If $A=\{x: x=3 n, n \in Z\}$ and $B=\{x: x=4 n, n \in Z\}$ then $A \cap B$ is

$\{{x}: {x}={n}, {n} \in {Z}\}$

$\{{x}: {x}={n} / 2, {n} \in {Z}\}$

$\{{x}: {x}={n}-1, {n} \in {Z}\}$

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JEE • MATHEMATICS • SET THEORY

Set theory practice · Easy

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 151-180 of 194 questions

151 Easy

151 of 194

Understanding +4 | -1

X \cup\{3,4\}=\{1,2,3,4,5,6\}

then which of the following is true

View answer and explanation

Correct option: A

Smallest set

X=\{1,2,5,6\}

Option A is correct because the smallest set X is obtained by removing {3,4} from {1,2,3,4,5,6}.

152 Easy

152 of 194

Understanding +4 | -1

A=\{x: x=3 n, n \in Z\}

and

B=\{x: x=4 n, n \in Z\}

then

A \cap B

View answer and explanation

Correct option: D

\{{x}: {x}=12 {n}, {n} \in {Z}\}

Option D is correct because A contains multiples of 3, B contains multiples of 4, and their intersection contains multiples of both 3 and 4, which are multiples of 12.

153 Easy

153 of 194

Analyzing +4 | -1

Which of the following sets is not finite?

View answer and explanation

Correct option: A

\left\{(x, y): x^2+y^2 \leq 1 \leq x+y, x, y \in R\right\}

Option A (

\left\{(x, y): x^2+y^2 \leq 1 \leq x+y, x, y \in R\right\}

) is correct because it represents an infinite set of real numbers within the specified region.

154 Easy

154 of 194

Understanding +4 | -1

A \subseteq B

, then

A \cup B

is equal to:

View answer and explanation

Correct option: C

B

Option C (

B

) is correct because if

A

is a subset of

B

, then the union of

A

and

B

is equal to

B

155 Easy

155 of 194

Understanding +4 | -1

A=\{8,9,10\}

and

B=\{1,2,3,4,5\}

, then the number of elements in

\mathrm{A} \times \mathrm{A} \times \mathrm{B}

are

View answer and explanation

Correct option: C

The correct answer is 45 because the cardinality of the Cartesian product of sets is the product of their individual cardinalities.

156 Easy

156 of 194

Analyzing +4 | -1

n\left( {A \cap B} \right) = 5,\;n\left( {A \cap C} \right) = 7

and

n\left( {A \cap B \cap C} \right) = 3,

then the minimum possible value of

n\left( {B \cap C} \right)

View answer and explanation

Correct option: C

Option C (3) is correct because the minimum possible value of

n(B \cap C)

is equal to

n(A \cap B \cap C)

when

A \cap B \cap C

is the entire intersection of B and C.

157 Easy

157 of 194

Understanding +4 | -1

X

and

Y

are two sets and

X^{\prime}

denotes the complement of

X

, then

X \cap(X \cup Y)^{\prime}

equals

View answer and explanation

Correct option: C

\phi

Option C (

\phi

) is correct because the intersection of a set with the complement of a superset is always the empty set.

158 Easy

158 of 194

Understanding +4 | -1

{N}

is the set of natural numbers. The relation

{R}

is defined on

{N} \times {N}

as follows

(a, b) {R}(c, d) \Leftrightarrow a+d=b+c

. Then,

{R}

View answer and explanation

Correct option: D

An equivalence relation

Option D (An equivalence relation) is correct because the relation is reflexive, symmetric, and transitive.

159 Easy

159 of 194

Understanding +4 | -1

A,B

and

C

are three non-empty sets such that any two of them are disjoint, then

\left( {A \cup B \cup C} \right) \cap \left( {A \cap B \cap C} \right) =

View answer and explanation

Correct option: D

\phi

Option D (

\phi

) is correct because if the sets are disjoint, their intersection is empty, making the intersection of all three sets also empty, and the intersection of any set with the empty set is empty.

160 Easy

160 of 194

Analyzing +4 | -1

Let

A = \left\{ {\left( {x,y} \right):y = {e^x},x \in R} \right\},

B = \left\{ {\left( {x,y} \right):y = {e^{ - x}},x \in R} \right\}.

Then,

View answer and explanation

Correct option: B

A \cap B \ne \phi

Option B (

A \cap B \ne \phi

) is correct because the two curves y = e^x and y = e^-x intersect at the point (0, 1), indicating a non-empty intersection.

161 Easy

161 of 194

Understanding +4 | -1

n\left( {{A_i}} \right) = i + 1

and

{A_1} \subset {A_2} \subset {A_3} \subset \ldots \subset {A_{99}},

then

n\left( {U_{i = 1}^{99}{A_i}} \right) =

View answer and explanation

Correct option: C

100

Option C (100) is correct because the union of nested sets is the largest set, and n(A_99) = 99 + 1 = 100.

162 Easy

162 of 194

Understanding +4 | -1

A=\{x, y\}

, then the power set of

A

View answer and explanation

Correct option: D

\{\phi,\{x\},\{y\}\{x, y\}\}

Option D (

\{\phi,\{x\},\{y\}\{x, y\}\}

) is correct because it lists all possible subsets of the set A, including the empty set and the set itself.

163 Easy

163 of 194

Analyzing +4 | -1

Let

P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}

and

Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}

be two sets. Then,

View answer and explanation

Correct option: D

P=Q

Option D (

P=Q

) is correct because both sets are equivalent, resulting in the same value for

\tan \theta

164 Easy

164 of 194

Application +4 | -1

A

and

B

are two sets such that

n\left( A \right) = 7,n\left( B \right) = 6

and

\left( {A \cap B} \right) \ne \phi .

The least possible value of

n\left( {A\;\mathrm{\Delta \;}B} \right),

View answer and explanation

Correct option: A

Option A (11) is correct because the least possible value of n(A Δ B) is achieved when n(A ∩ B) is maximized, given that A ∩ B is not empty.

165 Easy

165 of 194

Application +4 | -1

n(U)=700, n(A)=200, n(B)=300

, and

n(A \cap B)= 100 ,

then

n\left(A^{\prime} \cap B^{\prime}\right)

is equal to

View answer and explanation

Correct option: C

300

Option C (300) is correct because it is the number of elements in the complement of the union of A and B, calculated using De Morgan's law and set theory principles.

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166 Easy

166 of 194

Understanding +4 | -1

For any three sets

{A_1},{A_2},{A_3},

let

{B_1} = {A_1},{B_2} = {A_2} - {A_1}

and

{B_3} = {A_3} - \left( {{A_1} \cup {A_2}} \right),

then which one of the following statement is always true

View answer and explanation

Correct option: A

{A_1} \cup {A_2} \cup {A_3} \supset {B_1} \cup {B_2} \cup {B_3}

Option B (

{A_1} \cup {A_2} \cup {A_3} = {B_1} \cup {B_2} \cup {B_3}

) is correct because the sets B_i are constructed to be disjoint and their union is equal to the union of the A_i.

167 Easy

167 of 194

Analyzing +4 | -1

A = \left\{ {\left( {x,y} \right):{y^2} = x;x,y \in R} \right\}

and

B = \left\{ {\left( {x,y} \right):y = \left| x \right|;x,y \in R} \right\},

then

View answer and explanation

Correct option: D

A \cap B

contains three elements only

Option D (

A \cap B

contains three elements only) is correct because the graphs of y² = x and y = |x| intersect at three points: (0, 0), (1, 1), and (1, -1).

168 Easy

168 of 194

Analyzing +4 | -1

n\left( {A \cap B} \right) = 10,n\left( {B \cap C} \right) = 20{\text{ }}n\left( {A \cap C} \right) = 30,

then the greatest possible value of

n\left( {A \cap B \cap C} \right)

View answer and explanation

Correct option: C

Option C (10) is correct because the greatest possible value of

n(A \cap B \cap C)

is limited by the smallest of

n(A \cap B)

n(B \cap C)

, and

n(A \cap C)

, which is 10.

169 Easy

169 of 194

Analyzing +4 | -1

Let

A

and

B

be two sets defined as given below:

A=\{(x, y):|x-3|<1

and

|y-3|<1\}

B=\left\{(x, y): 4 x^2+9 y^2-32 x-54 y+109 \leq 0\right\}

Then

View answer and explanation

Correct option: A

A \subset B

Option A (

A \subset B

) is correct because the region defined by A (a square) is entirely contained within the region defined by B (an ellipse).

170 Easy

170 of 194

Understanding +4 | -1

The collection of intelligent students in a class is:

View answer and explanation

Correct option: D

not a well defined collection

Option D is correct because the term "intelligent" is subjective and lacks a clear, objective definition, making the collection not well-defined.

171 Easy

171 of 194

Application +4 | -1

Let

\bigcup\limits_{i\, = \,1}^{50} {{X_i}} \, = \,\bigcup\limits_{i\, = \,1}^n {{Y_i}} \, = \,T,

where each

{X_i}

contains 10 elements and each

{Y_i}

contains 5 elements. If each element of the set

T

is an element of exactly 20 of sets

{X_i}^\prime

s

and exactly 6 of sets

{Y_i}^\prime

s

, then

n

is equal to

View answer and explanation

Correct option: D

The correct answer is D (30) because it is derived by equating the total number of elements in the union of sets X_i and Y_i, considering the multiple occurrences of each element.

172 Easy

172 of 194

Understanding +4 | -1

a N=\{a x: x \in N\}

, then the set

3 N \cap 7 N

View answer and explanation

Correct option: A

21 {N}

Option A (

21N

) is correct because the intersection of multiples of 3 and multiples of 7 are multiples of 21.

173 Easy

173 of 194

Analyzing +4 | -1

Let R1, R2 are relations defined on Z such that a

a{R_1}b \Leftrightarrow (a - b)

is divisible by 3 and a R2

b \Leftrightarrow (a - b)

is divisible by 4. Then which of the two relations

\left( {{R_1} \cup {R_2}} \right),\left( {{R_1} \cap {R_2}} \right)

is an
equivalence relation?

View answer and explanation

Correct option: B

\left(R_1 \cap R_2\right)

only

Option B (

\left(R_1 \cap R_2\right)

only) is correct because the intersection of equivalence relations is always an equivalence relation, while the union is not necessarily so.

174 Easy

174 of 194

Understanding +4 | -1

A=\{x: x=2 n+1, n \in Z

and

B=\{x: x=2 n, n \in Z\}

, then

A \cup B

View answer and explanation

Correct option: C

Set of integers

Option C is correct because A contains all odd integers, B contains all even integers, and their union is the set of all integers.

175 Easy

175 of 194

Analyzing +4 | -1

For any three sets

A, B

and

C

the set

(A \cup B \cup C) \cap\left(A \cap B^{\prime} \cap C^{\prime}\right)^{\prime} \cap C^{\prime}

is equal to

View answer and explanation

Correct option: A

B \cap C^{\prime}

Option A (

B \cap C^{\prime}

) is the correct answer because after simplifying the set operations, the expression reduces to this intersection.

176 Easy

176 of 194

Application +4 | -1

A = \left\{ {1,\;2,\;3,\;4} \right\},

then the number of subsets of set A containing element 3, is

View answer and explanation

Correct option: C

Option C (8) is correct because there are 2³ = 8 subsets that contain the element 3, formed from the remaining elements 1, 2, and 4.

177 Easy

177 of 194

Understanding +4 | -1

X

and

Y

are two sets, then

X \cap {\left( {Y \cup X} \right)^\prime }

equals

View answer and explanation

Correct option: C

\phi

Option C (

\phi

) is correct because

X \cap (Y \cup X)' = X \cap (Y' \cap X') = (X \cap X') \cap Y' = \phi \cap Y' = \phi

178 Easy

178 of 194

Application +4 | -1

In a class of 45 students, 22 can speak Hindi and 12 can speak English only. The number of students, who can speak both Hindi and English, is

View answer and explanation

Correct option: B

The correct answer is B (11) because it is calculated by using set theory principles including union, intersection, and complement of sets.

179 Easy

179 of 194

Application +4 | -1

A survey shows that 63% of the Americans like cheese whereas 76% like apples. If

x\%

of the Americans like both cheese and apples, then

View answer and explanation

Correct option: C

39 \leqslant x \leqslant 63

The correct answer is C (

39 \leqslant x \leqslant 63

) because the percentage of Americans who like both cheese and apples must be within the range where the total percentage does not exceed 100%.

180 Easy

180 of 194

Understanding +4 | -1

The number of non-empty subsets of the set

\{1,2,3,4\}

View answer and explanation

Correct option: A

Option A (15) is correct because the number of non-empty subsets of a set with n elements is 2^n - 1.

Revision notes and key takeaways

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