JEE Main 2023 (13 Apr Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: Let $A = {- 4, - 3, - 2, 0, 1, 3, 4}$ and $R = {(a, b) \in A \times A : b = ∣ a ∣ or b^{2} = a + 1}$ be a relation on $A$ . Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is

Q: Wait, why do we reject the negative root for $b^2 = a + 1$ when $a=0$?

That is a fantastic observation! When $a=0$, $b^2 = 1$, so $b$ could be $1$ or $-1$. We only include the pair $(0, 1)$ because $1$ is in set $A$, but $-1$ is not. The relation $R$ is defined on $A \times A$, so both coordinates must belong to $A$. Always check the set membership for every potential coordinate!

Q: Why is $(-2, -2)$ not already in the relation?

Let's look at the conditions. For $(-2, -2)$ to be in $R$, it must satisfy $b = |a|$ or $b^2 = a + 1$. Here, $a = -2$ and $b = -2$. Does $-2 = |-2|$? No, that is $2$. Does $(-2)^2 = -2 + 1$? That is $4 = -1$, which is also false. Since neither condition is met, $(-2, -2)$ is not in the initial relation $R$ and must be added to satisfy reflexivity.

Enter Numerical Value:

Progressive Hints

Hint 1

Hint 1 (The Spark): Visualize the set

A

as a

7 \times 7

grid of points on a Cartesian plane where both coordinates are from

A

. The relation

R

is simply a collection of specific points on this grid. Your goal is to identify which points are already present and which are missing to satisfy the properties of reflexivity and symmetry.

Hint 2

Hint 2 (The Arsenal): Recall that a relation is reflexive if

(a, a) \in R

for every

a \in A

. This means all points on the line

y = x

must be in the set. A relation is symmetric if

(a, b) \in R

implies

(b, a) \in R

. This means for every point

(a, b)

in the set, its reflection across the line

y = x

must also be present.

Hint 3

Hint 3 (The Execution): First, list all elements of

R

by testing each

a \in A

against the conditions

b = ∣ a ∣

and

b^{2} = a + 1

. Once you have the set

R

, identify the missing

(a, a)

pairs for reflexivity, and then for every

(a, b) \in R

where

a \neq = b

, check if

(b, a)

is in

R

. Count the total number of unique pairs you need to add.

Visualized Solution

The Set $A$ and the Cartesian Grid

Set $A = {- 4, - 3, - 2, 0, 1, 3, 4}$
We are building a relation $R$ on $A \times A$ .
Let's visualize the possible coordinate space.

The First Condition: $b = ∣ a ∣$

Relation $R$ has two conditions.
Condition 1: $b = ∣ a ∣$
This means the y-coordinate is the absolute value of the x-coordinate.

Plotting $b = ∣ a ∣$

For $a = - 4 ⟹ b = ∣ - 4∣ = 4 ⟹ (- 4, 4) \in R$
For $a = - 3 ⟹ b = ∣ - 3∣ = 3 ⟹ (- 3, 3) \in R$
For $a = 0, 1, 3, 4 ⟹ b = a ⟹ (0, 0), (1, 1), (3, 3), (4, 4) \in R$
Note: For $a = - 2$ , $b = 2 \in / A$ .

The Second Condition: $b^{2} = a + 1$

Condition 2: $b^{2} = a + 1$
We need to find pairs $(a, b)$ such that $b^{2} - 1 = a$ .
Both $a$ and $b$ must belong to set $A$ .

Plotting $b^{2} = a + 1$

Let's test values of $a \in A$ :
$a = 0 ⟹ b^{2} = 1 ⟹ b = 1 \in A ⟹ (0, 1) \in R$
$a = 3 ⟹ b^{2} = 4 ⟹ b = - 2 \in A ⟹ (3, - 2) \in R$
Other values of $a$ do not yield a valid $b \in A$ .

Understanding Reflexivity

A relation is reflexive if $(a, a) \in R$ for all $a \in A$ .
Graphically, all points on the line $y = x$ must be present.

Adding Elements for Reflexivity

Points already on $y = x$ : $(0, 0), (1, 1), (3, 3), (4, 4)$
Missing points: $(- 4, - 4), (- 3, - 3), (- 2, - 2)$
Number of elements to add = $3$

Understanding Symmetry

A relation is symmetric if $(a, b) \in R ⟹ (b, a) \in R$ .
Graphically, every point must have a mirror image across the line $y = x$ .

Adding Elements for Symmetry (Part 1)

Check points from Condition 1:
$(- 4, 4) \in R ⟹$ we must add $(4, - 4)$
$(- 3, 3) \in R ⟹$ we must add $(3, - 3)$

Adding Elements for Symmetry (Part 2)

Check points from Condition 2:
$(0, 1) \in R ⟹$ we must add $(1, 0)$
$(3, - 2) \in R ⟹$ we must add $(- 2, 3)$
Total elements added for symmetry = $4$

Final Conclusion

Elements added for Reflexivity = $3$
Elements added for Symmetry = $4$
Total minimum elements to add = $3 + 4 = 7$

00:00 / 00:00

The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Setup

We are given the set

A = {- 4, - 3, - 2, 0, 1, 3, 4}

. This set contains

7

distinct elements. A relation

R

A \times A

is defined by the conditions

b = ∣ a ∣

b^{2} = a + 1

Our goal is to determine the minimum number of elements that must be added to

R

to make it both reflexive and symmetric.

Mapping the Existing Territory

First, we identify the elements currently in

R

based on the given conditions. For the condition

b = ∣ a ∣

, we test each

a \in A

(- 4, 4), (- 3, 3), (0, 0), (1, 1), (3, 3), (4, 4)

Note that for

a = - 2

∣ - 2∣ = 2

, but $2 otin A$. Thus, this pair is excluded.

For the condition

b^{2} = a + 1

, we test each

a \in A

a = 0

, then

b^{2} = 1

, which gives

b = 1

(since

1 \in A

). Thus,

(0, 1) \in R

a = 3

, then

b^{2} = 4

, which gives

b = - 2

(since

- 2 \in A

). Thus,

(3, - 2) \in R

The current set

R

is:

R = {(- 4, 4), (- 3, 3), (0, 0), (1, 1), (3, 3), (4, 4), (0, 1), (3, - 2)}

The Quest for Reflexivity

A relation is reflexive if

(a, a) \in R

for all

a \in A

. Currently, we have

(0, 0), (1, 1), (3, 3),

and

(4, 4)

We are missing the diagonal elements for

- 4, - 3,

and

- 2

. To satisfy reflexivity, we must add the following

3

elements:

(- 4, - 4), (- 3, - 3), (- 2, - 2)

The Mirror of Symmetry

A relation is symmetric if for every

(a, b) \in R

, the point

(b, a)

must also be in

R

. We examine our current set

R

(including the reflexive additions):

(- 4, 4)

requires

(4, - 4)

(- 3, 3)

requires

(3, - 3)

(0, 1)

requires

(1, 0)

(3, - 2)

requires

(- 2, 3)

The diagonal elements are already symmetric. Thus, we must add these

4

new pairs to ensure symmetry.

Final Calculation

To make the relation reflexive, we added

3

elements. To make the relation symmetric, we added

4

elements.

The total number of elements that must be added to the relation

R

is:

3 + 4 = 7

The minimum number of elements required to complete the network is 7.

Common Traps & FAQs

Wait, why do we reject the negative root for

b^{2} = a + 1

when

a = 0

That is a fantastic observation! When

a = 0

b^{2} = 1

, so

b

could be

1

- 1

. We only include the pair

(0, 1)

because

1

is in set

A

, but

- 1

is not. The relation

R

is defined on

A \times A

, so both coordinates must belong to

A

. Always check the set membership for every potential coordinate!

Do I need to add the reflexive elements before checking for symmetry?

Yes, it is highly recommended! Once you add the elements to make the relation reflexive, your set

R

grows. Since the definition of symmetry applies to *all* elements in the relation, you must ensure that your new reflexive pairs (which are symmetric by nature) and the original pairs are all accounted for. It keeps your bookkeeping clean and prevents missing any mirror images.

Why is

(- 2, - 2)

not already in the relation?

Let's look at the conditions. For

(- 2, - 2)

to be in

R

, it must satisfy

b = ∣ a ∣

b^{2} = a + 1

. Here,

a = - 2

and

b = - 2

. Does

- 2 = ∣ - 2∣

? No, that is

2

. Does

(- 2)^{2} = - 2 + 1

? That is

4 = - 1

, which is also false. Since neither condition is met,

(- 2, - 2)

is not in the initial relation

R

and must be added to satisfy reflexivity.

Enter Numerical Value:

Visualized Solution

The Set $A$ and the Cartesian Grid

The First Condition: $b = ∣ a ∣$

Plotting $b = ∣ a ∣$

The Second Condition: $b^{2} = a + 1$

Plotting $b^{2} = a + 1$

Understanding Reflexivity

Adding Elements for Reflexivity

Understanding Symmetry

Adding Elements for Symmetry (Part 1)

Adding Elements for Symmetry (Part 2)

Final Conclusion

The Sigma Insight: Types of Relations

Analyzing the Setup

Mapping the Existing Territory

The Quest for Reflexivity

The Mirror of Symmetry

Final Calculation

Similar Questions

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is:

Let $A = {0, 3, 4, 6, 7, 8, 9, 10}$ and $R$ be the relation defined on $A$ such that $R = {(x, y) \in A \times A : x - y is odd positive integer or x - y = 2}$ . The minimum number of elements that must be added to the relation $R$ , so that it is a symmetric relation, is equal to

Let the relation $R$ on the set $M = {1, 2, 3, \dots, 16}$ be given by $R = {(x, y) : 4 y = 5 x - 3, x, y \in M}$ . Then the minimum number of elements required to be added in $R$ , in order to make the relation symmetric, is equal to

Let $A = {2, 3, 5, 7, 9}$ . Let $R$ be the relation on $A$ defined by $x R y$ if and only if $2 x \leq 3 y$ . Let $l$ be the number of elements in $R$ , and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to :

Enter Numerical Value:

Visualized Solution

The Set .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and the Cartesian Grid

The First Condition: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b=∣a∣

Plotting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b=∣a∣

The Second Condition: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b2=a+1

Plotting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b2=a+1

Understanding Reflexivity

Adding Elements for Reflexivity

Understanding Symmetry

Adding Elements for Symmetry (Part 1)

Adding Elements for Symmetry (Part 2)

Final Conclusion

The Sigma Insight: Types of Relations

Analyzing the Setup

Mapping the Existing Territory

The Quest for Reflexivity

The Mirror of Symmetry

Final Calculation

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The minimum number of elements that must be added to the relation R={(a,b),(b,c)} on the set {a,b,c} so that it becomes symmetric and transitive is:

The Set $A$ and the Cartesian Grid

The First Condition: $b = ∣ a ∣$

Plotting $b = ∣ a ∣$

The Second Condition: $b^{2} = a + 1$

Plotting $b^{2} = a + 1$

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is: