Mark the correct alternative in the following question:
Question:

Mark the correct alternative in the following question:

Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $I R m$ iff $/$ is perpendicular to $m$ for all $I, m \in L$. Then, $R$ is

(a) reflexive

(b) symmetric

(c) transitive

(d) none of these

[NCERT EXEMPLAR]

Solution:

We have,

$R=\{(l, m): l$ is perpendicular to $m ; l, m \in L\}$

As, $l$ is not perpencular to $l$

$\Rightarrow(l, l) \notin R$

So, $R$ is not reflexive relation

Let $(l, m) \in R$

$\Rightarrow l$ is perpendicular to $m$

$\Rightarrow m$ is also perpendicular to $l$

$\Rightarrow(m, l) \in R$

So, $R$ is symmetric relation

Let $(l, m) \in R$ and $(m, n) \in R$

$\Rightarrow l$ is perpendicular to $m$ and $m$ is perpendicular to $n$

$\Rightarrow l$ is parallel to $n$                                         (Lines perpendicular to same line are parallel)

$\Rightarrow(m, l) \notin R$

So, $R$ is not transitive relation

Hence, the correct alternative is option (b).