Section 1.4: Composition of Functions

Solution
The smallest equivalence relation \(R_1\) containing \((1, 2)\) and \((2, 1)\) is
\[ \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}. \]

Now we are left with only 4 pairs namely (2, 3), (3, 2), (1, 3) and (3, 1).
If we add any one, say (2, 3) to \(R_1\), then for symmetry we must add (3, 2) also and now for transitivity we are forced to add (1, 3) and (3, 1).
Thus, the only equivalence relation bigger than \(R_1\) is the universal relation.
This shows that the total number of equivalence relations containing \((1, 2)\) and \((2, 1)\) is two.