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

Define a relation $R$ over a class of $n \times n$ real matrices $A$ and $B$ as $A R B$ if and only if there exists a non-singular matrix $P$ such that $P A P^{-1} = B$. Then which of the following is true?

Show that the number of equivalence relations on the set $A = \{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is $2$.

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 $R = \{(a, a)\}$ be a relation on a set $A$. Then $R$ is:

Let $S$ be the set of all real numbers. Then on the set $S$,the relation $R$ defined as $R = \{ (a, b) : 1 + ab > 0 \}$ is