Let \( R \) be a relation on the set \( N \) of natural numbers defined by \( nRm \) if \( n \) divides \( m \). Then \( R \) is (A) Reflexive and symmetric (B) Transitive and symmetric (C) Equivalence (D) Reflexive, transitive but not symmetric

Question

Let \( R \) be a relation on the set \( N \) of natural numbers defined by \( nRm \) if \( n \) divides \( m \). Then \( R \) is (A) Reflexive and symmetric (B) Transitive and symmetric (C) Equivalence (D) Reflexive, transitive but not symmetric

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Answer: (D) Reflexive, transitive but not symmetric
Let there be a natural number n,
We know that n divides n, which implies nRn.
So, Every natural number is related to itself in relation R.
Thus relation R is reflexive .

Let there be three natural numbers a,b,c and let aRb,bRc
aRb implies a divides b and bRc implies b divides c, which combinedly implies that a divides c i.e. aRc.
So, Relation R is also transitive .

Let there be two natural numbers a,b and let aRb,
aRb implies a divides b but it can't be assured that b necessarily divides a.
For ex, 2R4 as 2 divides 4 but 4 does not divide 2 .
Thus Relation R is not symmetric .