A relation $R$ is said to be circular if $aRb$ and $bRc$ together imply $cRa$. Which of the following options is/are correct?
Show Hint
- If a relation $S$ is reflexive and symmetric, then $S$ is an equivalence relation.
- If a relation $S$ is circular and symmetric, then $S$ is an equivalence relation.
- If a relation $S$ is reflexive and circular, then $S$ is an equivalence relation.
- If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
The Correct Option is C
Solution and Explanation
Step 1: Conditions for an equivalence relation.
For a relation to qualify as an equivalence relation, it must satisfy all three properties:
reflexivity, symmetry, and transitivity.
Step 2: Understand the meaning of circularity.
Circularity indicates a backward linkage in the relation. Specifically, if
aRb and bRc, then cRa.
By itself, this condition does not directly ensure symmetry or transitivity unless additional properties are present.
Step 3: Examine each option.
Option (A): A relation that is reflexive and symmetric may still fail to be transitive. Therefore, this combination is insufficient.
Option (B): Circularity together with symmetry does not guarantee that every element is related to itself, so reflexivity is missing.
Option (C): Reflexivity gives aRa for all a. When combined with circularity, symmetry and transitivity can be logically deduced, satisfying all requirements of an equivalence relation.
Option (D): Even with transitivity and circularity, reflexivity is not ensured, so this option fails.
Final Conclusion:
The relation becomes an equivalence relation only under option (C).
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