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Question

Let R be a relation on the set N of natural numbers defined by nRmn is a factor of m(i.e.,n|m). Then R is


A

Reflexive and symmetric

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B

Transitive and symmetric

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C

Equivalence

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D

Reflexive, transitive but not symmetric

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Solution

The correct option is D

Reflexive, transitive but not symmetric


Explanation for the correct option:

Step-1: Checking reflexivity:

The relation is defined as

nRmn|m

Now consider m=n

nRnn|n

Since n=1×n

Therefore n is a factor of itself.

Thus R is reflexive.

Step-2: Checking Symmetricity:

Put n=2&m=4

2R42|4 is true

But 4R24|2

i.e 4 is not factor of 2

Thus R is not symmetric.

Step-3: Checking transtivity:

Let nRmn|m then assume m=nx,xN

And mRpm|p then assume p=my,y,mN

Now,

nRpn|pn|myn|nxy

Therefore, n is a factor of p

Thus R is transitive,

Hence, option (D) is correct.


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