Given,

$A=\left\{1, 2, 3, 4\right\}$ and $R=\left\{(1, 2), (2, 3), (1, 4)\right\}$ be a relation on $A$.

Now, given $S$ be the equivalence relation on $A$ and $R\subset S$

So, for $S$ to be equivalence it should be Reflexive, Symmetric and Transitive on given relation,

Now, for reflexive Relation we need, $\left\{\left(1,1\right), \left(2,2\right), \left(3,3\right), \left(4,4\right)\right\}\to 4$ elements,

For Symmetric relation we need, $\left\{\left(2,1\right), \left(3,2\right), \left(4,1\right)\right\}\to 3$ elements,

And for Transitive relation we need, $\left\{\left(1,3\right), \left(3,1\right), \left(3,4\right), \left(4,2\right), \left(2,4\right), \left(4,3\right)\right\}\to 6$ elements

So, minimum $13$ elements should be added in $R$

Hence , $n\left(S\right)=16$ elements.