The relation $R$ defined on the set of natural numbers as $\{(a, b) : a$ differs from $b$ by $3\}$, is given by

Let $A=\{1,2,3,4,5,6\} .$ Define a relation $R$ from $A$ to $A$ by $R=\{(x, y): y=x+1\}$

Depict this relation using an arrow diagram.

Determine the domain and range of the relation $R$ defined by $R =\{(x, x+5): x \in\{0,1,2,3,4,5\}\}$

Let $S=\{1,2,3,4,5,6\}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties:

$i$. $R$ has exactly $6$ elements.

$ii$. For each $(a, b) \in R$, we have $|a-b| \geq 2$.

Let $Y=\{R \in X$ : The range of $R$ has exactly one element $\}$ and $Z=\{R \in X: R$ is a function from $S$ to $S\}$.

Let $n(A)$ denote the number of elements in a Set $A$.

(There are two questions based on $PARAGRAPH " 1 "$, the question given below is one of them)

($1$) If $n(X)={ }^m C_6$, then the value of $m$ is. . . . 

($2$) If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is. . . . 

Give the answer or quetion ($1$) and ($2$)

The Fig shows a relationship between the sets $P$ and $Q .$ Write this relation

in set-builder form 

What is its domain and range?

Given two finite sets $A$ and $B$ such that $n(A) = 2, n(B) = 3$. Then total number of relations from $A$ to $B$ is