# 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.