Let $R$ be a reflexive relation on a finite set $A$ having $n$-elements, and let there be m ordered pairs in $R$. Then

Let $A = \{1, 2, 3\}, B = \{1, 3, 5\}$. If relation $R$ from $A$ to $B$ is given by $R =\{(1, 3), (2, 5), (3, 3)\}$. Then ${R^{ - 1}}$ is

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,2),(2,1)\}$ is symmetric but neither reflexive nor transitive.

Let a set $A=A_{1} \cup A_{2} \cup \ldots \cup A_{k,} \quad$ where $A_{ i } \cap A _{ j }=\phi$ for $i \neq j 1 \leq i , j \leq k$. Define the relation $R$ from $A$ to $A$ by $R=\left\{(x, y): y \in A_{i}\right.$ if and only if $\left.x \in A_{i}, 1 \leq i \leq k\right\}$. Then, $R$ is

Let $R_{1}$ and $R_{2}$ be relations on the set $\{1,2, \ldots, 50\}$ such that $R _{1}=\left\{\left( p , p ^{ n }\right)\right.$ : $p$ is a prime and $n \geq 0$ is an integer $\}$ and $R _{2}=\left\{\left( p , p ^{ n }\right)\right.$ : $p$ is a prime and $n =0$ or $1\}$. Then, the number of elements in $R _{1}- R _{2}$ is........

Let $X$ be a family of sets and $R$ be a relation on $X$ defined by $‘A$ is disjoint from $B’$. Then $R$ is