The minimum number of elements that must be added to relation R = \...

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)$</p><p><strong>Symmetric and Transitive:</strong></p><p>For symmetric and transitivity, we must add</p><p>$\left(b, a\right), \left(c, b\right), \left(d, b\right)$</p><p>\left(a, c\right), \left(a, d\right), \left(c, d\right), \left(d, c\right), \left(c, a\right), \left(d, a\right)$

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