Let $$R=\left\{(1,2),(2,3),(3,3)\right\}$$ be a relation defined on the set $$\left\{1,2,3,4\right\}$$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Given $$R = \{(1,2), (2,3), (3,3)\}$$ on the set $$\{1, 2, 3, 4\}$$, find the minimum number of elements to add so that R becomes an equivalence relation.

An equivalence relation must be reflexive, symmetric, and transitive.

Since $$(1,2)$$ and $$(2,3)$$ are in R, by symmetry and transitivity, 1, 2, and 3 must all be in the same equivalence class. Element 4 is in its own class.

So the equivalence classes are $$\{1, 2, 3\}$$ and $$\{4\}$$.

Reflexive: $$(1,1), (2,2), (3,3), (4,4)$$

Symmetric pairs for class $$\{1,2,3\}$$: $$(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)$$

Total required pairs: $$(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)$$ = 10 pairs

Already in R: $$(1,2), (2,3), (3,3)$$ = 3 pairs

Need to add: $$(1,1), (2,2), (4,4), (2,1), (1,3), (3,1), (3,2)$$ = 7 pairs

The correct answer is Option 2: 7.