Step 1: Identify the elements of the sets.

$A=\{1,2,3,6\}$
$B=\{2,4,8\}$

Step 2: Apply the condition to find ordered pairs.
The relation $R$ requires that the difference $(x-y)$ is an odd integer. We systematically check the difference for each $x\in A$ mapped with each $y\in B$:

• For $x=1$: $1-2=-1$ (odd), $1-4=-3$ (odd), $1-8=-7$ (odd). All are valid pairs.
• For $x=2$: $2-2=0$ (even), $2-4=-2$ (even), $2-8=-6$ (even). No valid pairs.
• For $x=3$: $3-2=1$ (odd), $3-4=-1$ (odd), $3-8=-5$ (odd). All are valid pairs.
• For $x=6$: $6-2=4$ (even), $6-4=2$ (even), $6-8=-2$ (even). No valid pairs.

Step 3: Arrow Diagram Representation

Step 1: Determine the domain values for $x$.
We are given that $x$ is a natural number ($N$) strictly less than $5$. The natural numbers start from $1$. Therefore: $$x\in \{1,2,3,4\}$$

Step 2: Calculate the corresponding values of $y$.
The relation maps each $x$ to $y$ using the equation $y=x+7$:

• Substitute $x=1$:   $y=1+7=8$  ⟹  Ordered pair is $(1,8)$
• Substitute $x=2$:   $y=2+7=9$  ⟹  Ordered pair is $(2,9)$
• Substitute $x=3$:   $y=3+7=10$  ⟹  Ordered pair is $(3,10)$
• Substitute $x=4$:   $y=4+7=11$  ⟹  Ordered pair is $(4,11)$

(i) Writing $R$ in roster form:
The condition states that $x$ perfectly divides $y$ without leaving a remainder. We systematically evaluate this for every element in $A$ pairing with every element in $A$.

• $x=1$ is a universal divisor. It divides: $1,2,3,4,5,8$.
• $x=2$ divides the even numbers in the set: $2,4,8$.
• $x=3$ divides only itself in this set: $3$.
• $x=4$ divides its multiples: $4,8$.
• $x=5$ divides only itself: $5$.
• $x=8$ divides only itself: $8$.

(ii) Finding Domain, Codomain, and Range:

• Domain: The set of all first components ($x$-values) of the ordered pairs in $R$.
  $\text{Domain}=\{1,2,3,4,5,8\}$
• Range: The set of all second components ($y$-values) of the ordered pairs in $R$.
  $\text{Range}=\{1,2,3,4,5,8\}$
• Codomain: Since the relation is defined "on set $A$" (which means from $A$ to $A$), the target set is entirely $A$.
  $\text{Codomain}=\{1,2,3,4,5,8\}$

(i) Set-builder form:
Let's analyze the mathematical pattern mapping the elements of $A$ to $B$. We see that:

• $7\to 4$ (Difference is $7-4=3$)
• $8\to 5$ (Difference is $8-5=3$)
• $9\to 6$ (Difference is $9-6=3$)

Every $y$-value is exactly 3 less than the corresponding $x$-value. Thus, the algebraic rule is $y=x-3$. Translating this into set-builder notation:

(ii) Roster form:
Extract the mapped coordinate pairs explicitly from the arrows in the diagram.

Domain and Range:

• Domain: $\{7,8,9\}$ (First elements of the pairs)
• Range: $\{4,5,6\}$ (Second elements of the pairs)

Step 1: Identify the domain elements.
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. We need all prime numbers strictly less than 12. $$x\in \{2,3,5,7,11\}$$

Step 2: Generate the ordered pairs.
The relation structure is $(x,x^2)$. We square each prime number to find its associated $y$-value.

• If $x=2$, then $x^2=4$  ⟹ $(2,4)$
• If $x=3$, then $x^2=9$  ⟹ $(3,9)$
• If $x=5$, then $x^2=25$  ⟹ $(5,25)$
• If $x=7$, then $x^2=49$  ⟹ $(7,49)$
• If $x=11$, then $x^2=121$  ⟹ $(11,121)$

Step 1: Understand the theorem for number of relations.
A relation from set $A$ to set $B$ is defined mathematically as a subset of the Cartesian product $A\times B$. The total number of subsets of any set with $k$ elements is $2^k$. Therefore, the number of relations from $A$ to $B$ is: $$2^{n(A\times B)}=2^{n\left(A\right)\times n\left(B\right)}$$

Step 2: Calculate the cardinalities.
Number of elements in $A$, $n\left(A\right)=2$
Number of elements in $B$, $n\left(B\right)=3$
Total elements in Cartesian product: $n(A\times B)=2\times 3=6$

Step 3: Calculate the total number of relations.
Number of relations = $2^6=64$.

Step 1: Understand the definition of a Cartesian product.
The Cartesian product $B\times A$ consists of all ordered pairs $(x,y)$ such that the first element $x\in B$ and the second element $y\in A$. Notice the order: since it is $B\times A$, the set $B$ comes first.

Step 2: Extract elements for Set $B$.
Set $B$ is the set of all unique first coordinates from the given ordered pairs.
First coordinates: $a,a,b,b$
Removing duplicates, we get: $B=\{a,b\}$

Step 3: Extract elements for Set $A$.
Set $A$ is the set of all unique second coordinates from the given ordered pairs.
Second coordinates: $1,5,1,5$
Removing duplicates, we get: $A=\{1,5\}$

Step 1: Understand the condition mathematically.
The difference between two integers, $a-b$, is even if and only if both $a$ and $b$ share the same parity. This means they must either be both even or both odd. (Note: $0$ is considered an even number, so pairs like $(1,1)$ where $1-1=0$ are valid).

Step 2: Group the elements of $A$ by parity.
The set $A=\{1,2,3,4\}$ can be split into:
Even subset: $A_{\text{even}}=\{2,4\}$
Odd subset: $A_{\text{odd}}=\{1,3\}$

Step 3: Form the valid ordered pairs.

• Pairs where both are odd: Combine elements of $A_{\text{odd}}$ with themselves.
  $(1,1),(1,3),(3,1),(3,3)$ (4 pairs)
• Pairs where both are even: Combine elements of $A_{\text{even}}$ with themselves.
  $(2,2),(2,4),(4,2),(4,4)$ (4 pairs)

Listing them all out gives the roster form of $R$: