The minimum number of elements that must be added to relation R = a , b , b , c , b , d on the set a , b , c…

Let

$A=\{a, b, c,d\}, R : A\to A$

$R=\left\{\left(a,b\right),\left(b,c\right),\left(b,d\right)\right\}$

Reflexive:

For reflexivity, we must add $\left(a,a\right), \left(b,b\right), \left(c,c\right), \left(d,d\right)$

Symmetric and Transitive:

For symmetric and transitivity, we must add

$\left(b,a\right), \left(c,b\right), \left(d,b\right)$

$\left(a,c\right), \left(a,d\right), \left(c,d\right), \left(d,c\right), \left(c,a\right), \left(d,a\right)$

So, we can say that minimum $13$ elements are required to make an equivalence relation.