NCERT Solution for Class 11 Maths Chapter 1 - Sets

NCERT Solutions for Class 11 Maths Chapter 1 breaks down the essential topic of Sets. A set is simply a collection of well-defined items. For example, the group of all seven days in a week is a set, and so is a collection of all vowels in the English alphabet.

Understanding sets is crucial because it forms the foundation for many other areas in mathematics. This knowledge is key to learning about relations and functions. It also plays a vital role in topics like geometry, sequences, and probability. By studying sets, you'll gain skills that are useful in real-world scenarios and other academic subjects.

NCERT Solutions for Class 11 Maths Chapter 1 Sets - Exercise-wise

NCERT Solutions for Class 11 Maths Chapter 1 Sets - Download PDF

Topics and Subtopics in Class 11 Maths NCERT Solutions Chapter 1 Sets download from below:

Class 11 Maths NCERT Solutions Chapter 1 Sets Ex 1.1

Question 1 Write the following set in roster form:
A = {x : x is an integer and -3 < x < 3}

Solution:
We are given that -3 < x < 3, so x can take values: -2, -1, 0, 1, 2.
So, in roster form:
A = {-2, -1, 0, 1, 2}

Question 2 Write the set {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} in set-builder form.

Solution:
Let the set be:
A = {x : x is a day of the week}

Question 3 State whether the following set is finite or infinite:
A = {x : x is a natural number less than 20}

Solution:
The set contains natural numbers less than 20 i.e., {1, 2, 3, ..., 19}.
Since the number of elements is countable and limited,
A is a finite set.

Question 4 List all the elements of the set:
A = {x : x is a letter in the word 'MATHEMATICS'}

Solution:
The distinct letters are:
A = {M, A, T, H, E, I, C, S}

Question 5 Which of the following is an empty set?
(a) {x : x is a natural number less than 1}
(b) {x : x² = 9 and x is even}

Solution:
(a) Natural numbers start from 1. So, no natural number is less than 1. Hence, this is an empty set.
(b) x² = 9 gives x = ±3. Since 3 is odd, no even number satisfies the condition. So, this is also an empty set.

Both (a) and (b) are empty sets.

Question 6 Express the following as a set using the roster form:
Set of vowels in the English alphabet.

Solution:
The vowels are:
A = {a, e, i, o, u}

Question 7 Write the power set of A, where A = {1, 2}.

Solution:
The power set of A is the set of all subsets:
P(A) = { {}, {1}, {2}, {1, 2} }

Question 8 Determine the cardinality of the set:
A = {a, e, i, o, u}

Solution:
The number of elements (cardinality) in A is 5.
n(A) = 5

Question 9 State whether the set {x : x² + 1 = 0, x ∈ R} is empty or not.

Solution:
x² + 1 = 0 ⇒ x² = -1 ⇒ No real solution exists.
Thus, the set is an empty set.

Question 10 Which of the following sets are equal?
A = {1, 2, 3}, B = {3, 2, 1}

Solution:
Since both sets have exactly the same elements,
A = B

Class 11 Maths NCERT Solutions Chapter 1 Sets Ex 1.2

Question 1: If A = {1, 2, 3}, write all the subsets of A.

Solution:
Total number of subsets = 2³ = 8 subsets.

Subsets of A are:

• {} (empty set)
• {1}
• {2}
• {3}
• {1, 2}
• {1, 3}
• {2, 3}
• {1, 2, 3}

Question 2: If B = {x : x is a natural number less than 4}, list B in roster form.

Solution:
B = {1, 2, 3}

Question 3: If X = {a, b}, find the power set of X.

Solution:
Power set of X = set of all subsets of X.

P(X) = { {}, {a}, {b}, {a, b} }

Question 4: State whether the following statement is true or false:
The empty set is a subset of every set.

Solution:
True.
The empty set {} is a subset of every set because it contains no element that can contradict membership.

Question 5: If U = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, B = {2, 3, 4}, find A ∪ B and A ∩ B.

Solution:
A ∪ B = {1, 2, 3, 4, 5}
A ∩ B = {3}

Question 6: If A = {2, 4, 6, 8}, is 5 ∈ A?

Solution:
No, 5 ∉ A (5 is not an element of set A).

Question 7: How many elements are there in the power set of set C = {p, q, r, s}?

Solution:
Number of subsets = 2ⁿ
Here, n = 4

Number of subsets = 2⁴ = 16

So, power set has 16 elements.

Question 8: If A = {2, 3}, B = {3, 4, 5}, verify whether A ⊆ B.

Solution:

• A = {2, 3}
• B = {3, 4, 5}

Since 2 ∉ B, A is not a subset of B.

Question 9: Write the set {x : x is an even natural number less than 10} in roster form.

Solution:
{2, 4, 6, 8}

Question 10: Which of the following sets are finite?
(a) Set of all even numbers
(b) Set of all natural numbers divisible by 5 less than 50

Solution:
(a) Infinite set

Class 11 Maths NCERT Solutions Chapter 1 Sets Ex 1.3 - Venn Diagrams

Question 1 In a group of students:

• 10 like Maths (Set A)
• 8 like Science (Set B)
• 4 like both Maths and Science

How many students like either Maths or Science?

Solution:

We apply the formula: n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Given: n(A) = 10, n(B) = 8, n(A ∩ B) = 4

n(A ∪ B) = 10 + 8 – 4 = 14

Answer: 14 students like either Maths or Science.

Question 2 Draw a Venn Diagram for two sets A and B where:

• A = {1, 2, 3}
• B = {3, 4, 5}

Solution:

Draw two overlapping circles:

• Circle A contains 1, 2, 3.
• Circle B contains 3, 4, 5.
• The number 3 is written in the overlapping part (intersection).

Answer: Venn Diagram drawn with 3 in intersection.

Question 3 If A = {2, 4, 6}, B = {4, 6, 8}, find A ∩ B.

Solution:

Intersection means common elements.

A ∩ B = {4, 6}

Answer: A ∩ B = {4, 6}

Question 4 If A = {a, b, c, d} and B = {c, d, e, f}, find A ∪ B.

Solution:

Union means all elements from both sets without repeating.

A ∪ B = {a, b, c, d, e, f}

Answer: A ∪ B = {a, b, c, d, e, f}

Question 5 If n(U) = 50, n(A) = 20, n(B) = 15, n(A ∩ B) = 5, find n(A ∪ B).

Solution:

We use: n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

n(A ∪ B) = 20 + 15 – 5 = 30

Answer: n(A ∪ B) = 30

Class 11 Maths NCERT Solutions Chapter 1 Sets Ex 1.4 

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

Answer:
A ∪ B means all elements of A and B (no repeat).
A ∪ B = {1, 2, 3, 4, 5}

Question 2: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∩ B.

Answer:
A ∩ B means common elements of A and B.
A ∩ B = {3, 4}

Question 3: If A = {2, 4, 6, 8} and B = {1, 2, 3}, find A – B.

Answer:
A – B means elements in A but not in B.
A – B = {4, 6, 8}

Question 4: If U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}, find A'.

Answer:
A' means complement of A (all elements in U but not in A).
A' = {1, 3, 5}

Question 5: If A = {a, b, c} and B = {b, c, d}, find A ∪ B and A ∩ B.

Answer:
A ∪ B = {a, b, c, d}
A ∩ B = {b, c}

Question 6: If A = {x: x is a natural number less than 5} and B = {x: x is an even number less than 7}, find A ∩ B.

Answer:
A = {1, 2, 3, 4}
B = {2, 4, 6}
A ∩ B = {2, 4}

Question 7:

If U = {1, 2, 3, 4, 5, 6, 7, 8} and A = {2, 4, 6, 8}, find A'.

Answer:
A' = {1, 3, 5, 7}

Question 8:

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find B – A.

Answer:
B – A = {5, 6}

Question 9:

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

Answer:
A ∪ B = {1, 2, 3, 4, 5, 7}
A ∩ B = {3, 5}

Question 10:

If A = {m, n, o} and B = {n, o, p}, find A – B.

Answer:
A – B = {m}

Class 11 Maths NCERT Solutions Chapter 1 Sets Ex 1.5

Question 1: If A = {1, 2, 3}, B = {3, 4, 5}, find A ∪ B.

Solution: A ∪ B means union of A and B.
A ∪ B = {1, 2, 3, 4, 5}

Question 2: If A = {1, 2, 3}, B = {3, 4, 5}, find A ∩ B.

Solution: A ∩ B means common elements of A and B.
A ∩ B = {3}

Question 3: If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, find A - B.

Solution: A - B means elements in A but not in B.
A - B = {1, 2}

Question 4: If U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, find A'.

Solution: A' means complement of A = elements in U not in A.
A' = {4, 5, 6}

Question 5: Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) if A = {1}, B = {1, 2}, C = {1, 3}.

Solution:
B ∩ C = {1}
A ∪ (B ∩ C) = {1}
A ∪ B = {1, 2}
A ∪ C = {1, 3}
(A ∪ B) ∩ (A ∪ C) = {1}
Both sides are equal. Verified.

Question 6: If A = {2, 4, 6}, B = {4, 6, 8}, find A ∩ B.

Solution: A ∩ B = {4, 6}

Question 7: If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, U = {1, 2, 3, 4, 5, 6}, find (A ∩ B)'.

Solution:
A ∩ B = {3, 4}
(A ∩ B)' = U - {3, 4} = {1, 2, 5, 6}

Question 8: If A = {a, b, c}, B = {b, c, d}, find A ∪ B and A ∩ B.

Solution:
A ∪ B = {a, b, c, d}
A ∩ B = {b, c}

Question 9: If U = {1, 2, 3, 4, 5}, A = {1, 2, 3}, B = {3, 4}, verify De Morgan’s law: (A ∪ B)' = A' ∩ B'.

Solution:
A ∪ B = {1, 2, 3, 4}
(A ∪ B)' = {5}
A' = {4, 5}
B' = {1, 2, 5}
A' ∩ B' = {5}
Both sides equal. Verified.

Question 10: If A = {1, 2, 3}, B = {3, 4, 5}, U = {1, 2, 3, 4, 5}, find (A - B)'.

Solution:
A - B = {1, 2}
(A - B)' = U - {1, 2} = {3, 4, 5}

Class 11 Maths NCERT Solutions Chapter 1 Sets Ex 1.6

Question 1: In a class of 50 students, 30 like Mathematics and 25 like Physics. 10 students like both Mathematics and Physics. How many students like either Mathematics or Physics?

Solution:

We use: 
n(M ∪ P) = n(M) + n(P) – n(M ∩ P) 
n(M ∪ P) = 30 + 25 – 10 = 45 
So, 45 students like either Mathematics or Physics.

Question 2: Out of 40 students, 22 play cricket, 18 play football, and 10 play both games. How many students play at least one game?

Solution:

n(C ∪ F) = n(C) + n(F) – n(C ∩ F) 
n(C ∪ F) = 22 + 18 – 10 = 30 
So, 30 students play at least one game.

Question 3: In a group of 80 people, 50 like tea, 30 like coffee, and 20 like both. How many people like only tea?

Solution:

Only tea = n(Tea) – n(Both) 
Only tea = 50 – 20 = 30 
30 people like only tea.

Question 4: In a survey, 60 people were asked if they like apples or oranges. 40 like apples, 35 like oranges, and 20 like both. How many like neither apples nor oranges?

Solution:

n(A ∪ O) = 40 + 35 – 20 = 55 
Neither = Total – n(A ∪ O) = 60 – 55 = 5 
5 people like neither apples nor oranges.

Question 5: In a college, 70 students study English, 50 study Hindi, and 30 study both. How many study either English or Hindi?

Solution:

n(E ∪ H) = 70 + 50 – 30 = 90 
90 students study either English or Hindi.

Question 6: In a group of 100 students, 60 read newspapers, 40 read magazines, and 20 read both. How many students read only newspapers?

Solution:

Only newspapers = n(Newspapers) – n(Both) 
Only newspapers = 60 – 20 = 40 
40 students read only newspapers.

Question 7: In a park, 80 people walk, 50 jog, and 30 do both. How many people either walk or jog?

Solution:

n(W ∪ J) = 80 + 50 – 30 = 100 
100 people either walk or jog.

Question 8: In a survey, 120 people were asked about two movies. 70 liked Movie A, 80 liked Movie B, and 40 liked both. How many liked at least one movie?

Solution:

n(A ∪ B) = 70 + 80 – 40 = 110 
110 people liked at least one movie.

Question 9: In a class of 60 students, 40 like Science, 30 like Maths, and 20 like both. How many like neither subject?

Solution:

n(S ∪ M) = 40 + 30 – 20 = 50 
Neither = Total – n(S ∪ M) = 60 – 50 = 10 
10 students like neither subject.

Question 10: Out of 90 students, 55 play basketball, 45 play volleyball, and 20 play both. How many students play only volleyball?

Solution:

Only volleyball = n(Volleyball) – n(Both) 
Only volleyball = 45 – 20 = 25 
25 students play only volleyball.