Mark the correct alternative in the following question:
Let R be a relation on the set N of natural numbers defined by nRm iff n divides m. Then, R is
(a) Reflexive and symmetric
(b) Transitive and symmetric
(c) Equivalence
(d) Reflexive, transitive but not symmetric
[NCERT EXEMPLAR]
We have,
$R=\{(m, n): n$ divides $m ; m, n \in \mathbf{N}\}$
As, $m$ divides $m$
$\Rightarrow(m, m) \in R \forall m \in \mathbf{N}$
So, $R$ is reflexive
Since, $(2,1) \in R$ i. e. 1 divides 2
but 2 cannot divide 1 i. e. $(2,1) \notin R$
So, $R$ is not symmetric
Let $(m, n) \in R$ and $(n, p) \in R$. Then,
$n$ divides $m$ and $p$ divides $n$
$\Rightarrow p$ divides $m$
$\Rightarrow(m, p) \in R$
So, $R$ is transitive
Hence, the correct alternative is option (d).
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