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Question

On the set N of all natural numbers define the relation R by aRb, if and only if the GCD of a and b is 2, then R is


A

reflexive but not symmetric

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B

only symmetric

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C

reflexive and transitive

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D

reflexive, symmetric and transitive

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Solution

The correct option is B

only symmetric


Explanation of correct option:

Determine the relation:
Given, aRbGCD of a and b is 2

To find if the relation is reflexive, the following conditions should satisfy

(a,a)R

aRaGCD(a,a)=a

Let's say,

(3,3)R

(4,4)R

Hence, it is not reflexive

For Symmetric

(a,b)RGCD(a,b)=2

(b,a)RGCD(b,a)=2

Hence, it is symmetric.

To find if the relation is transitive, the following conditions should satisfy

(a,b)RGCD(a,b)=2(b,c)RGCD(b,c)=2(a,c)RGCD(a,c)=2

Say

(4,6)RGCD(4,6)=2GCD(6,8)=2GCD(4,8)=4

Hence, it is not transitive.

Therefore, R is a symmetric relation.

Hence, the correct answer is option (B).


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