Question

Let A = {2, 3, 4} and B = {8, 9, 12}. Then the number of elements in the relation R = {((a₁, b₁), (a₂, b₂)) ∈ (A × B, A × B) : a₁ divides b₂ and a₂ divides b₁} is

• A. 12
• B. 18
• C. 24
• D. 36

Answer 1

The Correct Option is D

To find the number of elements in the relation \( R \), we first need to understand the condition given: 

We are given a relation \( R = \{ ((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1 \} \).

We have sets \( A = \{2, 3, 4\} \) and \( B = \{8, 9, 12\} \). The Cartesian product \( A \times B \) would yield all possible pairs \((a_i, b_j)\) where \( a_i \in A \) and \( b_j \in B \).

First, let's calculate the Cartesian product \( A \times B \). This will give us all pairs \((a, b)\) such that \( a \in A \) and \( b \in B \).

A	B	Pairs in \( A \times B \)
2	8	(2, 8)
2	9	(2, 9)
2	12	(2, 12)
3	8	(3, 8)
3	9	(3, 9)
3	12	(3, 12)
4	8	(4, 8)
4	9	(4, 9)
4	12	(4, 12)

There are \( 3 \times 3 = 9 \) pairs in the Cartesian product \( A \times B \). Now, construct the set \((A \times B, A \times B)\), which means we have to pair every element of \( A \times B \) with every other element, leading to \((9 \times 9) = 81\) possible pairs.

Next, count the number of tuple pairs ((a1, b1), (a2, b2)) which satisfy the condition:

1. \( a_1 \) divides \( b_2 \)
2. \( a_2 \) divides \( b_1 \)

Let's evaluate the division:

• For \( b_2 = 8 \), \( a_1 \) can be \( 2\) or \( 4\)
• For \( b_2 = 9 \), \( a_1 \) can be \( 3 \)
• For \( b_2 = 12 \), \( a_1 \) can be \( 2 \), \( 3 \), or \( 4 \)

For each \( b_1 \), calculate possible \( a_2 \):

• For \( b_1 = 8 \), \( a_2 \) can be \( 2 \) or \( 4 \)
• For \( b_1 = 9 \), \( a_2 \) can be \( 3 \)
• For \( b_1 = 12 \), \( a_2 \) can be \( 2 \), \( 3 \), or \( 4 \)

Now consider all combinations where each \( b_2 \) matches possible \( a_1 \), and each \( b_1 \) matches possible \( a_2 \):

• Each pair is matched, counting all combinations for each condition gives:
• A total combination count of 36:

Thus, the number of elements in the relation \( R \) is 36, confirming option 36 as the correct answer.