1. Given relation

On the set $A=\{a,b,c\}$,

We must add the minimum number of ordered pairs so that the new relation becomes both symmetric and transitive.

2. Condition for symmetry

A relation is symmetric if:

Since $(a,b)\in R$, we must add:

Since $(b,c)\in R$, we must add:

So symmetry forces at least these 2 additions.

Now relation becomes:

3. Condition for transitivity

A relation is transitive if:

Check all possible chains in $R_1$:

• $(a,b)$ and $(b,c)$ imply $(a,c)$ must be present.
• $(c,b)$ and $(b,a)$ imply $(c,a)$ must be present.
• $(a,b)$ and $(b,a)$ imply $(a,a)$ must be present.
• $(b,a)$ and $(a,b)$ imply $(b,b)$ must be present.
• $(b,c)$ and $(c,b)$ imply $(b,b)$ must be present (already needed).
• $(c,b)$ and $(b,c)$ imply $(c,c)$ must be present.

Thus we must add:

That is 5 more pairs.

4. Total additions so far

Already added for symmetry: 2 pairs

Added for transitivity: 5 pairs

Total added:

5. Check final relation

Now the relation contains:

This is actually the full relation on $\{a,b,c\}$, i.e.

which has $9$ elements. Since the original relation had $2$ elements, the number added is:

This is symmetric and transitive.

6. Conclusion

The minimum number of elements to be added is:

So the correct option is A.