GATE CS 2002 Q2.17 — Sets, Relations and Functions

To determine the properties of the empty relation $S = \emptyset$ on the set $A = \{1, 2, 3\}$, we test the standard definitions of binary relations:

1. Reflexivity: A relation $R$ on $A$ is reflexive if for every $a \in A$, $(a, a) \in R$.
   For $S = \emptyset$, the elements $(1,1), (2,2), (3,3)$ are not in $S$. Thus, $S$ is not reflexive.
2. Symmetry: A relation $R$ on $A$ is symmetric if for all $a, b \in A$, $(a, b) \in R \implies (b, a) \in R$.
   Since $S = \emptyset$, the premise $(a, b) \in S$ is always false. Therefore, the implication holds vacuously. Thus, $S$ is symmetric.
3. Transitivity: A relation $R$ on $A$ is transitive if for all $a, b, c \in A$, $((a, b) \in R \text{ and } (b, c) \in R) \implies (a, c) \in R$.
   Since $S = \emptyset$, the premise $((a, b) \in S \text{ and } (b, c) \in S)$ is always false. Therefore, this implication also holds vacuously. Thus, $S$ is transitive.

Since the relation $S$ is both transitive and symmetric (but not reflexive), the correct option is D.