Let $S$ be a set of all square matrices of order $2 .$ If a relation $\mathrm{R}$ defined on set $S$ such that $\mathrm{ARB}\Rightarrow \mathrm{AB}=\mathrm{O},$ where $\mathrm{O}$ is zero square matirx of order $2,$ then relation $\mathrm{R}$ is $(\mathrm{A},\mathrm{B}\in \mathrm{S})$

Let $\mathrm{A}=\mathrm{N}\times \mathrm{N}$ be the Cartesian product of $\mathrm{N}$ and $\mathrm{N}$. Let $\mathrm{S}=\left\{\left(\left(\mathrm{m},\mathrm{n}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{A}\times \mathrm{A}:\mathrm{m}+\mathrm{q}=\mathrm{n}+\mathrm{p}\right\}$.

 Consider the following statements

$I.$ If $\left(\left(\mathrm{m},\mathrm{n}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{S},$ and $\left(\left(\mathrm{p},\mathrm{q}\right),\left(\mathrm{r},\mathrm{s}\right)\right)\in \mathrm{S}$ then $\left(\left(\mathrm{r},\mathrm{s}\right),\left(\mathrm{m},\mathrm{n}\right)\right)\in \mathrm{S}$
$II.$ There exists at least one element $\left(\left(\mathrm{m},\mathrm{n}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{S}$ such that $\left(\left(\mathrm{p},\mathrm{q}\right),\left(\mathrm{m},\mathrm{n}\right)\right)\notin \mathrm{S}$ 
Which of the statements given above is/are correct?

Let $A=\left\{1,2,3,4\right\}$ and $R$ be a relation in $A$ given, then relation $R$ is by $R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),\left(1,2\right),\left(2,1\right),\left(3,1\right),\left(1,3\right)\right\}$

Consider the following relations:
$\mathrm{R}:\left\{\right(\mathrm{x}, \mathrm{y})\mid \mathrm{x}, \mathrm{y}$ are real numbers and $x=wy$ for some rational number $w\}$

$S=\{\left(\frac{m}{n},\frac{p}{q}\right)\mid m, n, p$ and $q$ are integers such that $n, q\ne 0$ and $qm=pn\}$
Then