# Set Theory Questions

Last Updated : 17 Jan, 2024

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.

## Set 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 âˆ© B) = 14.
n(H âˆª B) = n(H) + n(B) – n(H âˆ© B) = 22 + 26 – 14 = 34.
Students not playing either = Total students – n(H âˆª B) = 40 – 34 = 6.

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

Solution:

A âˆª B (union) is the set of elements that are in A, or B, or both.
A âˆª B = {1, 2, 3, 4, 5, 7, 9}.
A âˆ© B (intersection) is the set of elements that are in both A and B.
A âˆ© 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 â‰¤ x â‰¤ 6} and B = {x | x is an even integer, 2 â‰¤ x â‰¤ 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 Î” Y).

Solution:

X Î” Y is the set of elements in either X or Y, but not in their intersection.
Intersection X âˆ© Y = {b, d}.
X Î” Y = (X âˆª Y) – (X âˆ© 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 âˆ© D and C âˆª D?

Solution:

C âˆ© D (intersection) is the set of elements common to both C and D.
C âˆ© D = {6, 8}.
C âˆª D (union) is the set of all elements in C, or D, or both.
C âˆª 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 âˆª B) = n(P) + n(B) – n(P âˆ© B) = 70 + 75 – 60 = 85.
Students who like neither = Total students – n(P âˆª 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 âˆª F and the set E – F.

Solution:

E âˆª F (union) is the set of elements in E, or F, or both.
E âˆª 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 Î” H.

Solution:

G Î” H is the set of elements in either G or H, but not in their intersection.
Intersection G âˆ© H = {a, e}.
G Î” H = (G âˆª H) – (G âˆ© 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.

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 Ã— K.

Solution:

J Ã— K is the set of all ordered pairs (j, k) where j is in J and k is in K.
J Ã— 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 âˆ© N?

Solution:

M = {2, 3, 5, 7, 11, 13, 17, 19}, N = {1, 3, 5, 7, 9}.
M âˆ© N (intersection) is the set of elements that are in both M and N.
M âˆ© 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 âˆª B) = n(A) + n(B) – n(A âˆ© B).
n(A âˆª 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 âˆª Q and P – Q.

Solution:

P âˆª Q (union) is the set of elements in P, or Q, or both.
P âˆª 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 âˆ© Y.

Solution:

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

Solution:

A Ã— 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 Ã— 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 âˆª M and L âˆ© M.

Solution:

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

## Conclusion

To sum up, set theory is an important chapter for the students of Class 11. Through this article, we’ve explored a variety of set questions, offering clear solutions that follow the NCERT and CBSE guidelines. Practising these sets questions will help you develop a good foundation to solve more complex problems further.