For $\alpha \in N$, consider a relation $R$ on $N$ given by $R =\{( x , y ): 3 x +\alpha y$ is a multiple of 7$\}$.The relation $R$ is an equivalence relation if and only if.

If $R = \{(6, 6), (9, 9), (6, 12), (12, 12), (12,6)\}$ is a relation on set $A = \{3, 6, 9, 12\}$ , then relation $R$ is

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

If $R = \{ (x,\,y)|x,\,y \in Z,\,{x^2} + {y^2} \le 4\} $ is a relation in $Z$, then domain of $R$ 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 ………..