If a relation $R$ on the set $\{1, 2, 3\}$ is defined by $R = \{(1, 1)\}$,then $R$ is

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

Let $L$ be the set of all straight lines in a plane and the relation $R$ on $L$ is defined by $\alpha R \beta \Leftrightarrow \alpha \perp \beta$,where $\alpha, \beta \in L$. Then $R$ is:

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 . . . . . . .

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.

Let $A = \{1, 2, 3\}$. The number of equivalence relations on $A$ containing $(1, 2)$ is . . . . . . .