1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
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 circular, then S is an equivalence relation.
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
If a relation S is circular and symmetric, then S is an equivalence relation.
No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today!
C
If a relation S is transitive and circular, then S is an equivalence relation.
No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today!
D
If a relation S is reflexive and symmetric, then S is an equivalence relation.
No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today!
Open in App
Solution

## The correct option is A 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.

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Types of Relations
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program