Consider the relation $R$ on the set $\{-2, -1, 0, 1, 2\}$ defined by $(a, b) \in R$ if and only if $1 + ab > 0$. Then, among the statements:
$I$. The number of elements in $R$ is $17$
$II$. $R$ is an equivalence relation

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.

Consider set $A = \{1, 2, 3\}$. The number of symmetric relations that can be defined on $A$ containing the ordered pairs $(1, 2)$ and $(2, 1)$ is:

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$ be denoted by $S$,such that $R$ is reflexive and symmetric,and $R$ contains exactly $10$ elements. Then the number of elements in $S$ is $...$ .

Let a relation $R$ on $N \times N$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements:
$(I)$ $R$ is reflexive but not symmetric.
$(II)$ $R$ is transitive.
Then which one of the following is true?

Let $A = \{1, 2, 3\}$. The number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is: