Question

Le L be the set of all lines in a plane and R be the relation in L. defined as R = {(\(L_1, L_2\)): \(L_1\) is perpendicular to \(L_2\)} then R is:
A) Reflexive
B) Symmetric
C) Neither reflexive nor transitive
D) Transitive
E) Neither reflexive nor symmetric
Choose the correct answer from the options given below:

• A. B, C only
• B. A, D only
• C. C, D, E only
• D. E only

Answer 1

The Correct Option is A

To determine the nature of the relation \(R\), we need to evaluate whether it is reflexive, symmetric, or transitive. The relation is defined as \(R=\{(L_1,L_2) : L_1 \text{ is perpendicular to } L_2\}\).

Reflexive: A relation \(R\) is reflexive if every element is related to itself. For lines \(L_1\), to be reflexive, \(L_1\) must be perpendicular to itself. However, no line can be perpendicular to itself. Therefore, \(R\) is not reflexive.

Symmetric: A relation \(R\) is symmetric if whenever \((L_1, L_2) \in R\), then \((L_2, L_1) \in R\). If \(L_1 \perp L_2\), then \(L_2 \perp L_1\). Thus, if one line is perpendicular to another, the reverse is also true. Therefore, \(R\) is symmetric.

Transitive: A relation \(R\) is transitive if whenever \((L_1, L_2) \in R\) and \((L_2, L_3) \in R\), then \((L_1, L_3) \in R\). If \(L_1 \perp L_2\) and \(L_2 \perp L_3\), there is no guarantee that \(L_1\) is perpendicular to \(L_3\). Therefore, \(R\) is not transitive.

Therefore, the relation \(R\) is symmetric but neither reflexive nor transitive.
Correct answer: B, C only.