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JEE • MATHEMATICS • SET THEORY

Set theory practice · Medium

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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 1-30 of 78 questions

1 Medium

1 of 78

Analyzing +4 | -1

Let

S

be the set of all those 2007-place decimal integers

2 a_1 a_2 a_3 \ldots . . a_{2006}

which contain odd number of digit 9 in each sequence

a_1, a_2, a_3, \ldots ., a_{2006}

. The cardinal number of

S

View answer and explanation

Correct option: B

\frac{1}{2}\left(10^{2006}-8^{2006}\right)

Option A (

\frac{1}{2}(10^{2006}+8^{2006})

) is incorrect. The correct answer is

\frac{1}{2}(10^{2006}-8^{2006})

because it represents the number of 2007-place decimal integers with an odd number of 9s in each sequence.

2 Medium

2 of 78

Analyzing +4 | -1

a N=\{a n: n \in N: b N=\{n b: n \in N\}

c N=\{c N: n \in N\}

and

b N \cap c N=d N

, where

a, b, c \in N

and

b, c

are coprime, then

View answer and explanation

Correct option: C

d=b c

Option A (0) is correct because the equation

x^3 - 3x - k = 0

cannot have two distinct real roots in the interval (0, 1) based on Rolle's theorem.

3 Medium

3 of 78

Understanding +4 | -1

A=\{x: x

is a multiple of 3

\}

and

B=\{x: x

is a multiple of 5}, then

A-B

is equal to

View answer and explanation

Correct option: B

A \cap \bar{B}

Option B (

A \cap \bar{B}

) is correct because A - B represents elements in A but not in B, which is the intersection of A and the complement of B.

4 Medium

4 of 78

Application +4 | -1

F(n)

denotes the set of all divisors of

n

except 1 , what is the least value of y satisfying

[\mathrm{F}(20) \cap \mathrm{F}(16)] \subseteq \mathrm{F}(\mathrm{y})

View answer and explanation

Correct option: B

Option B (2) is correct because it's the smallest number whose divisors (excluding 1) contain the intersection of the divisors of 20 and 16 (excluding 1).

5 Medium

5 of 78

Understanding +4 | -1

A

and

B

are subsets of a set

X

, then what is

\{A \cap(X-B)\} \cup B

equal to?

View answer and explanation

Correct option: A

A \cup B

Option A (

A \cup B

) is correct because the expression simplifies to the union of sets A and B using set identities.

6 Medium

6 of 78

Understanding +4 | -1

A=\{1,2,3\}, B=\left\{x \in R: x^2-2 x+1=0, C=\{1,2,3\}\right.

and

D=\left\{x \in R: x^3-6 x^2+11 x-6=0 ;\right.

then the equal sets are

View answer and explanation

Correct option: D

A, C

and

D

Option D (

A, C

and

D

) is correct because sets A, C, and D all contain the elements {1, 2, 3}, making them equal.

7 Medium

7 of 78

Application +4 | -1

In a class of 30 pupils, 12 take needle work, 16 take physics and 18 take history. If all the 30 students take at least one subject and no one takes all three then the number of pupils taking 2 subjects is

View answer and explanation

Correct option: A

Option A (16) is correct because it represents the number of pupils taking exactly two subjects, calculated using the principle of inclusion-exclusion and the given conditions.

8 Medium

8 of 78

Application +4 | -1

In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. what is the minimum number of people who were having diabetes and high blood pressure?

View answer and explanation

Correct option: C

Option C (20) is correct because it is calculated using the formula for the number of elements in the union of two sets: n(D ∪ H) = n(D) + n(H) - n(D ∩ H), and considering the total number of people.

9 Medium

9 of 78

Application +4 | -1

n(A \cup B \cup C)+n(A \cup B \cup C)^{\prime}=50, n(A)=10, n(B)=15, n(C)=20, n(A \cap B)=5, n(B \cap C)=6, n(C \cap A)=10, n(A \cap B \cap C)=5,

then the value of

n(A \cup B \cup C)^{\prime}

View answer and explanation

Correct option: B

Option B (21) is correct because using the principle of inclusion-exclusion to find

n(A \cup B \cup C)

and subtracting it from the total number of elements gives the correct value for

n(A \cup B \cup C)'

10 Medium

10 of 78

Understanding +4 | -1

Let

A

and

B

have 3 and 6 elements, respectively. What can be minimum number of elements in

A \cup B

View answer and explanation

Correct option: B

Option B (6) is correct because the minimum number of elements in the union of two sets occurs when one set is a subset of the other.

11 Medium

11 of 78

Understanding +4 | -1

The Cartesian product of two sets

\mathrm{P}

and

\mathrm{Q}

, i.e.,

\mathrm{P} \times \mathrm{Q}=\phi

, if

View answer and explanation

Correct option: A

either

\mathrm{P}

\mathrm{Q}

is the null set

Option A is correct because the Cartesian product of two sets is empty if at least one of the sets is empty.

12 Medium

12 of 78

Application +4 | -1

n(A)=115, n(B)=326, n(A-B)=47

, then what is

n(A \cup B)

equal to?

View answer and explanation

Correct option: A

373

Option A (373) is correct because it is calculated using the formula for the number of elements in the union of two sets, given the number of elements in each set and their difference.

13 Medium

13 of 78

Analyzing +4 | -1

Which of the following is true?

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because none of the given set relations are true based on the definitions of set membership and subsets.

14 Medium

14 of 78

Application +4 | -1

In a B School there are 15 teachers who teach marketing or finance. Of these, 8 teach finance and 4 teach both marketing and finance. How many teach marketing but not finance?

View answer and explanation

Correct option: C

Option C (11) is correct because it is calculated by subtracting the number of teachers teaching both marketing and finance from the total number of teachers teaching marketing or finance.

15 Medium

15 of 78

Understanding +4 | -1

Consider the following statements.
I. If

A_{n}

is the set of first

n

prime numbers, then

\bigcup\limits_{{n=2}}^{10} A_{n}

is equal to

\{2,3,5,7,11,13,17,19,23,29\}

II. If

{A}

and

{B}

are two sets such that

{n}({A} \cup {B})=50

{n}({A})=28, {n}({B})=32

, then

{n}({A} \cap {B})=10

.
Which of these is correct?

View answer and explanation

Correct option: C

Both are true

Option C (Both are true) is correct because the union of the first n prime numbers up to n=10 is the set of the first 10 prime numbers, and the cardinality formula for the union of two sets is correctly applied.

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16 Medium

16 of 78

Application +4 | -1

In a group of 500 students, there are 475 students who can speak Hindi and 200 can speak Bengali. What is the number of students who can speak Hindi only?

View answer and explanation

Correct option: B

300

Option B (300) is correct because it is calculated by subtracting the number of students who speak Hindi or Bengali from the total number of students.

17 Medium

17 of 78

Analyzing +4 | -1

A=\{5 n / n \in \mathbb{N}, n \leq 100\}, B=\{3 n / n \in \mathbb{N}, n \geq 100\}

and

C=\{2 n / n \in \mathbb{N}\}

then number of elements in

(A \cap B) \cap C^1

View answer and explanation

Correct option: A

Option A (7) is correct because it correctly identifies the number of elements in the intersection of A and B that are odd multiples of 15 and less than or equal to 50*15.

18 Medium

18 of 78

Application +4 | -1

Let

X

and

Y

be the sets of all positive divisors of 400 and 1000 respectively (including 1 and the number). Then,

n\left( {X \cap Y} \right)

is equal to

View answer and explanation

Correct option: D

Option D (12) is correct because the intersection of the sets of divisors of 400 and 1000 is the set of divisors of their greatest common divisor (GCD), which is 200, and 200 has 12 divisors.

19 Medium

19 of 78

Analyzing +4 | -1

Which of the following is not correct?

View answer and explanation

Correct option: C

A \cup B=A \cup C

, then

B=C

The statement \"If

A \cup B=A \cup C

, then

B=C

\" (Option C) is not always correct, making it the correct answer.

20 Medium

20 of 78

Application +4 | -1

Let

n(\mathrm{U})=700, n(\mathrm{A})=200, n(\mathrm{B})=300, n(\mathrm{A} \cap \mathrm{B})

=100

, then

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

is equal to

View answer and explanation

Correct option: C

300

Option C (300) is correct because it is calculated using De Morgan's law and the formula for the number of elements in the union of two sets.

21 Medium

21 of 78

Analyzing +4 | -1

In a certain town,

25 \%

of the families own a phone and

15 \%

own a car;

65 \%

families own neither a phone nor a car and 2,000 families own both a car and a phone. Consider the following three statements:
(A)

5 \%

families own both a car and a phone
(B)

35 \%

families own either a car or a phone
(C) 40,000 families live in the town
Then,

View answer and explanation

Correct option: C

All (A), (B) and (C) are correct.

Option C (All (A), (B) and (C) are correct) is correct because the calculations based on the given percentages and the number of families owning both a car and a phone confirm all three statements.

22 Medium

22 of 78

Understanding +4 | -1

A,B

and

C

are three sets such that

A \supset B \supset C,

then

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

View answer and explanation

Correct option: C

A - C

Option C (

A - C

) is correct because given

A \supset B \supset C

A \cup B \cup C = A

and

A \cap B \cap C = C

, so the expression simplifies to

A - C

23 Medium

23 of 78

Analyzing +4 | -1

From 50 students taking examinations in Mathematics, Physics and Chemistry, 37 passed Mathematics, 24 Physics and 43 Chemistry. At most 19 passed Mathematics and Physics, at most 29 passed Mathematics and Chemistry and at most 20 passed Physics and Chemistry. The largest possible number that could have passed all three examinations is

View answer and explanation

Correct option: D

Option D (14) is correct because it represents the largest possible number of students who could have passed all three examinations, derived from the given constraints on the number of students passing in pairs of subjects.

24 Medium

24 of 78

Analyzing +4 | -1

Let

A

and

B

be two sets, then

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

is equal to

View answer and explanation

Correct option: A

A^{\prime}

Option A (

A^{\prime}

) is correct because the expression simplifies to the complement of set A using set theory rules.

25 Medium

25 of 78

Application +4 | -1

P(A)

denotes the power set of

A

and

A

is the void set, then what is number of elements in

P\{P\{P\{P(A)\}\}\} ?

View answer and explanation

Correct option: D

Option D (16) is correct because the power set of the power set is calculated iteratively, doubling the number of elements at each step.

26 Medium

26 of 78

Application +4 | -1

If there are three atheletic teams in a school, 21 are in the basketball team, 26 in hockey team and 29 in the football team. 14 play hockey and basketball, 15 play hockey and football, 12 play football and basketball and 8 play all the games. The total number of members is

View answer and explanation

Correct option: B

Option B (43) is correct because it is calculated using the principle of inclusion-exclusion to find the total number of members in all three teams.

27 Medium

27 of 78

Understanding +4 | -1

Let

P

be the set of prime numbers and

S=\left\{t: 2^t-1\right.

is a prime\}. Then,

View answer and explanation

Correct option: A

S \subset P

Option A (

S \subset P

) is correct because if

2^t - 1

is prime, then t must be prime, but not all prime numbers t will result in

2^t - 1

being prime. Therefore, S is a subset of P.

28 Medium

28 of 78

Understanding +4 | -1

Consider the following statements.

I. If (a, 1), (b, 2) and (c, 1) are in

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

and

\mathrm{n}(\mathrm{A})

=3, n(B)=2

, then

A=\{a, b, c\}

and

B=\{1,2\}

II. If

A=\{1,2\}

and

B=\{3,4\}

, then

A \times(B \cap \phi)

is equal to

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

.

Choose the correct option.

View answer and explanation

Correct option: A

Only I is true

Option A is correct because statement I is true based on the definition of Cartesian product, while statement II is false since the intersection of any set with the empty set is the empty set.

29 Medium

29 of 78

Application +4 | -1

In a class of 30 pupils 12 take needls work, 16 take physics and 18 take history. If all the 30 students take at least one subject and no one takes all three, then the number of pupils taking 2 subjects is

View answer and explanation

Correct option: A

The correct answer is A (16) because it is calculated by using the principle of inclusion-exclusion and the given conditions.

30 Medium

30 of 78

Analyzing +4 | -1

A=\left\{(x, y): x^2+y^2 \leq 9\right\}

B=\left\{(x, y):\left|x-1-x^2\right| \leq\left|x^2-3 x+4\right|\right\} \text { where }

x

and

y

are integers.

Number of integral

(\mathrm{x}, \mathrm{y})

in set

\mathrm{A} \cap \mathrm{B}

are

View answer and explanation

Correct option: D

The number of integral (x, y) in set A ∩ B is 23. This is found by identifying the integer values of x and y that satisfy both inequalities.

Revision notes and key takeaways

Review SET THEORY in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 78 single-correct-answer MCQs on SET THEORY.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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