JEE Main 2023 (08 Apr Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: 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

Q: Wait, why can't we just use the formula for the total number of relations, $2^{n^2}$?

That is a common trap! The formula $2^{n^2}$ gives you the total number of possible relations on a set, but this problem is not asking for the total number of relations. It is asking for the minimum number of elements to add to a *specific* existing relation to make it symmetric. You must analyze the specific pairs already in $R$ and see which ones are missing their symmetric counterparts.

Q: Why is the condition $x - y = 2$ included separately?

The condition $x - y = 2$ is included because it is an even number, while the other condition ($x - y$ is an odd positive integer) covers the odd differences. The problem defines the relation using a union of these two conditions. You must ensure you count all pairs that satisfy *either* condition.

Q: Is it possible that some $(y, x)$ pairs are already in $R$?

Excellent question! Let's look at the condition again: $x - y > 0$. This means for any $(x, y) \in R$, $x$ must be strictly greater than $y$. If we look for the symmetric counterpart $(y, x)$, we would need $y - x$ to satisfy the same condition. But if $x > y$, then $y - x$ is negative. Since the relation $R$ only contains pairs where the difference is positive, no symmetric counterpart $(y, x)$ can possibly be in the original relation. Thus, we must add every single one of them.

Enter Numerical Value:

Progressive Hints

Hint 1

Visualize the set

A = {0, 3, 4, 6, 7, 8, 9, 10}

on a Cartesian plane. Notice that for any pair

(x, y) \in R

, the condition

x - y \in {1, 3, 5, 7, 9} \cup {2}

implies that

x - y > 0

. What does this tell you about the position of all points

(x, y)

in the relation relative to the line

y = x

Hint 2

Recall the definition of a symmetric relation: a relation

R

is symmetric if and only if for every

(x, y) \in R

, the pair

(y, x)

must also be in

R

. If all your current points lie strictly below the line

y = x

, where must their symmetric counterparts lie, and are they already present in your set?

Hint 3

Calculate the number of pairs

(x, y)

that satisfy the condition

x - y \in {1, 3, 5, 7, 9} \cup {2}

. Since all these pairs satisfy

x > y

, and the condition

x - y > 0

is never satisfied if

x < y

, how many of these symmetric counterparts

(y, x)

are already present in

R

? This will lead you directly to the number of elements that must be added.

Visualized Solution

Understanding Set $A$ and Relation $R$

Given set $A = {0, 3, 4, 6, 7, 8, 9, 10}$
Relation $R = {(x, y) \in A \times A : x - y is odd positive or x - y = 2}$
Goal: Find minimum elements to add to make $R$ symmetric.

Analyzing the Conditions

Condition 1: $x - y \in {1, 3, 5, 7, 9}$
Condition 2: $x - y = 2$
In both cases, $x - y > 0 ⟹ x > y$

Finding Pairs for $x = 10$

For $x = 10$ , we check $y \in A$ such that $10 - y$ is odd or $2$ .
$y = 9 ⟹ 10 - 9 = 1$ (Odd)
$y = 8 ⟹ 10 - 8 = 2$ (Equals $2$ )
$y = 7 ⟹ 10 - 7 = 3$ (Odd)
$y = 3 ⟹ 10 - 3 = 7$ (Odd)
Total $4$ pairs for $x = 10$ .

Finding Pairs for $x = 9$

For $x = 9$ , check $9 - y$ .
$y = 8 ⟹ 9 - 8 = 1$
$y = 7 ⟹ 9 - 7 = 2$
$y = 6 ⟹ 9 - 6 = 3$
$y = 4 ⟹ 9 - 4 = 5$
$y = 0 ⟹ 9 - 0 = 9$
Total $5$ pairs for $x = 9$ .

Finding Pairs for $x = 8$ and $x = 7$

For $x = 8$ : $y \in {7, 6, 3} ⟹ 3$ pairs.
For $x = 7$ : $y \in {6, 4, 0} ⟹ 3$ pairs.

Finding Pairs for Remaining Elements

For $x = 6$ : $y \in {4, 3} ⟹ 2$ pairs.
For $x = 4$ : $y \in {3} ⟹ 1$ pair.
For $x = 3$ : $y \in {0} ⟹ 1$ pair.

Total Elements in Relation $R$

Total elements in $R = 4 + 5 + 3 + 3 + 2 + 1 + 1$
Total elements $= 19$

Symmetry Condition

A relation is symmetric if $(x, y) \in R ⟹ (y, x) \in R$ .
Graphically, this means points must be symmetric about the line $y = x$ .

Adding Elements for Symmetry

All current pairs have $x > y$ (below the line $y = x$ ).
Their symmetric counterparts $(y, x)$ will have $y > x$ (above the line).
None of the required $(y, x)$ pairs are currently in $R$ .
We must add exactly $19$ elements to make $R$ symmetric.

00:00 / 00:00

The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Setup

We are working with the set

A = {0, 3, 4, 6, 7, 8, 9, 10}

and a relation

R

defined by the condition that

x - y

is an odd positive integer or

x - y = 2

. Our objective is to determine the minimum number of elements required to make this relation symmetric.

The definition states that

(x, y) \in R

x - y \in {1, 3, 5, 7, 9}

x - y = 2

. Note that in all these cases,

x - y > 0

, which implies

x > y

This indicates that every pair

(x, y)

currently in

R

lies strictly below the line

y = x

. Consequently, the relation is currently asymmetric.

The Systematic Count

To find the total number of elements in

R

, we iterate through each

x \in A

and identify all

y \in A

such that

x - y \in {1, 2, 3, 5, 7, 9}

For

x = 10

10 - 9 = 1

10 - 8 = 2

10 - 7 = 3

10 - 3 = 7

. This yields

4

pairs:

(10, 9), (10, 8), (10, 7), (10, 3)

For

x = 9

9 - 8 = 1

9 - 7 = 2

9 - 6 = 3

9 - 4 = 5

9 - 0 = 9

. This yields

5

pairs:

(9, 8), (9, 7), (9, 6), (9, 4), (9, 0)

For

x = 8

8 - 7 = 1

8 - 6 = 2

8 - 3 = 5

. This yields

3

pairs:

(8, 7), (8, 6), (8, 3)

For

x = 7

7 - 6 = 1

7 - 4 = 3

7 - 0 = 7

. This yields

3

pairs:

(7, 6), (7, 4), (7, 0)

For

x = 6

6 - 4 = 2

6 - 3 = 3

. This yields

2

pairs:

(6, 4), (6, 3)

For

x = 4

4 - 3 = 1

. This yields

1

pair:

(4, 3)

For

x = 3

3 - 0 = 3

. This yields

1

pair:

(3, 0)

Summing these values, the total number of elements in

R

is:

4 + 5 + 3 + 3 + 2 + 1 + 1 = 19

The Symmetry Requirement

A relation is symmetric if, for every

(x, y) \in R

, the pair

(y, x)

is also in

R

. Since all

19

pairs in our current relation satisfy

x > y

, their symmetric counterparts

(y, x)

satisfy

y > x

Because our original relation

R

contains only pairs where the first element is greater than the second, none of the reverse pairs

(y, x)

are currently present in

R

To achieve symmetry, we must add the reverse of every existing pair. Since there are

19

such pairs and none of their inverses are already in the set, we must add exactly

19

new elements.

The minimum number of elements to add to make the relation symmetric is 19.

Common Traps & FAQs

Wait, why can't we just use the formula for the total number of relations,

2^{n^{2}}

That is a common trap! The formula

2^{n^{2}}

gives you the total number of possible relations on a set, but this problem is not asking for the total number of relations. It is asking for the minimum number of elements to add to a *specific* existing relation to make it symmetric. You must analyze the specific pairs already in

R

and see which ones are missing their symmetric counterparts.

Why is the condition

x - y = 2

included separately?

The condition

x - y = 2

is included because it is an even number, while the other condition (

x - y

is an odd positive integer) covers the odd differences. The problem defines the relation using a union of these two conditions. You must ensure you count all pairs that satisfy *either* condition.

Is it possible that some

(y, x)

pairs are already in

R

Excellent question! Let's look at the condition again:

x - y > 0

. This means for any

(x, y) \in R

x

must be strictly greater than

y

. If we look for the symmetric counterpart

(y, x)

, we would need

y - x

to satisfy the same condition. But if

x > y

, then

y - x

is negative. Since the relation

R

only contains pairs where the difference is positive, no symmetric counterpart

(y, x)

can possibly be in the original relation. Thus, we must add every single one of them.

Enter Numerical Value:

Visualized Solution

Understanding Set $A$ and Relation $R$

Analyzing the Conditions

Finding Pairs for $x = 10$

Finding Pairs for $x = 9$

Finding Pairs for $x = 8$ and $x = 7$

Finding Pairs for Remaining Elements

Total Elements in Relation $R$

Symmetry Condition

Adding Elements for Symmetry

The Sigma Insight: Types of Relations

Analyzing the Setup

The Systematic Count

The Symmetry Requirement

Similar Questions

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 :

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

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

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:

Enter Numerical Value:

Visualized Solution

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

Analyzing the Conditions

Finding Pairs for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=10

Finding Pairs for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=9

Finding Pairs for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=8 and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=7

Finding Pairs for Remaining Elements

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

Symmetry Condition

Adding Elements for Symmetry

The Sigma Insight: Types of Relations

Analyzing the Setup

The Systematic Count

The Symmetry Requirement

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:

Understanding Set $A$ and Relation $R$

Finding Pairs for $x = 10$

Finding Pairs for $x = 9$

Finding Pairs for $x = 8$ and $x = 7$

Total Elements in Relation $R$

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: