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]
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).