in some respect" — for example, "has the same remainder on division by $5$" is an equivalence relation on the integers.

Some named relations.

• The identity relation on $A$ is $I_A = \{(a, a) : a \in A\}$ — each element related only to itself. It is reflexive, symmetric and transitive, hence an equivalence relation.
• The universal relation on $A$ is the whole of $A \times A$ (everything related to everything). It too is an equivalence relation.
• The empty relation $\varnothing$ (no element related to any element) is symmetric and transitive but not reflexive (unless $A$ itself is empty).

How to test a given relation. To classify a relation given in roster form, check each property against its definition:

• For reflexive, confirm that $(a, a)$ is present for every element $a$ of the underlying set — a single missing self-pair breaks it.
• For symmetric, confirm that for every pair $(a, b)$ present, the reversed pair $(b, a)$ is also present. …