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 :

Sets and Relations PYQ — JEE Main 2023 · 11 Apr · Shift 2 · Q28 with Solution

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Correct answer: 256

Understand the relation

The relation $R$ is defined on $A \times B$ .

So elements related by $R$ are ordered pairs of the form

(a_1,b_1), (a_2,b_2) \in A\times B

with conditions

a_1 \le b_2 \quad \text{and} \quad b_1 \le a_2.

We need to count the number of quadruples $(a_1,b_1,a_2,b_2)$ satisfying these conditions.

Given:

A=\{1,3,4,6,9\}, \qquad B=\{2,4,5,8,10\}

Rewrite the counting problem

The two conditions are independent in the following sense:

$a_1 \le b_2$ involves only $(a_1,b_2)$
$b_1 \le a_2$ involves only $(b_1,a_2)$

Hence,

|R| = \big(\#\{(a,b)\in A\times B: a\le b\}\big)^2

Let

N = \#\{(a,b)\in A\times B: a\le b\}

Then

|R|=N^2.

Count pairs $(a,b)$ such that $a\le b$

We count for each $a\in A$ how many $b\in B$ satisfy $a\le b$ .

For $a=1$ : all elements of $B$ satisfy $1\le b$ $5 \text{ pairs}$
For $a=3$ : $b\in\{4,5,8,10\}$ $4 \text{ pairs}$
For $a=4$ : $b\in\{4,5,8,10\}$ $4 \text{ pairs}$
For $a=6$ : $b\in\{8,10\}$ $2 \text{ pairs}$
For $a=9$ : $b\in\{10\}$ $1 \text{ pair}$

Therefore,

N = 5+4+4+2+1 = 16.

Compute $|R|$

Thus,

|R| = N^2 = 16^2 = 256.

Check options

The obtained answer is

256

which is not present in the options.

So none of the given options matches the correct count.

Compare with stored correct answer

Stored correct answer: D = 160

But our derived answer is

256.

Hence the stored answer appears to be incorrect.

More from Sets and Relations

View written solutionFree

Correct answer: 256

Understand the relation

The relation $R$ is defined on $A \times B$ .

So elements related by $R$ are ordered pairs of the form

(a_1,b_1), (a_2,b_2) \in A\times B

with conditions

a_1 \le b_2 \quad \text{and} \quad b_1 \le a_2.

We need to count the number of quadruples $(a_1,b_1,a_2,b_2)$ satisfying these conditions.

Given:

A=\{1,3,4,6,9\}, \qquad B=\{2,4,5,8,10\}

Rewrite the counting problem

The two conditions are independent in the following sense:

$a_1 \le b_2$ involves only $(a_1,b_2)$
$b_1 \le a_2$ involves only $(b_1,a_2)$

Hence,

|R| = \big(\#\{(a,b)\in A\times B: a\le b\}\big)^2

Let

N = \#\{(a,b)\in A\times B: a\le b\}

Then

|R|=N^2.

Count pairs $(a,b)$ such that $a\le b$

We count for each $a\in A$ how many $b\in B$ satisfy $a\le b$ .

For $a=1$ : all elements of $B$ satisfy $1\le b$ $5 \text{ pairs}$
For $a=3$ : $b\in\{4,5,8,10\}$ $4 \text{ pairs}$
For $a=4$ : $b\in\{4,5,8,10\}$ $4 \text{ pairs}$
For $a=6$ : $b\in\{8,10\}$ $2 \text{ pairs}$
For $a=9$ : $b\in\{10\}$ $1 \text{ pair}$

Therefore,

N = 5+4+4+2+1 = 16.

Compute $|R|$

Thus,

|R| = N^2 = 16^2 = 256.

Check options

The obtained answer is

256

which is not present in the options.

So none of the given options matches the correct count.

Compare with stored correct answer

Stored correct answer: D = 160

But our derived answer is

256.

Hence the stored answer appears to be incorrect.

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