Let $$S = \{1, 2, 3, \ldots, 10\}$$. Suppose $$M$$ is the set of all the subsets of $$S$$, then the relation $$R = \{(A, B) : A \cap B \neq \phi; \; A, B \in M\}$$ is :

$$M$$ is the power set of $$S = \{1, 2, ..., 10\}$$. The relation $$R = \{(A, B) : A \cap B \neq \phi\}$$.

Reflexive? Is $$(A, A) \in R$$ for all $$A \in M$$? We need $$A \cap A \neq \phi$$, i.e., $$A \neq \phi$$. But $$\phi \in M$$ and $$\phi \cap \phi = \phi$$. So $$(\phi, \phi) \notin R$$. Not reflexive.

Symmetric? If $$A \cap B \neq \phi$$, then $$B \cap A \neq \phi$$. Yes, symmetric.

Transitive? Consider $$A = \{1\}$$, $$B = \{1, 2\}$$, $$C = \{2\}$$. Then $$A \cap B = \{1\} \neq \phi$$ and $$B \cap C = \{2\} \neq \phi$$, but $$A \cap C = \phi$$. Not transitive.

The relation is symmetric only. The answer corresponds to Option (4).