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

## The correct option is D If a relation S is reflexive and circular, then S is an equivalence relation.Let S be reflexive and circular. Let us checking symmetry: Symmetry: Let xSy Now since S is reflexive ySy true. So xSy and ySy is true Now by circular property we get, ySx So xSy⇒ySx So S is symmetric Transitive: Let xSy and ySz Now by circular property we get zSx and by symmetry property proved above, we get zSx⇒xSz So xSy and ySz⇒xSz So S is transitive. So S is reflexive, symmetric and transitive and hence an equivalence relation. So option(a) is true. Option(b):Let S be circular and symmetric. Let S be defined on set {1,2,3} Now empty relation is circular and symmetric but not reflexive. So S need not be an equivalence relation. So option(b) is false. Option(c):Let S be transitive and circular. Let S be defined on the set {1,2,3}. Now empty relation again satisfies transitive and circular but is not reflexive. So S need not be an equivalence relation. So option(c) is false. Option(d):Reflexive and symmetric need not be transitive for example on {1,2,3} S={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)} is reflexive and symmetric. But it is not transitive because (1,2) and (2,3) belong to S but (1,3) does not. So option(d) is false.

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