The number of relations, on the set \ 1, 2, 3\ containing (1, 2)...

Given,

Set $A = \left\{1, 2, 3\right\}$

Now Cartesian product $A \times A = \left\{\left(1, 1\right), \left(2, 1\right), \left(1, 2\right), . . . . . . . . , \left(3, 3\right)\right\}$

Now, given the relation is reflexive,

So, \left(1, 1\right), \left(2, 2\right), \left(3, 3\right) \in R$</p><p>Also given \left(1, 2\right), \left(2, 3\right) \in R, \left(1, 3\right)$ must $\in R$

Now finding, Possible cases :

Case-1: All of $\left(2, 1\right), \left(3, 2\right), \left(3, 1\right) \notin R \to 1$ relation.

Case-2: Only one of \left(2, 1\right), \left(3, 2\right), \left(3, 1\right) \in R \to 3$relations.</p><p>For example if relation is \left\{\left(1, 1\right), \left(2, 2\right), \left(3, 3\right), \left(2, 1\right)\right\}$ then it is reflexive as well as transitive as \left(2, 2\right), \left(2, 1\right) \to \left(2, 1\right)\text{is present in relation}$</p><p>Note that exactly two of \left(2, 1\right), \left(3, 2\right), \left(3, 1\right) \in R$ is not possible because if two of these $\in R$, third must $\in R$ to make relation transitive.

Total number of relations $= 4$