JEE Main 2024 (08 Apr Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is

Q: Does the order of the sets $A$ and $B$ matter in the Cartesian product $A \times B$?

Absolutely! The notation $A \times B$ strictly defines the domain. The first element of the pair must come from $A$, and the second from $B$. When constructing your counter-example, always double-check that your chosen pairs $(a, b)$ satisfy $a \in A$ and $b \in B$. Ignoring this is a classic trap!

Select Answer:

Progressive Hints

Hint 1

The core of this relation lies in the parity of the expression

3 a d - 7 b c

. Recall that an integer is even if it is a multiple of

2

. When checking for reflexivity, substitute

c = a

and

d = b

and simplify the expression

3 ab - 7 ba

. What do you notice about the resulting coefficient?

Hint 2

For symmetry, assume the condition holds for

(a, b) R (c, d)

, meaning

3 a d - 7 b c = 2 k

. To check

(c, d) R (a, b)

, evaluate

3 c b - 7 d a

. Use the algebraic trick of rewriting

3 c b - 7 d a

- (7 a d - 3 b c)

and then manipulate the terms to isolate the original expression

3 a d - 7 b c

Hint 3

Transitivity is the most common place for a trap in JEE problems. If you cannot find a general proof, do not force it. Instead, use the provided sets

A

and

B

to construct a counter-example. Test the chain

(3, 4) R (6, 4)

and

(6, 4) R (3, 1)

to see if

(3, 4) R (3, 1)

holds true.

Visualized Solution

Understanding the Relation $R$

Given sets: $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$
Relation $R$ on $A \times B$ is defined as:
$(a, b) R (c, d) ⟺ 3 a d - 7 b c$ is an even integer

Checking for Reflexivity

For reflexivity, we check if $(a, b) R (a, b)$ for all $(a, b) \in A \times B$
Substitute $c = a$ and $d = b$ into the condition $3 a d - 7 b c$

Reflexivity Conclusion

Expression becomes: $3 (a) (b) - 7 (b) (a)$
$3 ab - 7 ba = - 4 ab$
Since $- 4 ab$ is a multiple of $2$ , it is always even.
Therefore, $R$ is reflexive

Checking for Symmetry

For symmetry, assume $(a, b) R (c, d)$
This means $3 a d - 7 b c = 2 k$ (where $k$ is an integer)

Algebraic Proof for Symmetry

We need to check if $(c, d) R (a, b)$
This requires checking if $3 c b - 7 d a$ is even
Rewrite: $3 c b - 7 d a = - (7 a d - 3 b c)$
Split the terms: $7 a d = 3 a d + 4 a d$ and $- 3 b c = - 7 b c + 4 b c$
$= - (3 a d - 7 b c + 4 a d + 4 b c)$

Symmetry Conclusion

Substitute $3 a d - 7 b c = 2 k$ :
Expression becomes $- (2 k + 4 a d + 4 b c)$
$= - 2 (k + 2 a d + 2 b c)$
This is a multiple of $2$ , hence an even integer.
Therefore, $R$ is symmetric

Checking for Transitivity

For transitivity, if $(a, b) R (c, d)$ and $(c, d) R (e, f)$ , then $(a, b) R (e, f)$ must hold
We will test this using a counter-example from sets $A$ and $B$

Counter-example: Step 1

Let $(a, b) = (3, 4)$ and $(c, d) = (6, 4)$
Check $(3, 4) R (6, 4)$ :
$3 (3) (4) - 7 (4) (6) = 36 - 168 = - 132$
$- 132$ is even, so $(3, 4) R (6, 4)$ is true

Counter-example: Step 2

Let $(e, f) = (3, 1)$
Check $(6, 4) R (3, 1)$ :
$3 (6) (1) - 7 (4) (3) = 18 - 84 = - 66$
$- 66$ is even, so $(6, 4) R (3, 1)$ is true

Transitivity Failure

Now check $(3, 4) R (3, 1)$ :
$3 (3) (1) - 7 (4) (3) = 9 - 84 = - 75$
$- 75$ is an odd integer!
Therefore, $(3, 4)$ is NOT related to $(3, 1)$

Final Verdict

$R$ is reflexive and symmetric but not transitive
The correct option is (2)
Key takeaway: Always test transitivity with specific values if a general proof seems difficult

00:00 / 00:00

The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Setup

We are given a relation

R

A \times B

defined by the condition that

3 a d - 7 b c

is an even integer. To determine the properties of this relation, we must test for reflexivity, symmetry, and transitivity.

The Reflexivity Check

To check for reflexivity, we must determine if

(a, b) R (a, b)

holds true for all

(a, b) \in A \times B

. By substituting

c = a

and

d = b

into our condition, we obtain:

3 (a) (b) - 7 (b) (a) = 3 ab - 7 ab = - 4 ab

Since

- 4

is a multiple of

2

, the expression

- 4 ab

is always even, regardless of the values of

a

and

b

. Therefore, reflexivity is satisfied.

The Symmetry Dance

Next, we test for symmetry. We assume

(a, b) R (c, d)

is true, meaning

3 a d - 7 b c = 2 k

for some integer

k

. We must now evaluate the condition for the reverse pair

(c, d) R (a, b)

, which is

3 c b - 7 d a

We can rewrite

3 c b - 7 d a

- (7 a d - 3 b c)

. By splitting

7 a d

into

3 a d + 4 a d

and

- 3 b c

into

- 7 b c + 4 b c

, we transform the expression:

- (3 a d - 7 b c + 4 a d + 4 b c) = - (2 k + 4 a d + 4 b c) = - 2 (k + 2 a d + 2 b c)

Since this result is a multiple of

2

, it is necessarily even. Thus, symmetry holds.

The Transitivity Trap

Finally, we examine transitivity: if

(a, b) R (c, d)

and

(c, d) R (e, f)

are true, does it imply

(a, b) R (e, f)

? Rather than attempting a complex general proof, we test for a counter-example.

Let

(a, b) = (3, 4)

(c, d) = (6, 4)

, and

(e, f) = (3, 1)

First, check

(3, 4) R (6, 4)

3 (3) (4) - 7 (4) (6) = 36 - 168 = - 132 (Even)

Next, check

(6, 4) R (3, 1)

3 (6) (1) - 7 (4) (3) = 18 - 84 = - 66 (Even)

Finally, check

(3, 4) R (3, 1)

3 (3) (1) - 7 (4) (3) = 9 - 84 = - 75 (Odd)

Because the result is odd, the chain is broken. Transitivity fails.

Conclusion

We have systematically proven that the relation is reflexive and symmetric, but not transitive. This problem highlights the importance of using counter-examples to efficiently disprove properties that do not hold universally.

Common Traps & FAQs

Wait, why is

- 4 ab

always even? Does it depend on the values in

A

and

B

That is a fantastic question! Remember the definition of an even integer: any integer

n

is even if

n = 2 k

for some integer

k

. Since

- 4 ab = 2 (- 2 ab)

, and

- 2 ab

is clearly an integer (because

a

and

b

are integers), the expression

- 4 ab

is always even, regardless of the specific values of

a

and

b

. It is universally even!

How do I know when to stop trying to prove transitivity and switch to a counter-example?

Great intuition! In JEE Advanced, if you have verified reflexivity and symmetry, and the transitivity condition involves a complex expression like

3 a d - 7 b c

, try to test a few small values first. If your gut tells you it might fail, spend two minutes hunting for a counter-example. If you find one, you have saved yourself ten minutes of algebraic struggle!

Does the order of the sets

A

and

B

matter in the Cartesian product

A \times B

Absolutely! The notation

A \times B

strictly defines the domain. The first element of the pair must come from

A

, and the second from

B

. When constructing your counter-example, always double-check that your chosen pairs

(a, b)

satisfy

a \in A

and

b \in B

. Ignoring this is a classic trap!

Animated Solution for Mathematics - Sets and Relations: Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is

Select Answer:

Visualized Solution

Understanding the Relation $R$

Checking for Reflexivity

Reflexivity Conclusion

Checking for Symmetry

Algebraic Proof for Symmetry

Symmetry Conclusion

Checking for Transitivity

Counter-example: Step 1

Counter-example: Step 2

Transitivity Failure

Final Verdict

The Sigma Insight: Types of Relations

Analyzing the Setup

The Reflexivity Check

The Symmetry Dance

The Transitivity Trap

Conclusion

Similar Questions

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Select Answer:

Visualized Solution

Understanding the Relation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R

Checking for Reflexivity

Reflexivity Conclusion

Checking for Symmetry

Algebraic Proof for Symmetry

Symmetry Conclusion

Checking for Transitivity

Counter-example: Step 1

Counter-example: Step 2

Transitivity Failure

Final Verdict

The Sigma Insight: Types of Relations

Analyzing the Setup

The Reflexivity Check

The Symmetry Dance

The Transitivity Trap

Conclusion

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