Let R be a non-empty relation on a collection of sets defined by A R B if and only if A ∩ B = ∅. Then, (pick the true statement) A. R is reflexive and transitive B. R is symmetric and not transitive C. R is an equivalence relation D. R is not reflexive and not symmetric

GATE 1996 · Discrete Mathematics · Relations · medium

Answer: B. R is symmetric and not transitive.

1. Check reflexivity: For any non-empty set A in the collection, A ∩ A = A ≠ ∅, so (A, A) is not in R. R is NOT reflexive.
2. Check symmetry: If A ∩ B = ∅ then B ∩ A = ∅, so A R B implies B R A. R is symmetric.
3. Check transitivity with counterexample: Let A = {1}, B = {2}, C = {1, 3}. A ∩ B = ∅ so A R B. B ∩ C = ∅ so B R C. But A ∩ C = {1} ≠ ∅, so A R C fails. R is NOT transitive.