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Question

Let I be the set of integers and R be a relation on I defined by R={(x,y):(x-y)isdivisibleby11,x,yI}. Then R is


A

An equivalence relation

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B

Symmetric only

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C

Reflexive only

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D

Transitive only

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Solution

The correct option is A

An equivalence relation


Explanation of the correct Option:

The correct option is A:An equivalence relation

To verify the equivalence relation we need to show that the relation is reflexive , symmetric and transitive

Reflexive: For all aI,a-a=0, which is divisible by 11.
Thus, (a,a)R for all aIR is reflexive

Symmetric: Let (a,b)R(a-b) is divisible by 11
-(a-b) is divisible by 11
(b-a) is divisible by 11
(b,a)R
R is symmetric.

Transitive : Let (x,y)R and (y,z)R
(x-y) is divisible by 11 and (y-z) is divisible by 11
(x-y)+(y-z) is divisible by 11
(x-z) is divisible by 11
(x,z)R
R is transitive

Hence R is an equivalence relation on I×I.

Therefore, the correct option is (A)


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