The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation,is $.........$

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

Let $R_{1}$ and $R_{2}$ be two relations defined as follows:
$R_{1} = \{(a, b) \in \mathbb{R}^{2} : a^{2} + b^{2} \in \mathbb{Q}\}$ and $R_{2} = \{(a, b) \in \mathbb{R}^{2} : a^{2} + b^{2} \notin \mathbb{Q}\}$
where $\mathbb{Q}$ is the set of all rational numbers. Then:

Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.

In the set $N$,a relation $R$ is defined as $aRb \Leftrightarrow b$ is divisible by $a$. Then $R$ is:

Show that the relation $R$ in the set of real numbers $\mathbb{R}$,defined as $R = \{(a, b) : a \leq b^2\}$,is neither reflexive,nor symmetric,nor transitive.