Embibe Experts Solutions for Chapter: Set Theory and Relations, Exercise 2: Exercise-2

For real numbers $x$ and $y$, we write $x R y\Rightarrow x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is-

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 $\mathrm{A}=\mathrm{Z},$ the set of integers. Let ${\mathrm{R}}_1=\left\{\right(\mathrm{m},\mathrm{n})\in \mathrm{Z}\times \mathrm{Z}:(\mathrm{m}+4\mathrm{n})$ is divisible by $5$ in $\mathrm{Z}\}$ , Let ${\mathrm{R}}_2=\left\{\right(\mathrm{m},\mathrm{n})\in \mathrm{Z}\times \mathrm{Z}:(\mathrm{m}+9\mathrm{n})$ is divisibile by $5$ in $Z$
Which one of the following is correct?

Let $\mathrm{X}$ be the set of all persons living in a state. Elements $\mathrm{x}, \mathrm{y}$ in $\mathrm{X}$ are said to be related if ${}^{\text{'}}\mathrm{x}<{\mathrm{y}}^{\text{'}}$ whenever $\mathrm{y}$ is 5 years older than $\mathrm{x} .$ Which one of the following is correct?

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

For $n, m\in N, n\mid m$ means that $n$ is a factor of $m,$ then relation$\mid$is