Question

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

• A. If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation.
• B. If a relation $S$ is circular and symmetric, then $S$ is an equivalence relation.
• C. If a relation $S$ is reflexive and circular, then $S$ is an equivalence relation.
• D. If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.

Hint:

To verify equivalence relations, always check reflexivity first; without it, equivalence cannot be guaranteed.

Answer 1

The Correct Option is C

Step 1: Recall properties of an equivalence relation.
A relation must be reflexive, symmetric, and transitive to be an equivalence relation.

Step 2: Analyse circularity.
Circularity implies a form of reverse implication: if $aRb$ and $bRc$, then $cRa$. This property alone does not guarantee symmetry or transitivity unless combined with reflexivity.

Step 3: Evaluate each option.
Option (A): Reflexive and symmetric does not guarantee transitivity, so this is false.
Option (B): Circularity and symmetry do not ensure reflexivity, hence false.
Option (C): Reflexivity ensures $aRa$. Combined with circularity, symmetry and transitivity can be derived, making $S$ an equivalence relation.
Option (D): Transitivity and circularity do not imply reflexivity, so this is false.

Step 4: Conclusion.
Only option (C) correctly leads to an equivalence relation.