The number of equivalence relations in the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is 

The smallest equivalence relation  ${\mathrm{R}}_1$ containing $(1,2)$ and $(2, 1)$ is $\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\}$.

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 ${\mathrm{R}}_1\text{, }$ then for symmetry we must add  $(3,2)$ also and now for transitivity we are forced to  add $(1,3)\text{ and }(3,1)$.

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.