Miscellaneous Exercise on Chapter 1

Consider the binary operations $*$: $R\times R\to R$ and $o$ : $R\times R\to R$ defined as $a*b=|a-b|$ and $a o b=a,\forall a,b\in R.$ Show that $*$ is commutative but not associative, $o$ is associative but not commutative. Further, show that $\forall a,b,c\in R,a*(b o c)=\left(a*b\right) o \left(a*c\right)$. [If it is so, we say that the operation $*$ distributes over the operation $o$]. Does $o$ distribute over $*$? Justify your answer.

Given a non‐empty set $X,$ let $*$ : $P\left(X\right)\times P\left(X\right)\to P\left(X\right)$ be defined as $A*B=(A-B)\cup (B-A),{\forall }^{}A,B\in P\left(X\right)$. Show that the empty set $\varphi$ is the identity for the operation $*$ and all the elements $A$ of $P\left(X\right)$ are invertible with $A^{-1}=A$.

Define a binary operation $*$ on the set $\left\{0,1,2,3,4,5\right\}$ as
$a*b=\left\{\begin{array}{ll}a+b& \text{if} a+b<6\\ a+b-6& \text{if} a+b\ge 6\end{array}\right.$
Show that zero is the identity for this operation and each element $a\ne 0$ of the set is invertible with $6-a$ being the inverse of $a.$

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 $\left(\mathrm{0,1}\right]$?

Number of binary operations on the set $\left\{a,b\right\}$ are