The number of equivalence relation on the set \(\{1,2, 3\}\) containing (1, 2) and (2, 1) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
An equivalence relation must be reflexive, symmetric and transitive. Every such relation here contains (1, 1), (2, 2), (3, 3) (reflexive) and (1, 2), (2, 1) (given), so the smallest one is
\(R_1 = \{(1,1),(2,2),(3,3),(1,2),(2,1)\}\)
\(R_1\) is reflexive, symmetric and transitive, so it is one equivalence relation.
The only pairs left are (1, 3), (3, 1), (2, 3) and (3, 2). Adding any one of them forces all the others:
\((1,3)\) needs \((3,1)\) by symmetry, then \((2,1),(1,3)\) need \((2,3)\) by transitivity, and \((3,2)\) by symmetry.
The same happens starting from any of the four pairs. So the only other equivalence relation is the whole set
\(R_2 = \{1, 2, 3\} \times \{1, 2, 3\}\)
Hence there are 2 such equivalence relations.
NCERT (2026-27 edition):
Relations and Functions
