Let $M$ denotes set of all $3 \times 3$ non singular matrices. Define the relation $R$ by

$R = \{ (A,B) \in M \times M$ : $AB = BA\} ,$ then $R$ is-

The number of relations $R$ from an $m$-element set $A$ to an $n$-element set $B$ satisfying the condition$\left(a, b_1\right) \in R,\left(a, b_2\right) \in R \Rightarrow b_1=b_2$ for $a \in A, b_1, b_2 \in B$ is

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on A $\times$ B by $\left(a_1, b_1\right) R\left(a_2, b_2\right)$ is and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $\mathrm{R}$ is ...........

Let the relations $R_1$ and $R_2$ on the set $\mathrm{X}=\{1,2,3, \ldots, 20\}$ be given by $\mathrm{R}_1=\{(\mathrm{x}, \mathrm{y}): 2 \mathrm{x}-3 \mathrm{y}=2\}$ and $\mathrm{R}_2=\{(\mathrm{x}, \mathrm{y}):-5 \mathrm{x}+4 \mathrm{y}=0\}$. If $\mathrm{M}$ and $\mathrm{N}$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $\mathrm{M}+\mathrm{N}$ equals

Let $R$ be a reflexive relation on a finite set $A$ having $n$-elements, and let there be m ordered pairs in $R$. Then

Let $A=\{1,2,3, \ldots 20\}$. Let $R_1$ and $R_2$ two relation on $\mathrm{A}$ such that $\mathrm{R}_1=\{(\mathrm{a}, \mathrm{b}): \mathrm{b}$ is divisible by $\mathrm{a}\}$ $\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}$ is an integral multiple of $\mathrm{b}\}$. Then, number of elements in $R_1-R_2$ is equal to_____.