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

We need to find the minimum number of elements to add to $$R = \{(a,b), (b,c)\}$$ on $$\{a, b, c\}$$ to make it symmetric and transitive.

To begin,

For symmetry, if $$(x,y) \in R$$, then $$(y,x) \in R$$.

Add: $$(b,a)$$ and $$(c,b)$$.

$$R = \{(a,b), (b,a), (b,c), (c,b)\}$$

Next,

For transitivity, if $$(x,y) \in R$$ and $$(y,z) \in R$$, then $$(x,z) \in R$$.

From $$(a,b)$$ and $$(b,a)$$: need $$(a,a)$$ ✓ Add

From $$(a,b)$$ and $$(b,c)$$: need $$(a,c)$$ ✓ Add

From $$(b,a)$$ and $$(a,b)$$: need $$(b,b)$$ ✓ Add

From $$(c,b)$$ and $$(b,a)$$: need $$(c,a)$$ ✓ Add

From $$(c,b)$$ and $$(b,c)$$: need $$(c,c)$$ ✓ Add

We also need to check new pairs: $$(a,c)$$ and $$(c,a)$$ → $$(a,a)$$ ✓ already added. $$(a,c)$$ and $$(c,b)$$ → $$(a,b)$$ ✓ already in R.

Total elements added: $$(b,a), (c,b), (a,a), (a,c), (b,b), (c,a), (c,c) = 7$$

The correct answer is Option 2: $$7$$.