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.

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