JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Sets Relations and Functions

Let

A = {1, 3, 4, 6, 9}

and

B = {2, 4, 5, 8, 10}

. Let R be a relation defined on

A \times B

such that

R = {((a_{1}, b_{1}), (a_{2}, b_{2})) : a_{1} \leq b_{2}

and

b_{1} \leq a_{2}}

. Then the number of elements in the set R is

Answer: (B)

Solution

A member of

R

is a pair

((a_{1}, b_{1}), (a_{2}, b_{2}))

with

a_{1} \leq b_{2}

and

b_{1} \leq a_{2}

. The first condition involves only

a_{1} \in A

and

b_{2} \in B

, and the second only

b_{1} \in B

and

a_{2} \in A

, so the two are independent and the total is the product of the separate counts. As

a_{1}

runs through

1, 3, 4, 6, 9

, the number of

b_{2} \geq a_{1}

B

5 + 4 + 4 + 2 + 1 = 16

. As

b_{1}

runs through

2, 4, 5, 8, 10

, the number of

a_{2} \geq b_{1}

A

4 + 3 + 2 + 1 + 0 = 10

. Hence the number of elements in

R

16 \times 10 = 160

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Single correct option