Let A = \ 1, 2, 3, 4\ and R = \ ( 1, 2 ) , ( 2, 3 ) , ( 1, 4 )\ be...

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 $A and R ⊂ S$

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

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

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