Let R be the relation over the set of all straight lines in a plane such that

Question:

Let $R$ be the relation over the set of all straight lines in a plane such that $I_{1} R I_{2} \Leftrightarrow I_{1} \perp I_{2}$. Then, $R$ is

(a) symmetric
(b) reflexive
(c) transitive
(d) an equivalence relation

Solution:

(a) symmetric

A = Set of all straight lines in the plane

$R=\left\{\left(l_{1}, l_{2}\right): l_{1}, l_{2} \in A: l_{1} \perp l_{2}\right\}$

Reflexivity: $l_{1}$ is not $\perp l_{1}$

$\Rightarrow\left(l_{1}, l_{1}\right) \notin R$

So, $R$ is not reflexive on $A$.

Symmetry : Let $\left(l_{1}, l_{2}\right) \in R$

$\Rightarrow l_{1} \perp l_{2}$

$\Rightarrow l_{2} \perp l_{1}$

$\Rightarrow\left(l_{2}, l_{1}\right) \in R$

So, $R$ is symmetric on $A$.

Transitivity : Let $\left(l_{1}, l_{2}\right) \in R,\left(l_{2}, l_{3}\right) \in R$

$\Rightarrow l_{1} \perp l_{2}$ and $l_{2} \perp l_{3}$

But $l_{1}$ is not $\perp l_{3}$

$\Rightarrow\left(l_{1}, l_{3}\right) \notin R$

So, $R$ is not transitive on $A$.

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