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

$(a, a) \in R$ for all $a \in Q$

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.

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 $(b, a) \in R$

Define a relation $R$ on the set $N$ of natural numbers by $R=\{(x, y): y=x+5$ $x $ is a natural number less than $4 ; x, y \in N \} .$ Depict this relationship using roster form. Write down the domain and the 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

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 ($2$)