Karnataka Board Solutions for Chapter: Relations and Functions, Exercise 5: Miscellaneous Exercise on Chapter 1

Given a non-empty set $X,$ consider $P\left(X\right)$ which is the set of all subsets of $X.$ Define the relation $R$ in $P\left(X\right)$ as follows:
For subsets $A,$ $B$ in $P\left(X\right),$ $A\mathrm{R}B$ if and only if $A\subset B$. Is $R$ an equivalence relation on $P\left(X\right)$? Justify your answer.

Find the number of all onto functions from the set $\{1,2,3,\dots ,n\}$ to itself

Let $S=\left\{a,b,c\right\}$ and $T=\left\{1,2,3\right\}.$ Find $F^{-1}$ of the given function $F$ from $S$ to $T$, if it exists.
 $F=\left\{\left(a,3\right),\left(b,2\right),\left(c,1\right)\right\}$

$F=\left\{\left(a,2\right),\left(b,1\right),\left(c,1\right)\right\}$

Let $A=\left\{-1,0,1,2\right\}$, $B=\left\{-4,-2,0,2\right\}$ and $f,$ $g:A\to B$ be functions defined by $f\left(x\right)=x^2-x,x\in A$ and $g\left(x\right)=2\left|x-\frac{1}{2}\right|-1,$ $x\in A$. Are $f$ and $g$ equal? Justify your answer.

Let $A=\left\{1,2,3\right\}.$Then number of relations containing $\left(1,2\right)$ and $\left(1,3\right)$ which are reflexive and symmetric but not transitive is

Let $A=\left\{1,2,3\right\}.$ Then number of equivalence relations containing $\left(1,2\right)$ is

Let $f:R\to R$ be the Signum Function defined as
$f\left(x\right)=\left\{\begin{array}{ll}1,& x>0\\ -1,& x<0\end{array}\right.$ and $0$ if $x=0$
and $g:R\to R$ be the Greatest Integer Function given by $g\left(x\right)=\left[x\right]$, where $\left[x\right]$ is greatest integer less than or equal to $x$. Then, does $fog$ and $gof$ coincide in $\left(\mathrm{0,1}\right]$?