Let \( L \) be the set of all lines in a plane and \( R \) be the relation on \( L \) defined as
\( R=\{(L_{1},L_{2}) : L_{1}\text{ is perpendicular to }L_{2}\} \). Then \( R \) is
(A) Symmetric
(B) Reflexive
(C) Transitive
(D) neither reflexive nor transitive
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
Two lines are related when they are perpendicular.
If \( L_{1}\perp L_{2} \), then \( L_{2}\perp L_{1} \).
No line is perpendicular to itself.
If \( L_{1}\perp L_{2} \) and \( L_{2}\perp L_{3} \), then \( L_{1}\parallel L_{3} \), not perpendicular.
So R is symmetric, but it is neither reflexive nor transitive.
Checking the statements
(A) Symmetric — Correct. \( L_{1}\perp L_{2} \) gives \( L_{2}\perp L_{1} \).
(B) Reflexive — Incorrect. No line is perpendicular to itself.
(C) Transitive — Incorrect. \( L_{1}\perp L_{2} \) and \( L_{2}\perp L_{3} \) make \( L_{1}\parallel L_{3} \).
(D) neither reflexive nor transitive — Correct. R fails both reflexivity and transitivity.
NCERT (2026-27 edition):
Types of Relations
