1. Key Concepts and Formulas

• Cartesian Product: For sets $X$ and $Y$, the Cartesian product $X \times Y$ is the set of all ordered pairs $(x, y)$ where $x \in X$ and $y \in Y$.
• Divisibility: An integer $a$ divides an integer $b$ (denoted $a|b$) if there exists an integer $k$ such that $b = ak$.
• Multiplication Principle of Counting: If there are $n_1$ ways to perform the first task and $n_2$ ways to perform the second task, and these tasks are independent, then there are $n_1 \times n_2$ ways to perform both tasks.

2. Step-by-Step Solution

The problem defines a relation $R$ on the set $(A \times B, A \times B)$. An element of $R$ is an ordered pair of ordered pairs, denoted as $((a_1, b_1), (a_2, b_2))$, where $(a_1, b_1) \in A \times B$ and $(a_2, b_2) \in A \times B$. The condition for an element to be in $R$ is that $a_1$ divides $b_2$ and $a_2$ divides $b_1$.

Given sets: $A = \{2, 3, 4\}$ $B = \{8, 9, 12\}$

The relation $R$ is given by: $R = \{((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1 | b_2 \text{ and } a_2 | b_1\}$

Here, $a_1 \in A$, $b_1 \in B$, $a_2 \in A$, and $b_2 \in B$.

We need to count the number of quadruplets $(a_1, b_1, a_2, b_2)$ that satisfy the given conditions. The conditions $a_1 | b_2$ and $a_2 | b_1$ are independent of each other because the variables involved in one condition are distinct from the variables involved in the other condition. We can therefore use the Multiplication Principle.

Step 1: Count the number of pairs $(a_1, b_2)$ such that $a_1 \in A$, $b_2 \in B$, and $a_1 | b_2$.

We systematically check each element of $A$ against each element of $B$ for divisibility.

• For $a_1 = 2$:
  • $2 | 8$ (True)
  • $2 | 9$ (False)
  • $2 | 12$ (True) This gives pairs $(2, 8)$ and $(2, 12)$.
• For $a_1 = 3$:
  • $3 | 8$ (False)
  • $3 | 9$ (True)
  • $3 | 12$ (True) This gives pairs $(3, 9)$ and $(3, 12)$.
• For $a_1 = 4$:
  • $4 | 8$ (True)
  • $4 | 9$ (False)
  • $4 | 12$ (True) This gives pairs $(4, 8)$ and $(4, 12)$.

The total number of pairs $(a_1, b_2)$ satisfying $a_1 | b_2$ is $2 + 2 + 2 = 6$. Let $N_1$ be this count, so $N_1 = 6$.

Step 2: Count the number of pairs $(a_2, b_1)$ such that $a_2 \in A$, $b_1 \in B$, and $a_2 | b_1$.

This is structurally identical to Step 1, as the sets $A$ and $B$ are the same, and the divisibility condition is the same. We are choosing $a_2$ from $A$ and $b_1$ from $B$.

• For $a_2 = 2$: $2 | 8$, $2 | 12$. Pairs: $(2, 8), (2, 12)$.
• For $a_2 = 3$: $3 | 9$, $3 | 12$. Pairs: $(3, 9), (3, 12)$.
• For $a_2 = 4$: $4 | 8$, $4 | 12$. Pairs: $(4, 8), (4, 12)$.

The total number of pairs $(a_2, b_1)$ satisfying $a_2 | b_1$ is $2 + 2 + 2 = 6$. Let $N_2$ be this count, so $N_2 = 6$.

Step 3: Calculate the total number of elements in $R$ using the Multiplication Principle.

An element of $R$ is determined by choosing a pair $(a_1, b_2)$ satisfying $a_1 | b_2$ and a pair $(a_2, b_1)$ satisfying $a_2 | b_1$. Since these choices are independent, the total number of elements in $R$ is the product of the number of ways for each independent choice.

Number of elements in $R = (\text{Number of ways to choose } (a_1, b_2) \text{ such that } a_1 | b_2) \times (\text{Number of ways to choose } (a_2, b_1) \text{ such that } a_2 | b_1)$ $|R| = N_1 \times N_2$ $|R| = 6 \times 6 = 36$.

3. Common Mistakes & Tips

• Confusing the structure of elements: Remember that elements of $R$ are pairs of ordered pairs, i.e., $((a_1, b_1), (a_2, b_2))$, not simple ordered pairs.
• Independence of conditions: The conditions $a_1 | b_2$ and $a_2 | b_1$ involve disjoint sets of variables ($a_1, b_2$ vs. $a_2, b_1$), making them independent. This is key to applying the multiplication principle.
• Systematic enumeration: For divisibility problems with small sets, listing all valid pairs is a robust method to ensure no possibilities are missed or double-counted.

4. Summary

The problem requires us to count the number of elements in a relation $R$, which are quadruplets $(a_1, b_1, a_2, b_2)$ satisfying $a_1 \in A$, $b_1 \in B$, $a_2 \in A$, $b_2 \in B$, and the conditions $a_1 | b_2$ and $a_2 | b_1$. By observing that the two divisibility conditions are independent, we can count the number of ways each condition can be met separately and then multiply these counts. We found there are 6 pairs $(a_1, b_2)$ where $a_1 \in A$ divides $b_2 \in B$, and 6 pairs $(a_2, b_1)$ where $a_2 \in A$ divides $b_1 \in B$. Thus, the total number of elements in $R$ is $6 \times 6 = 36$. This corresponds to option (C).

The final answer is $\boxed{\text{36}}$.