Let A = {1, 2, 3}. The number of equivalence relations containing (1, 3) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
An equivalence relation must be reflexive, so it contains (1, 1), (2, 2), (3, 3). It is symmetric, so with (1, 3) it also contains (3, 1). The smallest such relation is
\(R_1 = \{(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)\}\)
\(R_1\) is also transitive, so it is an equivalence relation.
Any larger one must relate 2 to 1 or to 3. Then, by symmetry and transitivity, 2 is related to both, and every pair is in the relation:
\(R_2 = A \times A\)
So there are exactly 2 equivalence relations, \(R_1\) and \(R_2\).
NCERT (2026-27 edition):
Relations and Functions
