1. Given relation

We have $R=\{(1,2),(2,3),(3,3)\}$ on the set $A=\{1,2,3,4\}.$

We need to add the minimum number of ordered pairs so that $R$ becomes an equivalence relation on $A$.

An equivalence relation must be:

1. Reflexive
2. Symmetric
3. Transitive

2. Reflexive requirement

For reflexivity on $A=\{1,2,3,4\}$, we must have $(1,1),(2,2),(3,3),(4,4) \in R.$

Currently, only $(3,3)$ is present. So we must add: $(1,1),(2,2),(4,4)$ That is 3 pairs.

3. Symmetric requirement

If $(a,b)\in R$, then $(b,a)$ must also be in $R$.

Given:

• $(1,2)\in R$ so we need $(2,1)$
• $(2,3)\in R$ so we need $(3,2)$
• $(3,3)$ already satisfies symmetry by itself

So we must add: $(2,1),(3,2)$ That is 2 more pairs.

Current necessary pairs now are: $\{(1,2),(2,3),(3,3),(1,1),(2,2),(4,4),(2,1),(3,2)\}.$

4. Transitive requirement

Now check what transitivity forces.

Because $(1,2)\in R \quad \text{and} \quad (2,3)\in R,$ we must have $(1,3)\in R.$ So add $(1,3)$.

Also, since symmetry gave $(3,2)$ and $(2,1)$, $(3,2)\in R \quad \text{and} \quad (2,1)\in R$ forces $(3,1)\in R.$ So add $(3,1)$.

Now check if anything else is needed among $1,2,3$.

Once we have $(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),$ all ordered pairs among $\{1,2,3\}$ are present, so transitivity and symmetry are fully satisfied on this block.

Element $4$ has no relation with $1,2,3$, and only $(4,4)$ is needed for it to form its own equivalence class.

Thus the final equivalence relation must be:

• all $9$ pairs on $\{1,2,3\}$, and
• $(4,4)$.

So total pairs required in the equivalence relation are: $9+1=10.$

Initially, $R$ has $3$ pairs. Hence number of pairs to be added is: $10-3=7.$

5. Option check

• A: $9$ — incorrect
• B: $8$ — incorrect
• C: $7$ — correct
• D: $10$ — incorrect

6. Final answer

The minimum number of elements to be added is $\boxed{7}.$