Let \(A = \{1, 2, 3\}\). Then, the number of relations containing (1, 2) and (1, 3), which are reflexive and symmetric but not transitive, is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1
A reflexive relation must contain (1, 1), (2, 2), (3, 3). A symmetric relation containing (1, 2) and (1, 3) must also contain (2, 1) and (3, 1). So every such relation contains
\(R_1 = \{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1)\}\)
The only pairs of \(A \times A\) left are (2, 3) and (3, 2), and by symmetry they come together or not at all.
Without them: \(R_1\) is not transitive, since \((2,1), (1,3) \in R_1\) but \((2,3) \notin R_1\).
With them: the relation is all of \(A \times A\), which is transitive.
So exactly one relation, \(R_1\), is reflexive and symmetric but not transitive.
NCERT (2026-27 edition):
Relations and Functions
