JEE Main 2023 — Sets And Relations Question with Solution

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

We want to find the number of elements of the form $((a_1,b_1),(a_2,b_2))$ such that :

1. $a_1$ divides $b_2$
2. $a_2$ divides $b_1$

For the first condition :
$a_1$ divides $b_2$
Given $a_1\in A$ and $b_2\in B$, we can list the pairs:
$(a_1,b_2)\in (2,8),(2,12),(3,9),(3,12),(4,8),(4,12)$
This gives 6 pairs.

For the second condition, the pairs are the same, because it's just the reversed relation. So :
$a_2$ divides $b_1$
Again has 6 valid pairs.

Now, for every pair from the first condition, we can have any pair from the second condition. This leads to :
$6\times 6=36$ relations.