Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(x, y) \mid x, y \in R \text{ and } xy \text{ is an irrational number}\}$. Then,the relation $r$ is:

Let $A = \{0, 3, 4, 6, 7, 8, 9, 10\}$ and $R$ be the relation defined on $A$ such that $R = \{(x, y) \in A \times A : x - y \text{ is an odd positive integer or } x - y = 2\}$. The minimum number of elements that must be added to the relation $R$ so that it becomes a symmetric relation is equal to $...........$.

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

Define a relation $R$ on $A=\{1, 2, 3, 4\}$ as $x R y$ if $x$ divides $y$. $R$ is

In the set $A = \{1, 2, 3, 4, 5\}$,a relation $R$ is defined by $R = \{(x, y) | x, y \in A \text{ and } x < y\}$. Then $R$ is

If $R$ is the smallest equivalence relation on the set $\{1, 2, 3, 4\}$ such that $\{(1, 2), (1, 3)\} \subset R$,then the number of elements in $R$ is