The minimum number of elements that must be added to the relation R = {(1, 2), (2, 3)} on the 

set {1, 2, 3} so that it is an equivalence relation

New Relation = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (2, 1), (3, 1), (3, 2), (3, 3)}

$$\begin{gathered} \text{ For }\mathrm{R}\text{ to be equivalence relation, it should be reflexive, symmetric and transitive. } \\ \text{ Now, for }\mathrm{R}\text{ reflexive: } \\ \Rightarrow (\mathrm{x},\mathrm{x})\in \mathrm{R}\text{ for all }\mathrm{x}\in \{1,2,3\} \\ \Rightarrow (1,1),(2,2),(3,3)\in \mathrm{R} \\ \text{ For }\mathrm{R}\text{ is symmetric: } \\ \Rightarrow (1,2),(2,3)\in \mathrm{R} \\ \Rightarrow (2,1),(3,2)\in \mathrm{R} \\ \text{ For }\mathrm{R}\text{ is transitive: } \\ \Rightarrow (1,2),(2,3)\in \mathrm{R} \\ \Rightarrow (1,3)\in \mathrm{R} \\ \text{ Hence, minimum }7\text{ elements must be added. } \end{gathered}$$