# Transitive Relation

Let A be any set. A relation R on A is said to be a transitive relation if and only if,**(a,b) $\in$ R and (b,c) $\in$ R**

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

that means aRb and bRc

$\Rightarrow $ aRc for all a,b,c $\in$ A.

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

that means aRb and bRc

$\Rightarrow $ aRc for all a,b,c $\in$ A.

## Examples on Transitive Relation

**Example :1**Prove that the relation R on the set N of all natural numbers defined by (x,y) $\in$ R $\Leftrightarrow$ x divides y, for all x,y $\in$ N is transitive.

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

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

$\Rightarrow $ x divides y and y divides z

$\Rightarrow$ there exists p,q $\in$ N such that y = xp and z = yq

$\Rightarrow$ z = (xp)q

$\Rightarrow$ z = x(pq)

$\Rightarrow$ x divides z

**[ since pq $\in$ N]**

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

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

Hence, R is a transitive-relation on N.

**Example 2:**Let L be the set of all straight lines in a plane. Then the relation 'is parallel to' on L is a transitive-relation .

**Solution :**According to the relation l1 || l2 and l2 || l3 $\Rightarrow $ l1 || l3

And Let l1,l2,l3 $\in$ L.

$\Rightarrow $ L is a transitive-relation.

**Example 3 :**If R,S are both transitive,then prove that R∪S is not transitive.

**Solution :**Let A ={1, 2, 3}, let R = {(1, 2)}, let S = {(2, 1)}.

So R∪S = {(1, 2), (2, 1)}, which is not transitive, because, for instance,

1 is related to 2 and

2 is related to 1

but 1 is not related to 1.

∴ R∪S is not transitive.

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