Question

An integer m is said to be related to another integer n if m is an integral multiple of $n$. This relation in $Z$ is reflexive, symmetric and transitive.

Answer 1

Define a relation based on the given statement.
$R=\left\{\right(m,n):m=nk\text{where}m,n,kϵZ\}$

Check Reflexivity
If $(m,m)ϵR$, then $m=mk\Rightarrow k=1ϵZ$
$\therefore R$ is Reflexive.

Check whether $R$ is symmetric
Assuming that $m\ne n$
Let $(m,n)ϵR\Rightarrow m=nk$
$\Rightarrow n=\left(\frac{1}{k}\right)m(\because \frac{1}{k}\notin z,\text{if}m\ne n)$
$\Rightarrow (n,m)\notin R$
Therefore, $R$ is not symmetric

Check whether $R$ is transitive
Let $(m,n)ϵR$ and $(n,p)ϵR$ $\Rightarrow m=n{k}_1$ and $n=p{k}_2$
$\Rightarrow m=p{k}_1{k}_2$
$\Rightarrow m=pk(\text{Let}k=k_1{k}_2)$
$\Rightarrow (m,p)ϵR$
Therefore, $R$ is transitive

Hence, the given statement is false.