GATE CS 1996 Q2.2 — Sets, Relations and Functions

We are given a relation $R$ on a collection of sets defined by:

$$A R B \iff A \cap B = \emptyset$$

Let's analyze the properties of $R$:

1. Reflexivity: For $R$ to be reflexive, $A R A$ must hold for every set $A$ in the collection.
   $A R A \iff A \cap A = \emptyset \iff A = \emptyset$.
   For any non-empty set $A$ (e.g., $A = \{1\}$), $A \cap A = \{1\} \neq \emptyset$. Thus, $R$ is not reflexive.
2. Symmetry: If $A R B$, then $A \cap B = \emptyset$.
   Since set intersection is commutative, $B \cap A = A \cap B = \emptyset$, which implies $B R A$.
   Thus, $R$ is symmetric.
3. Transitivity: If $A R B$ and $B R C$, does it imply $A R C$?
   Let $A = \{1\}$, $B = \{2\}$, and $C = \{1\}$.
   • $A \cap B = \{1\} \cap \{2\} = \emptyset \implies A R B$
   • $B \cap C = \{2\} \cap \{1\} = \emptyset \implies B R C$
   • But $A \cap C = \{1\} \cap \{1\} = \{1\} \neq \emptyset \implies A \not R C$
   Thus, $R$ is not transitive.

Therefore, $R$ is symmetric and not transitive, which matches Option B.