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

If a relation S is reflexive and circular, then S is an equivalence relation.

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.

GATE CSE 2021 Set 1 Q53: Circular relations and equivalence relations

Engineering Mathematics
2 marks
MSQ
No negative marking

A relation $\mathbf{R}$ is said to be circular if $a\mathbf{R}b$ and $b\mathbf{R}c$ together imply $c\mathbf{R}a$ .

Which of the following options is/are correct?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

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Answer: C (official GATE 2021 final answer key)

Concept used: Reflexive + circular gives symmetry (take $c = b$ ) and then transitivity; the other pairs miss reflexivity or transitivity.

First step: With reflexivity, apply circularity to $a\mathbf{S}b$ and $b\mathbf{S}b$ ; for the other options, test the empty relation on a non-empty set.

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key

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GATE CSE 2021 Set 1 Q53: Circular relations and equivalence relations

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