The minimum number of elements that must be added to the relation $R=\left\{(a,b),(b,c)\right\}$ on the set $\left\{a,b,c\right\}$ so that it becomes symmetric and transitive is:

For transitive $(a,b) (b,c)\in R$  $\Rightarrow (\mathrm{a},\mathrm{c})\in \mathrm{R}$
For symmetry  $(\mathrm{b},\mathrm{a})(\mathrm{c},\mathrm{b})(\mathrm{c},\mathrm{a})\in \mathrm{R}$
 $\begin{array}{l}(\mathrm{c},\mathrm{a})\in \mathrm{R},(\mathrm{a},\mathrm{c})\in \mathrm{R}\Rightarrow (\mathrm{c},\mathrm{c})\in \mathrm{R}\\ (\mathrm{a},\mathrm{b})(\mathrm{b},\mathrm{a})\in \mathrm{R}\Rightarrow (\mathrm{a},\mathrm{a})\in \mathrm{R}\end{array}$

$\therefore \text{ Elements to be added }$

$\left\{(b,a),(c,b),(a,c),(c,a),(a,a),(b,b),(c,c)\right\}$
Total 7 elements to be added