JEE Main 2026 (22 January Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: 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

Q: Why do we reject the pair $(15, 18)$? It satisfies the equation $4y = 5x - 3$ perfectly!

That is a classic trap! While $(15, 18)$ satisfies the algebraic equation, the relation $R$ is defined on the set $M = \{1, 2, \dots, 16\}$. Since $18$ is not in $M$, the pair $(15, 18)$ cannot exist in $R$. Always check your boundaries!

Q: Do we need to add $(3, 3)$ again to make it symmetric?

Great question! The definition of symmetry is: if $(a, b) \in R$, then $(b, a) \in R$. For $(3, 3)$, the reverse is $(3, 3)$, which is already in $R$. So, adding it again would be redundant. We only add elements that are currently missing.

Select Answer:

Progressive Hints

Hint 1

The Spark: Imagine the set

M = {1, 2, \dots, 16}

as a discrete grid of points. The relation

R

is defined by the equation

4 y = 5 x - 3

. Your first task is to visualize this line cutting through the grid and identify which integer points

(x, y)

lie exactly on this line within the

16 \times 16

boundary.

Hint 2

The Arsenal: To find the points, you need to solve for

y

in terms of

x

. The condition

y \in M

is the filter. Since

y = \frac{5 x - 3}{4}

, you are looking for values of

x

that make

5 x - 3

a multiple of

4

. Think about modular arithmetic:

5 x - 3 \equiv 0 (mod 4)

, which simplifies to

x - 3 \equiv 0 (mod 4)

Hint 3

The Execution: Start by testing

x = 1, 2, 3, \dots

until you find the pattern. Once you have the set of pairs

(x, y)

, remember the definition of symmetry: for every

(a, b) \in R

, the pair

(b, a)

must also be in

R

. If

(b, a)

is missing, that is an element you must add.

Visualized Solution

Understanding the Set $M$ and Relation $R$

Given set $M = {1, 2, 3, \dots, 16}$ .
Relation $R = {(x, y) : 4 y = 5 x - 3, x, y \in M}$ .
We need to find the minimum number of elements to add to make $R$ symmetric.

Isolating $y$ in the Equation

Rearranging the equation: $y = \frac{5 x - 3}{4}$ .
Since $y \in M$ , $y$ must be an integer.
This implies $(5 x - 3)$ must be divisible by $4$ .

Finding the Condition for $x$

$5 x - 3 = 4 x + (x - 3)$ .
For divisibility by $4$ , $(x - 3)$ must be a multiple of $4$ .
Possible values for $x \in M$ : $3, 7, 11, 15$ .

Testing $x = 3$

Substitute $x = 3$ : $y = \frac{5 ( 3 ) - 3}{4} = \frac{12}{4} = 3$ .
$(3, 3) \in R$ since $3 \in M$ .

Testing $x = 7$

Substitute $x = 7$ : $y = \frac{5 ( 7 ) - 3}{4} = \frac{32}{4} = 8$ .
$(7, 8) \in R$ since $7, 8 \in M$ .

Testing $x = 11$

Substitute $x = 11$ : $y = \frac{5 ( 11 ) - 3}{4} = \frac{52}{4} = 13$ .
$(11, 13) \in R$ since $11, 13 \in M$ .

Checking the Boundary: $x = 15$

Substitute $x = 15$ : $y = \frac{5 ( 15 ) - 3}{4} = \frac{72}{4} = 18$ .
$18 \in / M$ , so $(15, 18) \in / R$ .

Listing the Elements of $R$

The relation $R = {(3, 3), (7, 8), (11, 13)}$ .

The Concept of Symmetry

A relation is symmetric if $(x, y) \in R ⟹ (y, x) \in R$ .
We need to check each element of $R$ .

Checking $(3, 3)$

For $(3, 3) \in R$ , the symmetric pair is $(3, 3)$ , which is already in $R$ .

Checking $(7, 8)$

For $(7, 8) \in R$ , we must add $(8, 7)$ to make it symmetric.

Checking $(11, 13)$

For $(11, 13) \in R$ , we must add $(13, 11)$ to make it symmetric.

Final Conclusion

Elements to be added: ${(8, 7), (13, 11)}$ .
Minimum number of elements = $2$ .
The correct option is (3).

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The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Setup

Imagine you are standing before a grid of numbers, a

16 \times 16

square where every point

(x, y)

is a potential candidate for our relation

R

. The relation is governed by the elegant linear equation:

4 y = 5 x - 3

Our mission is to find the points that lie on this line within the set

M = {1, 2, \dots, 16}

and determine the minimum number of 'mirror' points required to make the relation symmetric.

Decoding the Equation

To find our points, we first isolate

y

y = \frac{5 x - 3}{4}

Because

y

must be an integer within our set

M

, the numerator

5 x - 3

must be divisible by

4

. Using algebraic intuition, we observe:

5 x - 3 = 4 x + (x - 3)

Since

4 x

is always divisible by

4

, the condition for

y

to be an integer is simply that

(x - 3)

must be a multiple of

4

. Consequently, the possible values for

x

in our set

M

are

3, 7, 11,

and

15

The Search for Valid Pairs

Now, let us test these values to identify the valid coordinates

(x, y)

For

x = 3

y = \frac{5 ( 3 ) - 3}{4} = 3 \Rightarrow (3, 3)

For

x = 7

y = \frac{5 ( 7 ) - 3}{4} = 8 \Rightarrow (7, 8)

For

x = 11

y = \frac{5 ( 11 ) - 3}{4} = 13 \Rightarrow (11, 13)

For

x = 15

y = \frac{5 ( 15 ) - 3}{4} = 18

Since

18

is outside our set

M

, we must reject the pair

(15, 18)

. Thus, our relation

R

is defined as:

R = {(3, 3), (7, 8), (11, 13)}

The Symmetry Challenge

A relation is symmetric if, for every

(x, y) \in R

, the reverse pair

(y, x)

is also in

R

. We can visualize this as reflecting points across the line

y = x

For the point

(3, 3)

, the mirror image is

(3, 3)

, which is already present in

R

. However, for the point

(7, 8)

, the mirror image

(8, 7)

is missing.

Similarly, for the point

(11, 13)

, the mirror image

(13, 11)

is missing. To make

R

symmetric, we must add these two specific pairs to the set.

Therefore, the minimum number of elements to add is 2.

Common Traps & FAQs

Why do we reject the pair

(15, 18)

? It satisfies the equation

4 y = 5 x - 3

perfectly!

That is a classic trap! While

(15, 18)

satisfies the algebraic equation, the relation

R

is defined on the set

M = {1, 2, \dots, 16}

. Since

18

is not in

M

, the pair

(15, 18)

cannot exist in

R

. Always check your boundaries!

Can we just use symmetry to say

R

must be symmetric? Why do we need to add elements?

A relation is not automatically symmetric. Symmetry is a property that a relation *may* possess. If it doesn't, we are asked to *make* it symmetric by adding the minimum number of elements. We are essentially 'completing' the set to satisfy the definition.

Do we need to add

(3, 3)

again to make it symmetric?

Great question! The definition of symmetry is: if

(a, b) \in R

, then

(b, a) \in R

. For

(3, 3)

, the reverse is

(3, 3)

, which is already in

R

. So, adding it again would be redundant. We only add elements that are currently missing.

Animated Solution for Mathematics - Sets and Relations: 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

Select Answer:

Visualized Solution

Understanding the Set $M$ and Relation $R$

Isolating $y$ in the Equation

Finding the Condition for $x$

Testing $x = 3$

Testing $x = 7$

Testing $x = 11$

Checking the Boundary: $x = 15$

Listing the Elements of $R$

The Concept of Symmetry

Checking $(3, 3)$

Checking $(7, 8)$

Checking $(11, 13)$

Final Conclusion

The Sigma Insight: Types of Relations

Analyzing the Setup

Decoding the Equation

The Search for Valid Pairs

The Symmetry Challenge

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 $A = {- 4, - 3, - 2, 0, 1, 3, 4}$ 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 $A = {2, 3, 5, 7, 9}$ . 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 :

Select Answer:

Visualized Solution

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

Isolating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y in the Equation

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

Testing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=3

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

Testing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=11

Checking the Boundary: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=15

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

The Concept of Symmetry

Checking .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (3,3)

Checking .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (7,8)

Checking .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (11,13)

Final Conclusion

The Sigma Insight: Types of Relations

Analyzing the Setup

Decoding the Equation

The Search for Valid Pairs

The Symmetry Challenge

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 the Set $M$ and Relation $R$

Isolating $y$ in the Equation

Finding the Condition for $x$

Testing $x = 3$

Testing $x = 7$

Testing $x = 11$

Checking the Boundary: $x = 15$

Listing the Elements of $R$

Checking $(3, 3)$

Checking $(7, 8)$

Checking $(11, 13)$

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: