Embibe Experts Solutions for Chapter: Set Theory and Relations, Exercise 3: Exercise-3

Let $P=\left\{\theta :\sin \theta -\cos \theta =\sqrt{2}\cos \theta \right\}$ and $Q=\left\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \right\}$ be two sets. Then

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

Let $R$ be the set of real numbers.

Statement- $1: A=\left\{\right(x, y)\in R\times R: y-x$ is an integer} is an equivalence relation on $R.$

Statement- $2: B=\left\{\right(x, y)\in R\times R: x=\alpha y$ for some rational number $\alpha )$ is an equivalence relation on $R.$

Consider the following relation $\mathrm{R}$, on the set of real square matrices of order $3$.

$\mathrm{R}=\{\left(\mathrm{A}, \mathrm{B}\right)\mid \mathrm{A}={\mathrm{P}}^{-1}BP$, for some invertible matrix $P\}$.

Statement - $1 : R$, is equivalence relation.

Statement - $2 :$ For any two invertible $3\times 3$ matrices $\mathrm{M}$ and $\mathrm{N}, {\left(\mathrm{MN}\right)}^{-1}={\mathrm{N}}^{-1}{\mathrm{M}}^{-1}.$

Two sets $A$ and $B$, are as under $:A=\{\left(a, b\right)\in R\times R : \left|a-5\right|<1$ and $\left|b-5\right|<1\}$; $B=\left\{\left(a, b\right)\in R\times R:4\left(a-6^2\right)+9\left(b-5^2\right)\le 36\right\}$. Then.

Let $\mathrm{Z}$ be the set of integers. If $\mathrm{A}=\left\{\mathrm{x}\in \mathrm{Z} : 2^{\left(x+2\right)\left({\mathrm{x}}^2-5\mathrm{x}+6\right)}=1\right]$ and $\mathrm{B}=\left\{\mathrm{x}\in \mathrm{Z} : -3<2\mathrm{x}-1<9\right\}$ then the
number of subsets of the set $\mathrm{A}\times \mathrm{B}$, is

Let $S=\{1,2,3,\dots ,100\}.$ The number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even, is

Let $P=\left\{\theta :\sin \theta -\cos \theta =\sqrt{2}\cos \theta \right\}$ and $Q=\left\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \right\}$ be two sets. Then,