Consider set $A = \{1,2,3\}$ . Number of symmetric relations that can be defined on $A$ containing the ordered pair $(1,2)$ & $(2,1)$ is

If $R \subset A \times B$ and $S \subset B \times C\,$ be two relations, then ${(SoR)^{ – 1}} = $

Let L be the set of all lines in a plane and $\mathrm{R}$ be the relation in $\mathrm{L}$ defined as $\mathrm{R}=\left\{\left(\mathrm{L}_{1}, \mathrm{L}_{2}\right): \mathrm{L}_{1}\right.$ is perpendicular to $\left. \mathrm{L} _{2}\right\}$. Show that $\mathrm{R}$ is symmetric but neither reflexive nor transitive.

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on A $\times$ B by $\left(a_1, b_1\right) R\left(a_2, b_2\right)$ is and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $\mathrm{R}$ is ………..

Let $R$ and $S$ be two equivalence relations on a set $A$. Then