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.
