Question

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]

Answer 1

We have,

R = {(m, n) : n divides m; m, n $\in$ N}

$$\begin{gathered} \mathrm{As},m\mathrm{divides}m \\ \Rightarrow \left(m,m\right)\in R\forall m\in \mathbf{N} \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive} \\ \mathrm{Since},\left(2,1\right)\in R\mathrm{i}.\mathrm{e}.1\mathrm{divides}2 \\ \mathrm{but}2\mathrm{cannot}\mathrm{divide}1\mathrm{i}.\mathrm{e}.\left(2,1\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{symmetric} \\ \mathrm{Let}\left(m,n\right)\in R\mathit{}\mathrm{and}\left(n,p\right)\in R.\mathrm{Then}, \\ n\mathrm{divides}m\mathit{}\mathrm{and}p\mathrm{divides}n \\ \Rightarrow p\mathrm{divides}m \\ \Rightarrow \left(m,p\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{transitive} \end{gathered}$$

Hence, the correct alternative is option (d).