1. Understand the relation

We have $A=\{2,3,4\}, \qquad B=\{8,9,12\}.$

Relation $R$ is defined on $A\times B$ by $R=\left\{\big((a_1,b_1),(a_2,b_2)\big): a_1\mid b_2 \text{ and } a_2\mid b_1\right\}.$

So we must count the number of ordered pairs $\big((a_1,b_1),(a_2,b_2)\big)$ such that:

• $a_1$ divides $b_2$
• $a_2$ divides $b_1$

2. Find which elements of $A$ divide which elements of $B$

Check divisibility:

• For $2$: $2\mid 8$, $2\nmid 9$, $2\mid 12$
• For $3$: $3\nmid 8$, $3\mid 9$, $3\mid 12$
• For $4$: $4\mid 8$, $4\nmid 9$, $4\mid 12$

Thus for each $b\in B$, the possible $a\in A$ dividing it are:

• $8$: divisors from $A$ are $\{2,4\}$  choices
• $9$: divisors from $A$ are $\{3\}$  choice
• $12$: divisors from $A$ are $\{2,3,4\}$  choices

3. Count valid choices systematically

We choose $(a_1,b_1)$ and $(a_2,b_2)$ such that:

• $a_2\mid b_1$
• $a_1\mid b_2$

Notice these two conditions are independent once $b_1,b_2$ are fixed.

For a fixed $b_1$:

• number of choices of $a_2$ = number of elements of $A$ dividing $b_1$

For a fixed $b_2$:

• number of choices of $a_1$ = number of elements of $A$ dividing $b_2$

So for fixed $(b_1,b_2)$, number of valid pairs $(a_1,a_2)$ is $d(b_1)\,d(b_2),$ where $d(b)$ = number of elements of $A$ dividing $b$.

From above: $d(8)=2,\qquad d(9)=1,\qquad d(12)=3.$

Hence total number of elements in $R$ is