If R is the smallest equivalence relation on the set { 1 , 2 , 3 , 4 } such that { ( 1 , 2 ) , ( 1 , 3 ) } ⊂…

Given: $R=\left\{\left(1,3\right),\left(1,2\right)\right\}$

To make $R$ as Reflexive, the required elements are $\left(1,1\right), \left(2,2\right), \left(3,3\right) \text{and} \left(4,4\right)$

To make $R$ as Symmetric, the required elements are $\left(3,1\right) \text{and} \left(2,1\right)$

To make $R$ as Transitive, the required elements are $\left(3,2\right) \text{and} \left(2,3\right)$

So, $8$ elements are required to make $R$ an equivalence relation.

Hence, the total number of elements in $R$ is $10$.