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

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

in roster form

What is its domain and range ?

Let $A = \{1, 2, 3\}$. The total number of distinct relations that can be defined over $A$ is

Let $A=\{1,2,3,4\}, B=\{1,5,9,11,15,16\}$ and $f=\{(1,5),(2,9),(3,1),(4,5),(2,11)\}$
Are the following true?

$f$ is a relation from $A$ to $B$

Justify your answer in each case.

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 $N$ to $N$ defined by $R =\left\{(a, b): a, b \in N \text { and } a=b^{2}\right\} .$ Are the following true?

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