Wait, why do we only check $x \le 2/3$ for reflexivity?

Great question! For a relation to be reflexive, every element $x$ in set $A$ must satisfy the condition $(x, x) \in R$. Substituting $y = x$ into our inequality $2x + y \le 2$ gives $2x + x \le 2$, which simplifies to $3x \le 2$, or $x \le 2/3$. Only the elements in $A$ that are less than or equal to $2/3$ (which are $-2, -1, 0$) already satisfy this. The others ($1, 2, 3, 4$) fail, so we must add them to make the relation reflexive.

JEE Main 2026 (23 January Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: Let $A = {- 2, - 1, 0, 1, 2, 3, 4}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $2 x + y \leq 2$ . Let $l$ be the number of elements in $R$ . Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to :

Q: How do I know if I missed any pairs for symmetry?

The best way is to be systematic. List all pairs $(x, y)$ that are in $R$. Then, for each pair, check if the swapped pair $(y, x)$ is also in $R$. If $(x, y)$ is in $R$ but $(y, x)$ is not, that's a pair you need to add. For example, $(-2, 3)$ is in $R$ because $2(-2) + 3 = -1 \le 2$, but $(3, -2)$ is not in $R$ because $2(3) + (-2) = 4 > 2$. Keep a checklist of these missing pairs to ensure you don't double-count or miss any.

Select Answer:

Progressive Hints

Hint 1

Hint 1: Visualize the relation

R

as a set of points

(x, y)

on a coordinate plane. The condition

2 x + y \leq 2

defines a region on or below the line

y = - 2 x + 2

. Your task is to count how many integer pairs from the set

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

fall into this region.

Hint 2

Hint 2: For reflexivity, you need

(x, x) \in R

for all

x \in A

. Check the condition

2 x + x \leq 2

for each

x

. For symmetry, if

(x, y) \in R

, then

(y, x)

must also be in

R

. Identify pairs

(x, y)

R

where

(y, x)

is not, and count them.

Hint 3

Hint 3: Start by listing the valid

y

values for each

x \in A

that satisfy

2 x + y \leq 2

. For

x = - 2

y \leq 6

(all 7 elements work). For

x = - 1

y \leq 4

(all 7 elements work). Continue this for

x = 0, 1, 2, 3, 4

to find

l

. Then, identify the missing reflexive pairs for

m

and the missing symmetric pairs for

n

Visualized Solution

Understanding the Relation $R$

Set $A = {- 2, - 1, 0, 1, 2, 3, 4}$
Relation $R$ on $A$ : $x R y ⟺ 2 x + y \leq 2$
Goal: Find $l$ (size of $R$ ), $m$ (to make reflexive), $n$ (to make symmetric), and calculate $l + m + n$ .

The Boundary Line $2 x + y = 2$

The condition is $2 x + y \leq 2$ .
Let's visualize the boundary line $2 x + y = 2$ .
Points in $R$ must lie on or below this line.

Finding Elements for $x = - 2$

For $x = - 2$ : $2 (- 2) + y \leq 2 ⟹ - 4 + y \leq 2 ⟹ y \leq 6$
Since all $y \in A$ are $\leq 4$ , all $7$ elements are valid.
Pairs: $(- 2, - 2), (- 2, - 1), \dots, (- 2, 4)$

Finding Elements for $x = - 1$

For $x = - 1$ : $2 (- 1) + y \leq 2 ⟹ - 2 + y \leq 2 ⟹ y \leq 4$
All $7$ elements of $A$ satisfy $y \leq 4$ .
Pairs: $(- 1, - 2), (- 1, - 1), \dots, (- 1, 4)$

Finding Elements for $x = 0$

For $x = 0$ : $2 (0) + y \leq 2 ⟹ y \leq 2$
Valid $y \in {- 2, - 1, 0, 1, 2}$ . ( $5$ elements)
Pairs: $(0, - 2), (0, - 1), (0, 0), (0, 1), (0, 2)$

Finding Elements for $x = 1$ and $x = 2$

For $x = 1$ : $2 (1) + y \leq 2 ⟹ y \leq 0 ⟹ y \in {- 2, - 1, 0}$ ( $3$ pairs)
For $x = 2$ : $2 (2) + y \leq 2 ⟹ y \leq - 2 ⟹ y \in {- 2}$ ( $1$ pair)
For $x = 3, 4$ : $2 x + y > 2$ for all $y \in A$ . ( $0$ pairs)

Calculating $l$ (Total Elements in $R$ )

Total elements $l = 7 + 7 + 5 + 3 + 1 + 0 + 0$
$l = 23$

Making it Reflexive ( $m$ )

Reflexive condition: $(x, x) \in R$ for all $x \in A$ .
Check: $2 x + x \leq 2 ⟹ 3 x \leq 2 ⟹ x \leq \frac{2}{3}$
Elements in $A$ satisfying this: ${- 2, - 1, 0}$ .
Missing reflexive pairs: $(1, 1), (2, 2), (3, 3), (4, 4)$ .
Minimum elements to add $m = 4$ .

Making it Symmetric ( $n$ )

Symmetric condition: If $(x, y) \in R$ , then $(y, x) \in R$ .
We must add $(y, x)$ if $2 x + y \leq 2$ but $2 y + x > 2$ .
Missing pairs: $(3, - 2), (4, - 2), (2, - 1), (3, - 1), (4, - 1), (2, 0)$ .
Minimum elements to add $n = 6$ .

Final Sum $l + m + n$

We found: $l = 23$ , $m = 4$ , $n = 6$ .
Calculate $l + m + n$ :
$23 + 4 + 6 = 33$
The correct option is (3).

00:00 / 00:00

The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Geometry of the Relation

Imagine you are standing on a coordinate plane, looking at the set

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

. We define a relation

R

where a pair

(x, y)

is connected if and only if

2 x + y \leq 2

This is a geometric boundary. If you draw the line

2 x + y = 2

, or

y = - 2 x + 2

, you see a line with a slope of

- 2

. Any point

(x, y)

that lies on this line or anywhere below it is part of our relation.

The Systematic Count

To find

l

, the total number of elements in

R

, we iterate through each possible value of

x

A

For

x = - 2

2 (- 2) + y \leq 2 \Rightarrow y \leq 6

. Since the maximum value in

A

4

, all

7

elements of

A

satisfy this. This gives us

7

pairs.

For

x = - 1

2 (- 1) + y \leq 2 \Rightarrow y \leq 4

. Again, all

7

elements of

A

work.

For

x = 0

2 (0) + y \leq 2 \Rightarrow y \leq 2

. The valid

y

values are

{- 2, - 1, 0, 1, 2}

, giving us

5

pairs.

For

x = 1

2 (1) + y \leq 2 \Rightarrow y \leq 0

. The valid

y

values are

{- 2, - 1, 0}

, which is

3

pairs.

For

x = 2

2 (2) + y \leq 2 \Rightarrow y \leq - 2

. Only

y = - 2

works, giving

1

pair.

For

x = 3

and

x = 4

: No

y

A

satisfies the condition. Summing these up, we get:

l = 7 + 7 + 5 + 3 + 1 = 23

The Quest for Reflexivity

A relation is reflexive if

(x, x) \in R

for every

x \in A

. We check the condition:

2 x + x \leq 2 \Rightarrow 3 x \leq 2 \Rightarrow x \leq \frac{2}{3}

In our set

A

, the values

{- 2, - 1, 0}

satisfy this. The values

{1, 2, 3, 4}

do not.

To make the relation reflexive, we must add the pairs

(1, 1), (2, 2), (3, 3),

and

(4, 4)

. Thus,

m = 4

The Mirror Image

A relation is symmetric if

(x, y) \in R

implies

(y, x) \in R

. We look for pairs

(x, y)

R

where

(y, x)

is missing.

The pairs

(3, - 2), (4, - 2), (2, - 1), (3, - 1), (4, - 1),

and

(2, 0)

are all in the region where their swapped counterparts are not. Adding these

6

pairs makes the relation symmetric, so

n = 6

Final Calculation

We have our values:

l = 23

m = 4

, and

n = 6

. The problem asks for the sum

l + m + n

23 + 4 + 6 = 33

The final result is 33.

Common Traps & FAQs

Wait, why do we only check

x \leq 2/3

for reflexivity?

Great question! For a relation to be reflexive, every element

x

in set

A

must satisfy the condition

(x, x) \in R

. Substituting

y = x

into our inequality

2 x + y \leq 2

gives

2 x + x \leq 2

, which simplifies to

3 x \leq 2

, or

x \leq 2/3

. Only the elements in

A

that are less than or equal to

2/3

(which are

- 2, - 1, 0

) already satisfy this. The others (

1, 2, 3, 4

) fail, so we must add them to make the relation reflexive.

How do I know if I missed any pairs for symmetry?

The best way is to be systematic. List all pairs

(x, y)

that are in

R

. Then, for each pair, check if the swapped pair

(y, x)

is also in

R

. If

(x, y)

is in

R

but

(y, x)

is not, that's a pair you need to add. For example,

(- 2, 3)

is in

R

because

2 (- 2) + 3 = - 1 \leq 2

, but

(3, - 2)

is not in

R

because

2 (3) + (- 2) = 4 > 2

. Keep a checklist of these missing pairs to ensure you don't double-count or miss any.

Is there a faster way to count the pairs without listing them all?

While listing is the safest method for small sets like this, you can use the boundary line

y = - 2 x + 2

to visualize the region. For each

x

, you are essentially counting how many

y

values are

\leq - 2 x + 2

. This is a quick way to verify your counts. Don't rush the counting; the most common mistakes in JEE are simple arithmetic errors in these steps!

Select Answer:

Visualized Solution

Understanding the Relation $R$

The Boundary Line $2 x + y = 2$

Finding Elements for $x = - 2$

Finding Elements for $x = - 1$

Finding Elements for $x = 0$

Finding Elements for $x = 1$ and $x = 2$

Calculating $l$ (Total Elements in $R$ )

Making it Reflexive ( $m$ )

Making it Symmetric ( $n$ )

Final Sum $l + m + n$

The Sigma Insight: Types of Relations

Analyzing the Geometry of the Relation

The Systematic Count

The Quest for Reflexivity

The Mirror Image

Final Calculation

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

The number of relations on the set $A = {1, 2, 3}$ containing at most 6 elements including $(1, 2)$ , which are reflexive and transitive but not symmetric, is ________

The number of relations, defined on the set ${a, b, c, d}$ , which are both reflexive and symmetric, is equal to:

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

The Boundary Line .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2x+y=2

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

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

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

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

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

Making it Reflexive ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m)

Making it Symmetric ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n)

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

The Sigma Insight: Types of Relations

Analyzing the Geometry of the Relation

The Systematic Count

The Quest for Reflexivity

The Mirror Image

Final Calculation

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of relations on the set A={1,2,3} containing at most 6 elements including (1,2), which are reflexive and transitive but not symmetric, is ________

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of relations, defined on the set {a,b,c,d}, which are both reflexive and symmetric, is equal to:

Understanding the Relation $R$

The Boundary Line $2 x + y = 2$

Finding Elements for $x = - 2$

Finding Elements for $x = - 1$

Finding Elements for $x = 0$

Finding Elements for $x = 1$ and $x = 2$

Calculating $l$ (Total Elements in $R$ )

Making it Reflexive ( $m$ )

Making it Symmetric ( $n$ )

Final Sum $l + m + n$

The number of relations on the set $A = {1, 2, 3}$ containing at most 6 elements including $(1, 2)$ , which are reflexive and transitive but not symmetric, is ________

The number of relations, defined on the set ${a, b, c, d}$ , which are both reflexive and symmetric, is equal to: