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Question

# Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $\alpha R\beta ⇔\alpha \perp \beta ,\alpha ,\beta \in L.$ Then $R$ is

A

Reflexive

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B

Symmetric

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C

Transitive

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D

None of these

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Solution

## The correct option is B SymmetricExplanation for the correct option:Finding relation of $R$:Checking for Reflexive Relation:Given: $\alpha \perp \alpha$If one line is perpendicular, then $R$ cannot be reflexive.Hence, $R$ is not reflexive.Checking for Transitive Relation: Given $\alpha \perp \beta \text{and}\beta \perp r$Let's consider $\left(L,M\right)$ and $\left(M,N\right)$ be ordered pairs of $R$. If $L\perp M$ and $M\perp N$, it does not mean that $L\perp N$.Similarly, if $\alpha \perp \beta \text{and}\beta \perp r$, it does not imply that $\alpha \perp r$Hence, $R$ is not transitive.Checking for Symmetric Relation: Given $\alpha \perp \beta$If one line is perpendicular, there is a possibility that the other line will also be perpendicular.Therefore, $R$ can be symmetric.Since, $R$ is symmetric, the correct answer is option (B)

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