Let L be the set of all lines in a plane and R be the relation on set L defined by $R = \{(L_1, L_2): L_1 ⊥ L_2\}$. Then R is
(A) an equivalence Relation
(B) a symmetric Relation
(C) not a transitive Relation
(D) a reflexive Relation
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (C) only
Relation $R=\{(L_{1},L_{2}) : L_{1}\perp L_{2}\}$ on the set of all lines in a plane.
Reflexive: a line cannot be perpendicular to itself, so $(L,L)\notin R$. Hence $R$ is not reflexive → (D) false.
Symmetric: if $L_{1}\perp L_{2}$ then $L_{2}\perp L_{1}$, so $R$ is symmetric → (B) true.
Transitive: if $L_{1}\perp L_{2}$ and $L_{2}\perp L_{3}$, then $L_{1}$ is parallel to $L_{3}$, not perpendicular, so $R$ is not transitive → (C) true.
Hence $R$ is not an equivalence relation → (A) false.
Final answer: (B) and (C)