Let $R$ and $S$ be two equivalence relations on a set $A$. Then

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{R}=\{(1,2),(2,3),(1,4)\}$ be a relation on $\mathrm{A}$. Let $\mathrm{S}$ be the equivalence relation on $A$ such that $\mathrm{R} \subset \mathrm{S}$ and the number of elements in $\mathrm{S}$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is...............

If $R$ is an equivalence relation on a Set $A$, then $R^{-1}$ is not :-

Let $\mathrm{A}=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x R y$ if and only if $4 x \leq 5 y$. Let $m$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $m+n$ is equal to:

Let $r$ be a relation from $R$ (Set of real number) to $R$ defined by $r$ = $\left\{ {\left( {x,y} \right)\,|\,x,\,y\, \in \,R} \right.$ and $xy$ is an irrational number $\}$ , then relation $r$ is

$R$ is a relation from $\{11, 12, 13\}$ to $\{8, 10, 12\}$ defined by $y = x - 3$. Then ${R^{ - 1}}$ is