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Question

On the set N of all natural numbers, define the relation R×R be such that the greatest common divisor of a and b is 2. Then R is

A
reflexive only
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B
symmetric only
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C
transitive only
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D
equivalence
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Solution

The correct option is C symmetric only
If R is a relation on set N, then for it to be reflexive,
(a,a)R for all aN. But (4,4)R. Hence not reflexive.
If greatest common divisor of a and b is 2,
then greatest common divisor of b and a is also 2.
So, the given relation is symmetric. (aRbbRa).
The given relation is not transitive as (6,10)R and (10,18)R, but (6,18)R (greatest common divisor of 6 and 18 is 6)

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