Let R be a relation defined over N, where N is set of natural numbers, defined as"$\displaystyle \mathrm{m} R n$ if and only if $\displaystyle \mathrm{m}$ is a multiple of $\displaystyle n, \mathrm{m}, n \in N$. "Find whether $\displaystyle R$ is reflexive, symmetric and transitive or not.

Let $\displaystyle x \in N$. Then we know that x is a multiple of itself. $\displaystyle \Rightarrow x R x$ Hence, R is reflexive. We have $\displaystyle 2,8 \in N$ such that $\displaystyle 8$ is a multiple of $\displaystyle 2$ $\displaystyle \Rightarrow 8 R 2$ But, $\displaystyle 2$ is not a multiple of $\displaystyle 8$ . Hence, $\displaystyle 2$ is not R -related to 8. Therefore, R is not symmetric. Let $\displaystyle x, y, z \in N$ such that $\displaystyle x R y, y R z$ Then $\displaystyle x=\mathrm{m} y, y=n z$ for some $\displaystyle \mathrm{m}, n \in N$ $\displaystyle \Rightarrow x=\mathrm{m} n z \Rightarrow x=p z$, where $\displaystyle p=\mathrm{m} n \in N$. Hence, $\displaystyle x R z$ Therefore, R is transitive.