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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 xSyySx
So S is symmetric

Transitive:

Let xSy and ySz
Now by circular property we get zSx
and by symmetry property proved above,
we get
zSxxSz
So xSy and ySzxSz
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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