1. Given relation

Let $S=\{1,2,3,\ldots,10\}$ and let $M$ be the set of all subsets of $S$.

The relation on $M$ is

So, $(A,B)\in R$ exactly when $A$ and $B$ have at least one common element.

We check whether $R$ is reflexive, symmetric, and transitive.

2. Reflexive?

A relation on $M$ is reflexive if for every $A\in M$, we have $(A,A)\in R$.

Now,

But $\varnothing\in M$ since it is a subset of $S$.

For $A=\varnothing$,

so

Hence $R$ is not reflexive.

3. Symmetric?

A relation is symmetric if

If $(A,B)\in R$, then

But intersection is commutative:

So,

which means

Therefore, $R$ is symmetric.

4. Transitive?

A relation is transitive if

We test by counterexample.

Take

Then

so $(A,B)\in R$.

Also,

so $(B,C)\in R$.

But

so $(A,C)\notin R$.

Hence $R$ is not transitive.

5. Conclusion

• Not reflexive
• Symmetric
• Not transitive

Therefore, the relation is symmetric only.

So the correct option is: