Does the order of $x$ and $y$ matter in $2x \le 3y$?

Absolutely! The relation is defined as $xRy \iff 2x \le 3y$. This is not a symmetric relation by default. For example, $2R5$ is true because $4 \le 15$, but $5R2$ is false because $10 \le 6$ is false. That is exactly why we need to add elements to force symmetry.

JEE Main 2026 (21 January Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: 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 :

Q: Wait, why do we include $(x, x)$ pairs in the count for $l$?

Great question! The definition of a relation $R$ on set $A$ is simply a subset of $A \times A$. If the condition $2x \le 3y$ is satisfied when $x = y$ (which is $2x \le 3x$, true for all positive $x$), then $(x, x)$ must be included in the set $R$. Always stick to the definition!

Q: How do I know if I've missed a pair for $m$?

The best way is to create a 5x5 table where rows are $x$ and columns are $y$. Mark an 'X' if $(x, y) \in R$. Symmetry means the table must be symmetric across the main diagonal. Any 'X' that doesn't have a corresponding 'X' at its mirror position $(y, x)$ is a pair that needs a partner.

Select Answer:

Visualized Solution

Defining the Relation $R$

Given set $A = {2, 3, 5, 7, 9}$
Relation $R$ on $A$ is defined as $x R y ⟺ 2 x \leq 3 y$
This can be rewritten as $y \geq \frac{2 x}{3}$

Pairs for $x = 2$

For $x = 2$ : $y \geq \frac{2 ( 2 )}{3} = \frac{4}{3} \approx 1.33$
Possible $y \in {2, 3, 5, 7, 9}$
Pairs: $(2, 2), (2, 3), (2, 5), (2, 7), (2, 9)$ (Total 5)

Pairs for $x = 3$

For $x = 3$ : $y \geq \frac{2 ( 3 )}{3} = 2$
Possible $y \in {2, 3, 5, 7, 9}$
Pairs: $(3, 2), (3, 3), (3, 5), (3, 7), (3, 9)$ (Total 5)

Pairs for $x = 5$

For $x = 5$ : $y \geq \frac{2 ( 5 )}{3} = \frac{10}{3} \approx 3.33$
Possible $y \in {5, 7, 9}$
Pairs: $(5, 5), (5, 7), (5, 9)$ (Total 3)

Pairs for $x = 7$

For $x = 7$ : $y \geq \frac{2 ( 7 )}{3} = \frac{14}{3} \approx 4.66$
Possible $y \in {5, 7, 9}$
Pairs: $(7, 5), (7, 7), (7, 9)$ (Total 3)

Pairs for $x = 9$

For $x = 9$ : $y \geq \frac{2 ( 9 )}{3} = 6$
Possible $y \in {7, 9}$
Pairs: $(9, 7), (9, 9)$ (Total 2)

Calculating $l$

Total number of elements $l = 5 + 5 + 3 + 3 + 2$
$l = 18$

Symmetry Condition

A relation is symmetric if $(x, y) \in R ⟹ (y, x) \in R$
Visually, the relation matrix must be symmetric about the diagonal $y = x$

Finding $m$

Pairs in $R$ needing partners: $(2, 5), (2, 7), (2, 9), (3, 5), (3, 7), (3, 9), (5, 9)$
Missing partners to add: $(5, 2), (7, 2), (9, 2), (5, 3), (7, 3), (9, 3), (9, 5)$
Minimum elements to add $m = 7$

Final Calculation $l + m$

We have $l = 18$ and $m = 7$
The value of $l + m = 18 + 7$
Final Answer: 25

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The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Setup

We are given the set

A = {2, 3, 5, 7, 9}

and a relation

R

defined on

A

such that

x R y

if and only if

2 x \leq 3 y

To find

l

, the number of elements in

R

, we must systematically evaluate the inequality

y \geq \frac{2 x}{3}

for every

x \in A

Mapping the Territory

We evaluate the condition

y \geq \frac{2 x}{3}

for each element

x

in the set

A

For

x = 2

, we require

y \geq \frac{4}{3} \approx 1.33

. All elements

{2, 3, 5, 7, 9}

satisfy this, yielding 5 pairs:

(2, 2), (2, 3), (2, 5), (2, 7), (2, 9)

For

x = 3

, we require

y \geq \frac{6}{3} = 2

. All elements

{2, 3, 5, 7, 9}

satisfy this, yielding 5 pairs:

(3, 2), (3, 3), (3, 5), (3, 7), (3, 9)

For

x = 5

, we require

y \geq \frac{10}{3} \approx 3.33

. The elements

{5, 7, 9}

satisfy this, yielding 3 pairs:

(5, 5), (5, 7), (5, 9)

For

x = 7

, we require

y \geq \frac{14}{3} \approx 4.66

. The elements

{5, 7, 9}

satisfy this, yielding 3 pairs:

(7, 5), (7, 7), (7, 9)

For

x = 9

, we require

y \geq \frac{18}{3} = 6

. The elements

{7, 9}

satisfy this, yielding 2 pairs:

(9, 7), (9, 9)

Summing these results, we find the total number of elements in

R

l = 5 + 5 + 3 + 3 + 2 = 18

The Quest for Symmetry

A relation is symmetric if

(x, y) \in R ⟹ (y, x) \in R

. We identify pairs

(x, y)

where $x eq y$ such that

(x, y) \in R

but $(y, x) otin R$.

The asymmetric pairs are: 1.

(2, 5) \in R

, but $(5, 2) otin R$ 2.

(2, 7) \in R

, but $(7, 2) otin R$ 3.

(2, 9) \in R

, but $(9, 2) otin R$ 4.

(3, 5) \in R

, but $(5, 3) otin R$ 5.

(3, 7) \in R

, but $(7, 3) otin R$ 6.

(3, 9) \in R

, but $(9, 3) otin R$ 7.

(5, 9) \in R

, but $(9, 5) otin R$

There are exactly 7 such pairs. To make the relation symmetric, we must add these 7 missing links to

R

. Thus,

m = 7

Final Calculation

Having determined the values of

l

and

m

, we perform the final summation:

l + m = 18 + 7 = 25

The final result is 25.

Common Traps & FAQs

Wait, why do we include

(x, x)

pairs in the count for

l

Great question! The definition of a relation

R

on set

A

is simply a subset of

A \times A

. If the condition

2 x \leq 3 y

is satisfied when

x = y

(which is

2 x \leq 3 x

, true for all positive

x

), then

(x, x)

must be included in the set

R

. Always stick to the definition!

Is there a faster way to find

l

without checking every single pair?

While you could use inequalities to bound the search, for a small set like

A = {2, 3, 5, 7, 9}

, the most robust method is systematic enumeration. It prevents silly mistakes and ensures you don't miss any boundary cases.

Why is

m

the number of missing pairs? Can't we just remove the ones that don't have a partner?

The question specifically asks for the minimum number of elements to be *added* to make it symmetric. Removing elements would change the relation entirely, whereas adding elements preserves the existing structure while enforcing the symmetry property.

How do I know if I've missed a pair for

m

The best way is to create a 5x5 table where rows are

x

and columns are

y

. Mark an 'X' if

(x, y) \in R

. Symmetry means the table must be symmetric across the main diagonal. Any 'X' that doesn't have a corresponding 'X' at its mirror position

(y, x)

is a pair that needs a partner.

Does the order of

x

and

y

matter in

2 x \leq 3 y

Absolutely! The relation is defined as

x R y ⟺ 2 x \leq 3 y

. This is not a symmetric relation by default. For example,

2 R 5

is true because

4 \leq 15

, but

5 R 2

is false because

10 \leq 6

is false. That is exactly why we need to add elements to force symmetry.

Select Answer:

Visualized Solution

Defining the Relation $R$

Pairs for $x = 2$

Pairs for $x = 3$

Pairs for $x = 5$

Pairs for $x = 7$

Pairs for $x = 9$

Calculating $l$

Symmetry Condition

Finding $m$

Final Calculation $l + m$

The Sigma Insight: Types of Relations

Analyzing the Setup

Mapping the Territory

The Quest for Symmetry

Final Calculation

Similar Questions

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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:

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

Select Answer:

Visualized Solution

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

Pairs for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=2

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

Pairs for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=5

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

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

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } l

Symmetry Condition

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

Final Calculation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } l+m

The Sigma Insight: Types of Relations

Analyzing the Setup

Mapping the Territory

The Quest for 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:

Defining the Relation $R$

Pairs for $x = 2$

Pairs for $x = 3$

Pairs for $x = 5$

Pairs for $x = 7$

Pairs for $x = 9$

Calculating $l$

Finding $m$

Final Calculation $l + m$

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: