Solution
The smallest relation \(R_1\)
containing \((1, 2)\) and \((2, 3)\) which is reflexive and transitive
but not symmetric is
\[
\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}.
\]
Now, if we add the pair \((2, 1)\)
to \(R_1\) to get \(R_2\), then the relation will be reflexive,
transitive but not symmetric.
Similarly, we can obtain \(R_3\) by
adding \((3, 2)\) to \(R_1\) to get the desired relation.
However, we cannot add two pairs \((2, 1), (3,
2)\) or single pair \((3, 1)\)
to \(R_1\) at a time, as by doing so,
we will be forced to add the remaining pair in order to maintain
transitivity and in the process, the relation will become symmetric also
which is not required.
Thus, the total number of desired relations is three.
