Let $A=\{1,2,3\} .$ Then number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is

The given set is $A=\{1,2,3\}$.

The smallest relation containing $(1,2)$ and $(1,3)$ which is reflexive and symmetric, but not transitive is given by:

$R=\{(1,1),\,(2,2),\,(3,3),\,(1,2),\,(1,3),\,(2,1),\,(3,1)\}$

This is because relation $R$ is reflexive as $(1,1),\,(2,2),\,(3,3) \in R$

Relation $R$ is symmetric since $(1,2),\,(2,1) \in R$ and $(1,3),\,(3,1) \in R$

But relation $R$ is not transitive as $(3,1),\,(1,2) \in R,$ but $(3,2)\notin R$

Now, if we add any two pairs $(3,2)$ and $(2,3) $ (or both) to relation $R$, then relation $R$ will become transitive.

Hence, the total number of desired relations is one.

The correct answer is $D$.