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 ($2$)

Let $R$ be a relation from $Q$ to $Q$ defined by $R=\{(a, b): a, b \in Q$ and $a-b \in Z \} .$ Show that

$(a, b) \in R$ implies that $(b, a) \in R$

Let $A=\{1,2,3, \ldots, 14\} .$ Define a relation $R$ from $A$ to $A$ by $R = \{ (x,y):3x – y = 0,$ where $x,y \in A\} .$ Write down its domain, codomain and range.

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

in set – bulider form,

What is its domain and range ?

Let $R$ be the relation on $Z$ defined by $R = \{ (a,b):a,b \in Z,a – b$ is an integer $\} $  Find the domain and range of $R .$