JEE Main 2025 April

LEVELJEE Main

Animated Solution for Mathematics - Sets and Relations: Let $A = {- 3, - 2, - 1, 0, 1, 2, 3}$ and $R$ be a relation on $A$ defined by $x R y$ if and only if $2 x - y \in {0, 1}$ . 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 $l + mn$ is equal to :-

Q: Wait, why do we reject the negative values for $x$ when calculating $y$?

We don't reject them! We test them. The condition $2x - y \in \{0, 1\}$ applies to the entire set $A$. When $x = -3$, $y$ would be $-6$ or $-7$. Since neither $-6$ nor $-7$ exists in our set $A = \{-3, -2, -1, 0, 1, 2, 3\}$, those pairs simply cannot exist in the relation $R$. Always check the boundary of your set!

Q: What if a pair is already symmetric, like $(0, 0)$?

That is a great observation! A pair $(a, a)$ is its own symmetric counterpart. Since $(0, 0)$ and $(1, 1)$ are already in $R$, they don't require any 'new' elements to satisfy the symmetry condition. We only focus on the non-reflexive pairs that lack their mirror images.

Select Answer:

Progressive Hints

Hint 1

Hint 1: The Spark. Do not be intimidated by the set notation. The relation

x R y

is simply a filter. For every element

x

in your set

A

, you are looking for a partner

y

that satisfies the linear equations

y = 2 x

y = 2 x - 1

. The real constraint is that

y

must also live within the set

A

Hint 2

Hint 2: The Arsenal. Remember the definitions of the properties you are testing. A relation is reflexive if every element

a \in A

has a self-loop

(a, a)

. A relation is symmetric if for every pair

(x, y)

you found, the reverse pair

(y, x)

must also exist in the set. Count what you have, then count what is missing.

Hint 3

Hint 3: The Execution. Start by systematically iterating through each

x \in {- 3, - 2, - 1, 0, 1, 2, 3}

. For each

x

, calculate the two potential

y

values. If

y

falls outside the set

A

, discard it. Once you have the full set

R

, count the missing reflexive pairs to find

m

and the missing symmetric counterparts to find

n

Visualized Solution

Defining Set $A$ and Relation $R$

Set $A = {- 3, - 2, - 1, 0, 1, 2, 3}$
Relation $R$ is defined on $A$ .
Condition: $2 x - y \in {0, 1}$

Splitting the Condition

The expression $2 x - y$ can take two values: $0$ or $1$ .
Case 1: $2 x - y = 0 ⟹ y = 2 x$
Case 2: $2 x - y = 1 ⟹ y = 2 x - 1$
For any $x \in A$ , we must check if the resulting $y \in A$ .

Testing $x = - 1, 0$

For $x = - 1$ :
$y = 2 (- 1) = - 2 \in A$
$y = 2 (- 1) - 1 = - 3 \in A$
Pairs: $(- 1, - 2), (- 1, - 3)$
For $x = 0$ :
$y = 2 (0) = 0 \in A$
$y = 2 (0) - 1 = - 1 \in A$
Pairs: $(0, 0), (0, - 1)$

Testing $x = 1, 2$

For $x = 1$ :
$y = 2 (1) = 2 \in A$
$y = 2 (1) - 1 = 1 \in A$
Pairs: $(1, 2), (1, 1)$
For $x = 2$ :
$y = 2 (2) = 4 \in / A$ (Reject)
$y = 2 (2) - 1 = 3 \in A$
Pair: $(2, 3)$

Boundary Checks

For $x = 3$ : $y = 6$ or $5$ (Both $\in / A$ )
For $x = - 2$ : $y = - 4$ or $- 5$ (Both $\in / A$ )
For $x = - 3$ : $y = - 6$ or $- 7$ (Both $\in / A$ )
No more pairs can be formed.

Total Elements in $R$ ( $l$ )

$R = {(- 1, - 2), (- 1, - 3), (0, 0), (0, - 1), (1, 2), (1, 1), (2, 3)}$
Total number of pairs, $l = 7$

Reflexive Additions ( $m$ )

A relation is reflexive if $(a, a) \in R$ for all $a \in A$ .
We need 7 self-pairs: $(- 3, - 3)$ to $(3, 3)$ .
Already present: $(0, 0)$ and $(1, 1)$ .
Missing pairs: $(- 3, - 3), (- 2, - 2), (- 1, - 1), (2, 2), (3, 3)$ .
Minimum elements to add, $m = 5$ .

Symmetric Additions ( $n$ )

A relation is symmetric if $(x, y) \in R ⟹ (y, x) \in R$ .
Existing non-symmetric pairs:
$(- 1, - 2) ⟹$ need $(- 2, - 1)$
$(- 1, - 3) ⟹$ need $(- 3, - 1)$
$(0, - 1) ⟹$ need $(- 1, 0)$
$(1, 2) ⟹$ need $(2, 1)$
$(2, 3) ⟹$ need $(3, 2)$
Minimum elements to add, $n = 5$ .

Final Calculation

We found: $l = 7$ , $m = 5$ , $n = 5$
The question asks for $l + m + n$ (Note: typo in original text as $l + mn$ ).
$l + m + n = 7 + 5 + 5 = 17$
Final Answer: 17

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The Sigma Insight: Types of Relations

Solution Diagram

Analyzing the Setup

We are given the set

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

. The relation

R

is defined on

A

such that

x R y

if and only if

2 x - y \in {0, 1}

This condition implies two possible equations for any pair

(x, y)

: 1.

y = 2 x

y = 2 x - 1

Mapping the Connections

To find

l

, the number of elements in

R

, we test each

x \in A

to see which

y

values fall within the set

A

For $x = - 3$ : $y = 2 (- 3) = - 6$ (not in $A$ ) or $y = 2 (- 3) - 1 = - 7$ (not in $A$ ). For

x = - 2

y = 2 (- 2) = - 4

(not in

A

) or

y = 2 (- 2) - 1 = - 5

(not in

A

). For $x = - 1$ : $y = 2 (- 1) = - 2$ or $y = 2 (- 1) - 1 = - 3$ . Pairs: $(- 1, - 2), (- 1, - 3)$ . For

x = 0

y = 2 (0) = 0

y = 2 (0) - 1 = - 1

. Pairs:

(0, 0), (0, - 1)

. For $x = 1$ : $y = 2 (1) = 2$ or $y = 2 (1) - 1 = 1$ . Pairs: $(1, 2), (1, 1)$ . For

x = 2

y = 2 (2) = 4

(not in

A

) or

y = 2 (2) - 1 = 3

. Pair:

(2, 3)

. * For

x = 3

y = 2 (3) = 6

(not in

A

) or

y = 2 (3) - 1 = 5

(not in

A

The relation is

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

. Counting these, we find $l = 7$ .

The Quest for Reflexivity

A relation is reflexive if

(a, a) \in R

for all

a \in A

. The set

A

contains 7 elements, requiring the pairs:

(- 3, - 3), (- 2, - 2), (- 1, - 1), (0, 0), (1, 1), (2, 2), (3, 3)

Currently, only

(0, 0)

and

(1, 1)

are in

R

. We must add the 5 missing pairs to satisfy reflexivity. Thus, $m = 5$ .

The Mirror of Symmetry

A relation is symmetric if

(x, y) \in R \Rightarrow (y, x) \in R

. We examine the non-reflexive pairs in

R

(- 1, - 2), (- 1, - 3), (0, - 1), (1, 2), (2, 3)

None of these pairs have their corresponding symmetric counterparts present in

R

. To satisfy symmetry, we must add the following 5 pairs:

(- 2, - 1), (- 3, - 1), (- 1, 0), (2, 1), (3, 2)

Thus, $n = 5$ .

Final Calculation

Given the values derived from our analysis: $l = 7$

m = 5

n = 5

The final result is calculated as:

l + mn = 7 + (5 \times 5) = 7 + 25 = 32

Common Traps & FAQs

Wait, why do we reject the negative values for

x

when calculating

y

We don't reject them! We test them. The condition

2 x - y \in {0, 1}

applies to the entire set

A

. When

x = - 3

y

would be

- 6

- 7

. Since neither

- 6

nor

- 7

exists in our set

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

, those pairs simply cannot exist in the relation

R

. Always check the boundary of your set!

Do I need to add

(0, 0)

and

(1, 1)

again when calculating

m

Absolutely not! The definition of a reflexive relation requires that for every

a \in A

, the pair

(a, a)

is in

R

. Since

(0, 0)

and

(1, 1)

are already present in our initial set

R

, we only need to add the remaining 5 pairs to complete the set of 7 required reflexive pairs.

m

is the count of *additional* elements needed.

Is the order of elements in the pair

(x, y)

important for symmetry?

Yes, it is everything. Symmetry means that if the relation 'points' from

x

y

, it must also 'point' back from

y

x

. If you have

(1, 2)

in your relation, you must have

(2, 1)

. If you don't, the relation is not symmetric. You are essentially looking for the 'missing mirrors' of your existing pairs.

Why is the calculation

l + mn

instead of

l + m + n

In competitive exams like JEE, always read the final expression carefully. While the logic for finding

l

m

, and

n

remains the same, the final step is just an algebraic evaluation. If the question asks for

l + mn

, you multiply

m

and

n

first, then add

l

. Always follow the prompt's specific arithmetic instruction.

What if a pair is already symmetric, like

(0, 0)

That is a great observation! A pair

(a, a)

is its own symmetric counterpart. Since

(0, 0)

and

(1, 1)

are already in

R

, they don't require any 'new' elements to satisfy the symmetry condition. We only focus on the non-reflexive pairs that lack their mirror images.

Select Answer:

Visualized Solution

Defining Set $A$ and Relation $R$

Splitting the Condition

Testing $x = - 1, 0$

Testing $x = 1, 2$

Boundary Checks

Total Elements in $R$ ( $l$ )

Reflexive Additions ( $m$ )

Symmetric Additions ( $n$ )

Final Calculation

The Sigma Insight: Types of Relations

Analyzing the Setup

Mapping the Connections

The Quest for Reflexivity

The Mirror of Symmetry

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

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 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 ________

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Visualized Solution

Defining 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

Splitting the Condition

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

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

Boundary Checks

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

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

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

Final Calculation

The Sigma Insight: Types of Relations

Analyzing the Setup

Mapping the Connections

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 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 ________

Defining Set $A$ and Relation $R$

Testing $x = - 1, 0$

Testing $x = 1, 2$

Total Elements in $R$ ( $l$ )

Reflexive Additions ( $m$ )

Symmetric Additions ( $n$ )

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 ________