Let $\mathrm{A}=\{2,3,4\}$ and $\mathrm{B}=\{8,9,12\}$. Then the number of elements in the relation \[R=\left\{\left(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\right) \in(A \times B, A \times B): a_{1}\text{ divides }\mathrm{b}_{2}\text{ and }\mathrm{a}_{2}\text{ divides }\mathrm{b}_{1}\right\}\] is :

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Sets Relations and Functions

Let

A = {2, 3, 4}

and

B = {8, 9, 12}

. Then the number of elements in the relation

R = {((a_{1}, b_{1}), (a_{2}, b_{2})) \in (A \times B, A \times B) : a_{1} divides b_{2} and a_{2} divides b_{1}}

is :

Answer: (A)

The two requirements

a_{1} ∣ b_{2}

and

a_{2} ∣ b_{1}

involve disjoint pairs of variables, so they can be counted independently. In

A \times B

the divisor pairs are

2 ∣ 8, 12

;

3 ∣ 9, 12

;

4 ∣ 8, 12

, which is

6

pairs. The pair

(a_{1}, b_{2})

contributes

6

choices and the pair

(a_{2}, b_{1})

another

6

. Therefore the number of elements is

6 \times 6 = 36

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Single correct option