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GATE 2021 CS – Question 53

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 2 marks · Multiple select

A relation R is said to be circular if aRb and bRc together imply cRa. Which of the following options is/are correct?

  1. If a relation S is reflexive and symmetric, then S is an equivalence relation.
  2. If a relation S is circular and symmetric, then S is an equivalence relation.
  3. If a relation S is reflexive and circular, then S is an equivalence relation.
  4. If a relation S is transitive and circular, then S is an equivalence relation.

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Show answer and explanation

Correct answer: (C) If a relation S is reflexive and circular, then S is an equivalence relation.

Explanation

Reflexive + circular: aRb and bRb give bRa (symmetric). Then aRb, bRc give cRa, hence aRc by symmetry (transitive). So it is an equivalence relation. The other options lack reflexivity or fail otherwise.