Let $A=\left\{1,2,3\right\}$ be any set the minimum number of elements that must be added to the relation $\left\{R=\left\{\right(1,2),(2,3\left)\right\}\right\}$ on the set A so that it is an equivalence relation :

The relation $R_1=\left\{\right(1,2),(2,3),(1,1),(2,2)$,$(3,3)\}\text{ has }5\text{ elements and is reflexive, while }{R}_2=\{(1\text{, }$$2),(2,3),(1,1),(2,2),(2,1),(3,2\left)\right(3,3\left)\right\}$ has 7 elements and it is reflexive and symmetric. 

$\text{ Finally, }{R}_3=\left\{\right(1,2),(2,3),(1,1),(2,2),(2,1),(3,2)\text{, }$$(3,3),(1,3),(3,1)\}\text{ has }9$ elements and it is reflexive, symmetric and transitive. Starting from 2 elements, therefore, the minimum number of elements to be added is 7.