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

For real numbers $x$ and $y$,let $xRy$ if and only if $x - y + \sqrt{2}$ is an irrational number. Then $R$ is:

The number of relations on the set $A = \{1, 2, 3\}$ containing at most $6$ elements including $(1, 2)$,which are reflexive and transitive but not symmetric,is . . . . . . .

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time is given by:
$R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$

Given the relation $R = \{(1, 2), (2, 3)\}$ on the set $A = \{1, 2, 3\}$,the minimum number of ordered pairs which when added to $R$ make it an equivalence relation is

Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $R$ be a relation on $A$ defined by $x R y$ if and only if $2x - y \in \{0, 1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations,respectively. Then $l + m + n$ is equal to :-