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-

Determine whether each of the following relations are reflexive, symmetric and transitive :

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3, \ldots, 13,14\}$ defined as $\mathrm{R}=\{(x, y): 3 x-y=0\}$

Let $X =\{1,2,3,4,5,6,7,8,9\} .$ Let $R _{1}$ be a relation in $X$ given by $R _{1}=\{(x, y): x-y$ is divisible by $3\}$ and $R _{2}$ be another relation on $X$ given by ${R_2} = \{ (x,y):\{ x,y\}  \subset \{ 1,4,7\} \} $ or $\{x, y\} \subset\{2,5,8\} $ or $\{x, y\} \subset\{3,6,9\}\} .$ Show that $R _{1}= R _{2}$.

Let $A =\{1,2,3,4, \ldots .10\}$ and $B =\{0,1,2,3,4\}$ The number of elements in the relation $R =\{( a , b )$ $\left.\in A \times A : 2( a – b )^2+3( a – b ) \in B \right\}$ is $………$.

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