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 transitive and circular, then S is an equivalence relation.b)If a relation S is reflexive 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 circular and symmetric, then S is an equivalence relation.Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Let's start with Option 3;

Option 3:if a relation S is reflexive and circular, then S is an equivalence relation.
It is correct.
- S is reflexive and circular; see that if are able to derive Symmetric and transitive property from these given properties.
aRa ∈ S and aRb ∈ S => bRa ∈ S //Using Reflexive and circular
aRb and bRa both exist then the relation is Symmetric.
- Now
aRb and bRc together => cRa and according to symmetricity ≡ aRc
aRb and bRc together => aRc
Hence Transitivity satisfied.
Hence S is an Equivalence Relation.
Example : S = { {a,a) (b.b) (c,c) } (diagonal relation)

Option 1: If a relation S is transitive and circular, then S is an equivalence relation.
It is not correct.
Consider a example of empty relation or S = {(a,b)}; it is transitive and circular but its not Equivalence Relation.
Hence S is not an equivalence Relation.

Option 2: If a relation S is reflexive and symmetric, then S is an equivalence relation.
It is not correct. For a Relation S to be Equivalence Reflexive, Symmetric and Transitive are required.

Option 4: if a relation S is circular and symmetric, then S is an equivalence relation.
It is not correct.
Consider a example of empty relation or S = {(a,b) , (b,a), (a,a)}; It is circular and symmetric but its not Equivalence relation.
Hence S is not an equivalence Relation.

Answer

Understanding Circular Relations
A circular relation R implies that if aRb and bRc hold true, then it must also follow that cRa. This property fundamentally shapes how we assess the nature of the relation.
Examining Option C: Reflexive and Circular
- Reflexive: A relation S is reflexive if for every element a in the set, aSa holds true.
- Circular: Given the definition, if we have aSa and aSb, then we can derive that bSa through the circular property.
Proof of Equivalence
1. Reflexivity: Since S is reflexive, for any element a, aSa holds true.
2. Symmetry: If aSb is true, then by the circular property, bSa must also hold true.
3. Transitivity: Assume aSb and bSc are true. By the circular property, we can derive cSa.
Since S satisfies all three properties—reflexivity, symmetry, and transitivity— it qualifies as an equivalence relation.
Why Other Options are Incorrect
- Option A (Transitive and Circular): A circular relation does not guarantee reflexivity or symmetry, which are essential for equivalence.
- Option B (Reflexive and Symmetric): Although these properties hold, transitivity is not ensured by circularity alone.
- Option D (Circular and Symmetric): Without reflexivity, we cannot conclude that S is an equivalence relation.
Conclusion
Thus, among the options provided, only option C correctly proves that a relation S, when both reflexive and circular, is indeed an equivalence relation.

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