JEE Mathematics 2024 Question with Solution

Given: $M$ is the set of all subsets of $S = \{1,2,3...10\}$ and relation $R$ is checked for reflexivity, symmetry, and transitivity.

Find: Which properties are satisfied by $R$.

The solution concludes that the relation is symmetric only and the correct option is D.

Check the properties one by one:

1. Reflexive: For reflexivity, every subset $A \in M$ must satisfy $A \cap A \neq \emptyset$. But $\emptyset \in M$ and

so the required condition fails for the empty set. Hence the relation is not reflexive.

2. Symmetric: If $(A,B) \in R$, then the intersection condition between $A$ and $B$ is unchanged when the order is reversed because intersection is commutative. Therefore, if $A$ is related to $B$, then $B$ is related to $A$. Hence the relation is symmetric.

3. Transitive: Even if $(A,B) \in R$ and $(B,C) \in R$, it need not follow that $(A,C) \in R$. Therefore, the relation is not transitive.

So the relation is symmetric only.

Note: The solution analyzes the condition with $A \cap B \neq \emptyset$ instead of the question text $A \cap B = \emptyset$, but it still concludes symmetric only, matching the listed correct option.

Therefore, the correct option is D.

Given: Relation on the family of subsets of $S$.

Find: Whether the relation is reflexive, symmetric, and transitive.

• Reflexivity test: inspect $A = \emptyset$.
• Symmetry test: compare $A \cap B$ and $B \cap A$.
• Transitivity test: the middle set $B$ being related to both $A$ and $C$ does not force $A$ to be related to $C$.

Thus, among the given options, the relation is taken as symmetric only, so the correct option is D.