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Question

If R is an equivalence relation on a set A, then R1 is

A
Reflexive relation only
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B
Symmetric but not transitive relation
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C
Reflexive, symmetric and transitive relation
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D
reflexive and transitive but not symmetric relation
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Solution

The correct option is C Reflexive, symmetric and transitive relation
Let (a,a)R, aA
as R is reflexive
Hence (a,a)R1
R1 is also reflexive relation.

Let (a,b)R, a,bA
(b,a)R, as R is symmetric
Hence (b,a),(a,b)R1
R1 is also symmetric relation

Let (a,b),(b,c)R, a,b,cA
(a,c)R as R is transitive
(c,b),(b,a)R1 and (c,a)R1
R1 is also transitive relation.

Hence R1 is equivalence relation.

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