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.

Let $R$ be a reflexive relation on a set $A$ and $I$ be the identity relation on $A$. Then

Let $n$ be a fixed positive integer. Define a relation $R$ on the set $Z$ of integers by, $aRb \Leftrightarrow n|a - b$|. Then $R$ is

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...............

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 $A =\{1,2,3,4, \ldots .10\}$ 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 $.........$.