Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $\alpha R\beta \Leftrightarrow \alpha \bot \beta ,\,\alpha ,\,\beta \in L$. Then $R$ is

Let $R$ be a relation on the set of all natural numbers given by $\alpha b \Leftrightarrow \alpha$ divides $b^2$.

Which of the following properties does $R$ satisfy?

$I.$ Reflexivity   $II.$ Symmetry   $III.$ Transitivity

Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1 = \{(x,y) \in  N \times  N : 2x + y= 10\}$ and $R_2 = \{(x,y) \in  N\times  N : x+ 2y= 10\} $. Then

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \, | a,b \in R$  and  $a - b + \sqrt 3$ is an irrational number$\}$ The relation $r$ is

Let $n(A) = n$. Then the number of all relations on $A$ is

Let $A =\{1,2,3,4,5,6,7\}$. Then the relation $R =$ $\{( x , y ) \in A \times A : x + y =7\}$ is