JEE Mathematics 2023 Question with Solution

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

Find: The number of elements in the relation $R = \{((a_1,b_1),(a_2,b_2)) : a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\}$.

The solution states that for each element of $A$, there are exactly $2$ elements of $B$ satisfying the divisibility condition.

Check divisibility from $A$ to $B$:

• $2$ divides $8, 12$
• $3$ divides $9, 12$
• $4$ divides $8, 12$

So there are $6$ valid ordered pairs $(a,b) \in A \times B$ such that $a$ divides $b$.

Now the relation consists of ordered pairs $((a_1,b_1),(a_2,b_2))$ with conditions:

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

By the same counting, there are $6$ choices for one compatible pair and $6$ choices for the other compatible pair.

Therefore, the number of elements in the relation is $36$. The correct option is D.

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

Find: The cardinality of the relation $R$.

First count all ordered pairs $(a,b)$ with $a \in A$, $b \in B$, and $a \mid b$.

Hence the valid pairs are:

So the number of such pairs is $6$.

In $((a_1,b_1),(a_2,b_2)) \in R$, the conditions are cross-linked:

• $a_1 \mid b_2$
• $a_2 \mid b_1$

This means choosing $((a_1,b_2))$ as one valid divisible pair and $((a_2,b_1))$ as another valid divisible pair.

Each choice has $6$ possibilities, so

Therefore, the correct option is D.