What is Equivalence Relation – Definition and Examples

Here you will learn what is equivalence relation on a set with definition and examples.

Let’s begin –

What is Equivalence Relation ?

Definition : A relation R on a set A is said to be an equivalence relation on A iff it is

(i) reflexive i.e. (a, a) $\in$ R for all a $\in$ A.

(ii) symmetric i.e (a, b) $\in$ R $\implies$ (b, a) $\in$ R for all a, b $\in$ A.

(iii) transitive i.e. (a, b) $\in$ R and (b, c) $\in$ R $\implies$ (a, c) $\in$ R for all a, b, c $\in$ A.

Also Read : Types of Relations in Math

Example : If A = {1,2,3}, then the relation R = {(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(1,3).(3,2),(3,1)} is the equivalence relation on A,

because {(1,1),(2,2),(3,3)} $\in$ R hence it is reflexive,

{(2,1),(1,2),(2,3),(3,2),(1,3),(3,1)} $\in$ R  hence it is symmetric on A,

{(1,2),(2,3),(1,3)} and {(1,3),(3,2),(1,2)} $\in$ R hence it is transitive.

Example : Let R be a relation on the set of all lines in a plane defined by $(l_1, l_2)$ $\in$ R $\iff$ line $l_1$ is parallel to line $l_2$. Show that R is an equivalence relation.

Solution : Let L be the given set of all lines in a plane. Then, we observe the following properties.

Reflexive : For each line l $\in$ L, we have

l || l $\implies$ (l, l) $\in$ R for all l $\in$ L

$\implies$ R is reflexive.

Symmetric : Let $l_1$, $l_2$ $\in$ L such that $(l_1, l_2)$ $\in$ R. Then,

$(l_1, l_2)$ $\in$ R $\implies$ $(l_1$ || $l_2)$ $\implies$ $(l_2$ || $l_1)$ $\in$ R.

So, R is symmetric on L.

Transitive : Let $l_1$, $l_2$, $l_3$ $\in$ L such that $(l_1, l_2)$ $\in$ R and $(l_2, l_3)$  $\in$  R. Then,

$(l_1, l_2)$ $\in$ R and $(l_2, l_3)$ $\in$ R $\implies$ $l_1$ || $l_2$ and $l_2$ || $l_3$ $\implies$ $l_1$ || $l_3$ $\implies$ $(l_1, l_3)$ $\in$ R.

So, R is transitive on L.

Hence, R being reflexive, symmetric and transitive is an equivalence relation on L.