Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $x R y$ if and only if $0 \leq x^2 + 2y \leq 4$. Let $l$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added to $R$ to make it a reflexive relation. Then $l+m$ is equal to

If $R$ is a relation from set $A$ to set $B$ and $S$ is a relation from set $B$ to set $C$,then the relation $S \circ R$ is:

Let $A = \{1, 2, 3\}$. The relation $R$ on set $A$ is defined as $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$. Determine the nature of the relation $R$.

Let $R$ be an equivalence relation on a finite set $A$ having $n$ elements. Then the number of ordered pairs in $R$ 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.

Let $A = \{1, 2, 3, \ldots, 20\}$. Let $R_1$ and $R_2$ be two relations on $A$ such that $R_1 = \{(a, b) : b \text{ is divisible by } a\}$ and $R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}$. Then,the number of elements in $R_1 - R_2$ is equal to . . . . . . .