Set Theory Questions with Solutions

Set Questions have been provided here to simplify the concept of sets and relations for the students of Class 11. These set questions have been designed according to the latest CBSE syllabus. From basic sets and relations to set functions and subsets, we've covered key areas to help you develop a good understanding of set theory.

Learn about set theory in detail through this tutorial, which covers all the concepts of set theory, from basics to advanced topics. [Read here]

Set Theory Questions and Solutions

Question 1: In a class of 40 students, 22 play hockey, 26 play basketball, and 14 play both. How many students do not play either of the games?

Solution:

Let H be the set of students playing hockey, and B be the set playing basketball.
n(H) = 22, n(B) = 26, n(H ∩ B) = 14.
n(H ∪ B) = n(H) + n(B) - n(H ∩ B) = 22 + 26 - 14 = 34.
Students not playing either = Total students - n(H ∪ B) = 40 - 34 = 6.

Question 2: If set A = {1, 3, 5, 7, 9} and set B = {1, 2, 3, 4, 5}, find A ∪ B and A ∩ B.

Solution:

A ∪ B (union) is the set of elements that are in A, or B, or both.
A ∪ B = {1, 2, 3, 4, 5, 7, 9}.
A ∩ B (intersection) is the set of elements that are in both A and B.
A ∩ B = {1, 3, 5}.

Question 3: In a survey of 60 people, 25 liked tea, 30 liked coffee, and 10 liked both. How many people liked only tea?

Solution:

Number of people who liked only tea = Number who liked tea - Number who liked both.
= 25 - 10 = 15 people liked only tea.

Question 4: For sets A = {x | x is an integer, 1 ≤ x ≤ 6} and B = {x | x is an even integer, 2 ≤ x ≤ 8}, find the set A - B.

Solution:

A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}.
A - B (difference) is the set of elements in A that are not in B.
A - B = {1, 3, 5}.

Question 5: Given sets X = {a, b, c, d} and Y = {b, d, e, f}, find the symmetric difference of X and Y (denoted as X Δ Y).

Solution:

X Δ Y is the set of elements in either X or Y, but not in their intersection.
Intersection X ∩ Y = {b, d}.
X Δ Y = (X ∪ Y) - (X ∩ Y) = {a, b, c, d, e, f} - {b, d} = {a, c, e, f}.

Question 6: If set C = {2, 4, 6, 8} and set D = {6, 8, 10, 12}, what are the sets C ∩ D and C ∪ D?

Solution:

C ∩ D (intersection) is the set of elements common to both C and D.
C ∩ D = {6, 8}.
C ∪ D (union) is the set of all elements in C, or D, or both.
C ∪ D = {2, 4, 6, 8, 10, 12}.

Question 7: A survey of 100 students found that 70 students like pizza, 75 like burgers, and 60 like both. How many students like neither pizza nor burgers?

Solution:

Let P be the set of students who like pizza, and B be the set who like burgers.
n(P ∪ B) = n(P) + n(B) - n(P ∩ B) = 70 + 75 - 60 = 85.
Students who like neither = Total students - n(P ∪ B) = 100 - 85 = 15.

Question 8: For sets E = {1, 3, 5, 7, 9} and F = {0, 1, 2, 3, 4}, find the set E ∪ F and the set E - F.

Solution:

E ∪ F (union) is the set of elements in E, or F, or both.
E ∪ F = {0, 1, 2, 3, 4, 5, 7, 9}.
E - F (difference) is the set of elements in E that are not in F.
E - F = {5, 7, 9}.

Question 9: Given set G = {a, e, i, o, u} and set H = {a, e, y}, find the symmetric difference G Δ H.

Solution:

G Δ H is the set of elements in either G or H, but not in their intersection.
Intersection G ∩ H = {a, e}.
G Δ H = (G ∪ H) - (G ∩ H) = {a, e, i, o, u, y} - {a, e} = {i, o, u, y}.

Question 10: In a group of 50 people, 28 like tea, 26 like coffee, and 12 like both. Find the number of people who like only coffee.

Solution:

Number of people who like only coffee = Number who like coffee - Number who like both.
= 26 - 12 = 14 people like only coffee.

Related Links:

• Representation of a Set
• Types of Set
• Set Operations

Question 11: A class has 60 students. 35 students play football, 40 play cricket, and 15 play both. How many students play only one sport?

Solution:

Number of students playing only one sport = Number playing football only + Number playing cricket only.
= (Number playing football - Number playing both) + (Number playing cricket - Number playing both).
= (35 - 15) + (40 - 15) = 20 + 25 = 45 students.

Question 12: Sets J = {2, 4, 6, 8, 10} and K = {3, 6, 9, 12}. Find the Cartesian product J × K.

Solution:

J × K is the set of all ordered pairs (j, k) where j is in J and k is in K.
J × K = {(2, 3), (2, 6), (2, 9), (2, 12), (4, 3), (4, 6), (4, 9), (4, 12), (6, 3), (6, 6), (6, 9), (6, 12), (8, 3), (8, 6), (8, 9), (8, 12), (10, 3), (10, 6), (10, 9), (10, 12)}.

Question 13: If set M = {x | x is a prime number less than 20} and set N = {x | x is an odd number less than 10}, what is M ∩ N?

Solution:

M = {2, 3, 5, 7, 11, 13, 17, 19}, N = {1, 3, 5, 7, 9}.
M ∩ N (intersection) is the set of elements that are in both M and N.
M ∩ N = {3, 5, 7}.

Question 14: A survey of 200 people found that 120 read newspaper A, 150 read newspaper B, and 90 read both. How many read at least one of the newspapers?

Solution:

Use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
n(A ∪ B) = 120 + 150 - 90 = 180.
180 people read at least one of the newspapers.

Question 15: For sets P = {a, b, c, d} and Q = {c, d, e, f, g}, find P ∪ Q and P - Q.

Solution:

P ∪ Q (union) is the set of elements in P, or Q, or both.
P ∪ Q = {a, b, c, d, e, f, g}.
P - Q (difference) is the set of elements in P that are not in Q.
P - Q = {a, b}.

Question 16: In a group of 50 people, 28 have traveled to Europe, 31 have traveled to Asia, and 10 have traveled to both continents. How many people have not traveled to either continent?

Solution:

Use the principle of inclusion-exclusion.
Number of people who have traveled to either continent = (Number to Europe) + (Number to Asia) - (Number to both).
= 28 + 31 - 10 = 49.
Number of people who haven't traveled to either continent = Total people - Number who have traveled to either continent.
= 50 - 49 = 1.

Question 17: Set X contains all even numbers between 1 and 20, and Set Y contains all multiples of 3 between 1 and 20. Find X ∩ Y.

Solution:

X = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}, Y = {3, 6, 9, 12, 15, 18}.
X ∩ Y (intersection) is the set of elements common to both X and Y.
X ∩ Y = {6, 12, 18}.

Question 18: If Set A = {2, 4, 6, 8} and Set B = {1, 3, 5, 7, 9}, find the Cartesian product A × B and B × A.

Solution:

A × B = {(2, 1), (2, 3), (2, 5), (2, 7), (2, 9), (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), (6, 1), (6, 3), (6, 5), (6, 7), (6, 9), (8, 1), (8, 3), (8, 5), (8, 7), (8, 9)}.
B × A = {(1, 2), (1, 4), (1, 6), (1, 8), (3, 2), (3, 4), (3, 6), (3, 8), (5, 2), (5, 4), (5, 6), (5, 8), (7, 2), (7, 4), (7, 6), (7, 8), (9, 2), (9, 4), (9, 6), (9, 8)}.

Question 19: In a survey of 150 people, 95 like apples, 70 like bananas, and 60 like both apples and bananas. Find the number of people who like only apples.

Solution:

Number of people who like only apples = Number who like apples - Number who like both apples and bananas.
= 95 - 60 = 35 people.

Question 20: Given sets L = {x | x is a positive integer less than 6} and M = {x | x is a positive integer and a multiple of 2}, find L ∪ M and L ∩ M.

Solution:

L = {1, 2, 3, 4, 5}, M = {2, 4, 6, 8, 10, ...}.
L ∪ M (union) = {1, 2, 3, 4, 5, 6, 8, 10, ...} (all positive integers less than 6 and all multiples of 2).
L ∩ M (intersection) = {2, 4} (elements that are both less than 6 and multiples of 2).

Also check other Practice Questions

• Questions on Operations on Sets
• Union and Intersection of Sets Questions
• Problems on Finite and Infinite Sets

Also check the Quiz - [Quiz on Set Theory]

Practice Problem for Sets

Question 1: For sets P = {a, b, c, d} and Q = {c, d, e, f, g}, find P ∪ Q and P - Q.

Question 2: Set X contains all even numbers between 1 and 20, and Set Y contains all multiples of 3 between 1 and 20. Find X ∩ Y.

Question 3: If Set A = {2, 4, 6, 8} and Set B = {1, 3, 5, 7, 9}, find the Cartesian product A × B and B × A.

Question 4: In a group of 500 people, 350 people can speak English, and 400 people can speak Hindi. Find out how many people can speak both languages?

Question 5: In a class of 40 students, 20 have chosen Mathematics, 15 have chosen mathematics but not Biology. If every student has chosen either mathematics or biology or both, find the number of students who chose both mathematics and biology and the number of students who chose biology but not mathematics.

Question 6:In a survey among 140 students, 60 likes to play videogames, 70 likes to play indoor games, 75 likes to play outdoor games, 30 play indoor and outdoor games, 18 like to play video games and outdoor games, 42 play video games and indoor games and 8 likes to play all types of games. Use the Venn diagram to find

• (i) Students who play only outdoor games
• (ii) Students who play video games and indoor games, but not outdoor games

Question 7: In a group of 120 people, 54 like Coca-Cola and 84 like Pepsi and each person likes at least two beverages. How many like both Coca-Cola and Pepsi?

Question 8: In group of 300 students, 216 students study hindi and 129 study punjabi how many students study hindi only?

Question 9: A survey was conducted among certain people, it was found that certain number of persons like chocolate, only vanilla, both vanilla and chocolate and neither chocolate and vanilla of them are 120, 135, 54 & 66. Find the number of people surveyed.

Question 10: In a sports club of 530 members, each member play at least one of the three sports cricket, football and squash. 240 of them play cricket, 255 play football and squash. 80 play cricket and football, 85 play cricket and squash and 90 play football and squash. How many members play all three sports?

Question 11: In class of 100 students, 35 like science and 45 like maths and 10 like both. how many like either of them and how many like either?

Question 12: In class of 30 students, 18 students play football, 12 play basketball and 6 students play both. How many students play football only?

Question 13: In a survey of 150 people, 95 like apples, 70 like bananas, and 60 like both apples and bananas. Find the number of people who like only apples.

Question 14: Given sets L = {x | x is a positive integer less than 6} and M = {x | x is a positive integer and a multiple of 2}, find L ∪ M and L ∩ M.

Question 15: In a group of 50 people, 20 like reading, 30 like travelling and 15 like both reading and travelling. How many like either reading and travelling?

Answer Key:

1. P ∪ Q = {a, b, c, d, e, f, g} {P - Q = {a, b}
2. X ∩ Y = {6, 12, 18}
3. A × B and B × A each contain 2020​ ordered pairs.
4. Both languages: 250
5. Both subjects: 5​, Biology only: 20
6. (i) 35, (ii) 34
7. Both beverages: 18
8. Hindi only: 171
9. Total surveyed: 321
10. All three sports: 35
11. At least one: 70, Exactly one: 60
12. Football only: 12
13. Only apples: 35
14. L ∪ M = {1, 2, 3, 4, 5, 6, 8, 10, …}​, L ∩ M = {2,4}​
15. Either reading or traveling: 35