Question

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

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Answer 1

The correct option is B

Symmetric

Explanation 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)