A relation R on a set S is said to be circular if aRb and bRc together imply cRa. Which of the following options is/are correct? A. If a relation R is reflexive and symmetric, then R is an equivalence relation. B. If a relation R is circular and symmetric, then R is an equivalence relation. C. If a relation R is reflexive and circular, then R is an equivalence relation. D. If a relation R is transitive and circular, then R is an equivalence relation.

GATE 2021 · Discrete Mathematics · Relations · medium

Answer: C. If a relation R is reflexive and circular, then R is an equivalence relation.

1. Eliminate Option A (reflexive + symmetric is NOT enough): Reflexive + symmetric lacks transitivity. Counterexample: on {1,2,3}, R = {(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}. Not transitive since 1R2 and 2R3 but not 1R3.
2. Check Option B (circular + symmetric): A circular + symmetric relation need not be reflexive. Example: empty relation is both circular and symmetric but not reflexive. So B is FALSE.
3. Verify Option C (reflexive + circular => equivalence): Reflexive: given. Symmetric: aRa and aRb give bRa (circular). Transitive: aRb and bRc give cRa (circular); by symmetry aRc. All three hold => equivalence.
4. Check Option D (transitive + circular): Transitive + circular does not guarantee reflexivity. Counterexample: empty relation is transitive and circular but not reflexive. D is FALSE.