# What is Transitive Relation – Definition and Examples

Here you will learn what is transitive relation on set with definition and examples based on it.

Let’s begin –

## What is Transitive Relation ?

Definition : Let A be any set. A relation R on A is said to be a transitive relation iff

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

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

Note : The identity and the universal relations on a void set are transitive.

Also Read : Types of Relations in Math

Given below are some transitive relation examples.

Example : If A = {1,2,3}, then the relation R = {(1,2),(2,3),(1,3),(2,2)} is the transitive relation on A, But $$R_1$$ = {(1,2),(2,3),(2,2),(1,1)}  is not a transitive relation on A, because (1,3) and (2,3) $$\in$$ $$R_1$$ but (1,3) $$\notin$$ $$R_1$$.

Example : The relation R on the set N of all natural numbers defined by

(x, y)  $$\in$$  R  $$\iff$$  x divides y, for all x, y $$\in$$ N is transitive.

Solution : Let x, y, z $$\in$$ N be such that (x, y) $$\in$$ R and (y, z) $$\in$$ R. Then,

(x, y) $$\in$$ R and (y, z) $$\in$$ R

$$\implies$$ x divides y and, y divides z.

$$\implies$$ There exist p, q $$\in$$ N such that y = xp and z = yq

$$\implies$$ z = (xp) q

$$\implies$$ z = x (pq)

$$\implies$$ x divides z

(\implies\) (x, z) $$\in$$ R

Thus, (x, y) $$\in$$ R, (y, z) $$\in$$ R $$\implies$$ (x, z) $$\in$$ R for all x, y, z $$\in$$ N.

Hence, R is a transitive relation on N.