Let $R$ be a relation on the set of all natural numbers $\mathbb{N}$ defined by $aRb \iff a \text{ divides } b^2$. Which of the following properties does $R$ satisfy?
$I.$ Reflexivity
$II.$ Symmetry
$III.$ Transitivity

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

Let $A = \{-2, -1, 0, 1, 2, 3, 4\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x + y \le 2$. 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:

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 given by $R = \{(x, y): x \text{ is the father of } y\}$.

Let $A = \{0, 1, 2, \ldots, 9\}$. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $|x - y|$ is a multiple of $3$. Given below are two statements:
Statement $I$: $n(R) = 36$
Statement $II$: $R$ is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below:

If $R$ and $S$ are two non-empty relations on a set $A$,then which of the following statements is false?