Let $A = \{1, 2, 3\}, B = \{1, 3, 5\}$. $A$ relation $R:A \to B$ is defined by $R = \{(1, 3), (1, 5), (2, 1)\}$. Then ${R^{ – 1}}$ is defined by

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.

If $A = \left\{ {x \in {z^ + }\,:x < 10} \right.$& and $x$ is a multiple of $3$ or $4\}$, where $z^+$ is the set of positive integers, then the total number of symmetric relations on $A$ is

How many reflexive relation are there on a set ' with $3$ elements

If $R = \{(6, 6), (9, 9), (6, 12), (12, 12), (12,6)\}$ is a relation on set $A = \{3, 6, 9, 12\}$ , then relation $R$ is