Core Syllabus Module

Sets and Relations

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Chapter Elite-20 Challenge Zone

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JEE Main 2021 (22 July Shift 1)

LEVELJEE Advanced

The sum of all the elements in the set ${n \in {1, 2, \dots, 100} : H.C.F. of n and 2040 is 1}$ is equal to ____.

JEE Main 2022 (25 July Shift 2)

LEVELJEE Advanced

Let $A : {1, 2, 3, 4, 5, 6, 7}$ . Define $B = {T \subseteq A : either 1 \in / T or 2 \in / T}$ and $C = {T \subseteq A : the sum of all the elements of T is a prime number}$ . Then the number of elements in the set $B \cup C$ is ________

JEE Main 2024 (04 Apr Shift 1)

LEVELJEE Advanced

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m + n$ is equal to ______

JEE(ADVANCED)-202

LEVELJEE Main

Comprehension Passage

Let

S = {1, 2, 3, 4, 5, 6}

and

X

be the set of all relations

R

from

S

S

that satisfy both the following properties: i.

R

has exactly 6 elements. ii. For each

(a, b) \in R

, we have

∣ a - b ∣ \geq 2

. Let

Y = {R \in X : The range of R has exactly one element}

and

Z = {R \in X : R is a function from S to S}

. Let

n (A)

denote the number of elements in a set

A

Question 1:

If $n (X) =^{m} C_{6}$ , then the value of $m$ is ________.

Question 2:

If the value of $n (Y) + n (Z)$ is $k^{2}$ , then $∣ k ∣$ is ________.

JEE Main 2025 April

LEVELJEE Main

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 ________

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

Among the relations $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and $T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ ,

(A)

S is transitive but T is not

(B)

T is symmetric but S is not

(C)

Neither S nor T is transitive

(D)

Both S and T are symmetric

JEE Main 2024 (08 Apr Shift 2)

LEVELJEE Main

Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is

(A)

(1) an equivalence relation.

(B)

(2) reflexive and symmetric but not transitive.

(C)

(3) transitive but not symmetric.

(D)

(4) reflexive but not symmetric.

JEE Main 2026 (23 January Shift 1)

LEVELJEE Main

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 :

JEE Main 2023 (01 February Shift 1)

LEVELJEE Main

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is

JEE Main 2026 (23 January Shift 2)

LEVELJEE Main

Let $A = {0, 1, 2, \dots, 9}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $∣ x - y ∣$ is a multiple of 3. Given below are two statements: Statement I: $n (R) = 36$ . Statement II: $R$ is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below

(A)

(1) Statement I is incorrect but Statement II is correct

(B)

(2) Statement I is correct but Statement II is incorrect

(C)

(3) Both Statement I and Statement II are incorrect

(D)

(4) Both Statement I and Statement II are correct

JEE Main 2023 (13 April Shift 1)

LEVELJEE Main

The negation of the statement $((A \land (B \lor C)) \Rightarrow (A \lor B)) \Rightarrow A$ is

(A)

equivalent to

\sim C

(B)

equivalent to

B \lor \sim C

\sim A

JEE Main 2023 (24 January Shift 1)

LEVELJEE Main

The compound statement $(\sim (P \land Q)) \lor ((\sim P) \land Q) \Rightarrow ((\sim P) \land (\sim Q))$ is equivalent to

(A)

((\sim P) \vee Q) \wedge((\sim Q) \vee P)

(B)

(\sim Q) \vee P

(C)

((\sim P) \vee Q) \wedge(\sim Q)

(D)

(\sim P) \vee Q

JEE Main 2026 (22 January Shift 2)

LEVELJEE Main

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in Z}$ is

JEE Main 2021 (27 July Shift 2)

LEVELJEE Main

Let $A = {n \in N ∣ n^{2} \leq n + 10, 000}$ , $B = {3 k + 1 ∣ k \in N}$ and $C = {2 k ∣ k \in N}$ , then the sum of all the elements of the set $A \cap (B - C)$ is equal to

JEE Main 2021 (27 Aug Shift 1)

LEVELJEE Main

If $A = {x \in R : ∣ x - 2∣ > 1}$ , $B = {x \in R : x^{2} - 3 > 1}$ , $C = {x \in R : ∣ x - 4∣ \geq 2}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^{c} \cap Z$ is .

JEE Main 2021 (27 Aug Shift 2)

LEVELJEE Main

Let $Z$ be the set of all integers, $A = {(x, y) \in Z \times Z : (x - 2)^{2} + y^{2} \leq 4}$ , $B = {(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4}$ and $C = {(x, y) \in Z \times Z : (x - 2)^{2} + (y - 2)^{2} \leq 4}$ . If the total number of relation from $A \cap B$ to $A \cap C$ is $2^{p}$ , then the value of $p$ is :

JEE Main 2023 (01 February Shift 2)

LEVELJEE Main

Let $P (S)$ denote the power set of $S = {1, 2, 3, \dots, 10}$ . Define the relations $R_{1}$ and $R_{2}$ on $P (S)$ as $A R_{1} B$ if $(A \cap B^{c}) \cup (B \cap A^{c}) = \emptyset$ and $A R_{2} B$ if $A \cup B^{c} = B \cup A^{c}, \forall A, B \in P (S)$ . Then:

(A)

both

R_{1}

and

R_{2}

are equivalence relations

(B)

only

R_{1}

is an equivalence relation

(C)

only

R_{2}

is an equivalence relation

(D)

both

R_{1}

and

R_{2}

are not equivalence relations

JEE Main 2021 (31 Aug Shift 1)

LEVELJEE Main

Which of the following is not correct for relation $R$ on the set of real numbers?

(A)

(x, y) \in R \Leftrightarrow 0 < ∣ x ∣ - ∣ y ∣ \leq 1

is neither transitive nor symmetric.

(B)

(x, y) \in R \Leftrightarrow 0 < ∣ x - y ∣ \leq 1

is symmetric and transitive.

(C)

(x, y) \in R \Leftrightarrow ∣ x ∣ - ∣ y ∣ \leq 1

is reflexive but not symmetric.

(D)

(x, y) \in R \Leftrightarrow ∣ x - y ∣ \leq 1

is reflexive and symmetric.

JEE Main 2022 (29 July Shift 1)

LEVELJEE Main

Let $S = {4, 6, 9}$ and $T = {9, 10, 11, \dots, 1000}$ . If $A = {a_{1} + a_{2} + \dots + a_{k} : k \in N, a_{1}, a_{2}, \dots, a_{k} \in S}$ , then the sum of all the elements in the set $T - A$ is equal to ______.

JEE Main 2021 (31 Aug Shift 2)

LEVELJEE Main

Micro-Topic Strategy Matrix

Representation of Sets

Types of Sets and Set Operations

Cartesian Product of Sets

Domain and Range of a Relation

Types of Relations

Equivalence Relations

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all the elements in the set {n∈{1,2,…,100}:H.C.F. of n and 2040 is 1} is equal to ____.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If n(X)=mC6​, then the value of m is ________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the value of n(Y)+n(Z) is k2, then ∣k∣ is ________.

.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; } Among the relations S={(a,b):a,b∈R−{0},2+ba​>0} and T={(a,b):a,b∈R,a2−b2∈Z},

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={2,3,6,8,9,11} and B={1,4,5,10,15}. Let R be a relation on A×B defined by (a,b)R(c,d) if and only if 3ad−7bc is an even integer. Then the relation R is

The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The negation of the statement ((A∧(B∨C))⇒(A∨B))⇒A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The compound statement (∼(P∧Q))∨((∼P)∧Q)⇒((∼P)∧(∼Q)) is equivalent to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the relation R={(x,y):4x2+y2<52,x,y∈Z} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={n∈N∣n2≤n+10,000}, B={3k+1∣k∈N} and C={2k∣k∈N}, then the sum of all the elements of the set A∩(B−C) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A={x∈R:∣x−2∣>1}, B={x∈R:x2−3​>1}, C={x∈R:∣x−4∣≥2} and Z is the set of all integers, then the number of subsets of the set (A∩B∩C)c∩Z is .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following is *not* correct for relation R on the set of real numbers?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={4,6,9} and T={9,10,11,…,1000}. If A={a1​+a2​+⋯+ak​:k∈N,a1​,a2​,…,ak​∈S}, then the sum of all the elements in the set T−A is equal to ______.

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is .

The sum of all the elements in the set ${n \in {1, 2, \dots, 100} : H.C.F. of n and 2040 is 1}$ is equal to ____.

If $n (X) =^{m} C_{6}$ , then the value of $m$ is ________.

If the value of $n (Y) + n (Z)$ is $k^{2}$ , then $∣ k ∣$ is ________.

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 ________

Among the relations $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and $T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ ,

Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is

The negation of the statement $((A \land (B \lor C)) \Rightarrow (A \lor B)) \Rightarrow A$ is

The compound statement $(\sim (P \land Q)) \lor ((\sim P) \land Q) \Rightarrow ((\sim P) \land (\sim Q))$ is equivalent to

The number of elements in the relation $R = {(x, y) : 4 x^{2} + y^{2} < 52, x, y \in Z}$ is

Let $A = {n \in N ∣ n^{2} \leq n + 10, 000}$ , $B = {3 k + 1 ∣ k \in N}$ and $C = {2 k ∣ k \in N}$ , then the sum of all the elements of the set $A \cap (B - C)$ is equal to

If $A = {x \in R : ∣ x - 2∣ > 1}$ , $B = {x \in R : x^{2} - 3 > 1}$ , $C = {x \in R : ∣ x - 4∣ \geq 2}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^{c} \cap Z$ is .

Which of the following is not correct for relation $R$ on the set of real numbers?

Let $S = {4, 6, 9}$ and $T = {9, 10, 11, \dots, 1000}$ . If $A = {a_{1} + a_{2} + \dots + a_{k} : k \in N, a_{1}, a_{2}, \dots, a_{k} \in S}$ , then the sum of all the elements in the set $T - A$ is equal to ______.