Show that the relation $R$ in the set $A$ of all the books in a library of a college, given by $R =\{(x, y): x $  and $y$ have same number of pages $\}$ is an equivalence relation.

Let $A=\{1,2,3, \ldots \ldots .100\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $\mathrm{R} \subset \mathrm{R}_1$ and the number of elements in $\mathrm{R}_1$ is $\mathrm{n}$. Then, the minimum value of $n$ is……………………..

${x^2} = xy$ is a relation which is

Let $P = \{ (x,\,y)|{x^2} + {y^2} = 1,\,x,\,y \in R\} $. Then $P$ is

If $R \subset A \times B$ and $S \subset B \times C\,$ be two relations, then ${(SoR)^{ – 1}} = $