The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ 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 real numbers $x$ and $y$,let $xRy$ if and only if $x - y + \sqrt{2}$ is an irrational number. Then $R$ is:

The relation $R$ defined on the set $A = \{1, 2, 3, 4, 5\}$ by $R = \{(x, y) : |x^2 - y^2| < 16\}$ is:

Let $A$ be the set of all students of a boys' school. Show that the relation $R$ in $A$ given by $R = \{(a, b) : a \text{ is sister of } b\}$ is the empty relation and $R^{\prime} = \{(a, b) : \text{the difference between heights of } a \text{ and } b \text{ is less than } 3 \text{ meters}\}$ is the universal relation.

Define a relation $R$ on the interval $[0, \frac{\pi}{2})$ by $xRy$ if and only if $\sec^2 x - \tan^2 y = 1$. Then $R$ is :