NCERT Solutions for Sets Exercise 1.3 Class 11 Maths focus on the concept of subsets, proper subsets, and the number of subsets of a set. This exercise is important for understanding relationships between sets and forms the base for set operations and advanced topics. It is highly useful for CBSE exams as well as competitive exams of India.
Chapter 1 SETS Exercise 1.3 NCERT Solutions for Class 11 Maths
NCERT Question 1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:
(i) {2, 3, 4} . . . {1, 2, 3, 4, 5}
(ii) {a, b, c} . . . {b, c, d}
(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}
(iv) {x: x is a circle in the plane} . . .{x: x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}
Solution:
(i) {2, 3, 4} ⊂ {1, 2, 3, 4,5}
(ii) {a, b, c} ⊄ {b, c, d}
(iii) {x : x is a student of Class XI of your school} ⊂ {x : x student of your school}
(iv) {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} ⊂ {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} ⊂ {x : x is an integer}
NCERT Question 2. Examine whether the following statements are true or false:
(i) {a, b} ⊄ {b, c, a}
(ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}
(iii) {1, 2, 3} ⊂ {1, 3, 5}
(iv) {a} ⊂ {a, b, c}
(v) {a} ∈ {a, b, c}
(vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}
Solution:
(i) False. Each element of {a, b} is an element of {b, c, a}.
(ii) True. Since a, e are two vowels of the English alphabet.
(iii) False. 2 is subset of {1, 2, 3} but not subset of {1, 3, 5}
(iv) True. Each element of {a} is also an element of {a, b. c} .
(v) False. Elements of {a, b, c} are a, b, c. Hence, {a} ⊂ {a, b, c}
(vi) True
{x : x is an even natural number less than 6} = {2, 4}
{x: x is a natural number which divides 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Hence, {2, 4} ⊂ {1, 2, 3, 4, 6, 9, 12, 18, 36}
NCERT Question 3. Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
(ii) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) 1 ⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) ∅ ∈ A
(x) ∅ ⊂ A
(xi) {∅} ⊂ A
Solution:
Given A= {1, 2, {3, 4}, 5}
(i) {3, 4} ⊂ A is incorrect. Here 3 ∈ {3, 4}, where 3 ∉ A.
(ii) {3, 4} ∈ A is correct. {3, 4} is an element of A.
(iii) {{3, 4}} ⊂ A is correct. {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
(iv) 1 ∈ A is correct. 1 is an element of A.
(v) 1 ⊂ A is incorrect. An element of a set can never be a subset of itself.
(vi) {1, 2, 5} ⊂ A is correct. Each element of {1, 2, 5} is also an element of A.
(vii) {1, 2, 5} ∈ A is incorrect. { 1, 2, 5 } is not an element of A.
(viii) {1, 2, 3} ⊂ A is incorrect. 3 ∈ {1, 2, 3}; where, 3 ∉ A.
(ix) ∅ ∈ A is incorrect. ∅ is not an element of A.
(x) ∅ ⊂ A is correct. ∅ is a subset of every set.
(xi) {∅} ⊂ A is incorrect. {∅} is not present in A.
NCERT Question 4. Write down all the subsets of the following sets
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) ∅
Solution:
(i) Subsets of {a} are ∅ and {a}.
(ii) Subsets of {a, b} are {a}, {b}, and {a, b}.
(iii) Subsets of {1, 2, 3} are ∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}.
(iv) Only subset of ∅ is ∅.
NCERT Question 5. Write the following as intervals:
(i) {x : x ∈ R, – 4 < x ≤ 6} (ii) {x : x ∈ R, – 12 < x < –10}
(iii) {x : x ∈ R, 0 ≤ x < 7} (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Solution:
(i) {x : x ∈ R, – 4 < x ≤ 6} = (-4, 6]
(ii) {x : x ∈ R, – 12 < x < –10} = (-12, -10)
(iii) {x : x ∈ R, 0 ≤ x < 7} = [0, 7)
(iv) {x : x ∈ R, 3 ≤ x ≤ 4} = [3, 4]
NCERT Question 6. Write the following intervals in set-builder form :
(i) (– 3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [–23, 5)
Solution:
(i) (– 3, 0) = {x : x ∈ R, -3 < x < 0}
(ii) [6, 12] = {x : x ∈ R, 6 ≤ x ≤ 12}
(iii) (6, 12] = {x : x ∈ R, 6 < x ≤ 12}
(iv) [–23, 5) = {x : x ∈ R, -23 ≤ x < 5}
NCERT Question 7. What universal set(s) would you propose for each of the following :
(i) The set of right triangles
(ii) The set of isosceles triangles.
Solution:
(i) The universal set for the set of right triangles is the set of triangles or the set of polygons.
(ii) The universal set for the set of isosceles triangles is the set of triangles or the set of polygons or the set of two-dimensional figures.
NCERT Question 8. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) ∅
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
Solution:
(i) A ⊂ {0, 1, 2, 3, 4, 5, 6}
B ⊂ {0, 1, 2, 3, 4, 5, 6}
But, C ⊄ {0, 1, 2, 3, 4, 5, 6}
Hence, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.
(ii) A ⊄ ∅, B ⊄ ∅, C ⊄ ∅
Hence, ∅ cannot be the universal set for the sets A, B, and C.
(iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Hence, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B, and C.
(iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
But, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}
Hence, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.
NCERT Question 9. How many elements has P(A), if A = ∅?
Solution:
For a set A with n(A) = m, then it can be shown that
Number of elements of P(A) = n[P(A)] = 2m
If A = ∅, we get n (A) = 0
So, n[P(A)] = 2° = 1
Therefore, P(A) has one element.
Important Chapter Links
Sets Exercise 1.3 of Class 11 NCERT Mathematics covers important concepts related to subsets and proper subsets. In this exercise, students learn how to determine whether a set is a subset of another, count the total number of subsets, and understand properties of subsets. The formula for the number of subsets plays a key role in solving problems efficiently. This exercise builds a strong base for set operations and Venn diagrams and is highly useful for CBSE exams and competitive exams of India.
FAQs of Chapter-1 Sets Exercise 1.3 NCERT Solutions for Class 11 Maths
What is Exercise 1.3 about?
It focuses on subsets, proper subsets, and counting the number of subsets.
What is a subset?
A set $A$ is a subset of $B$ if every element of $A$ is in $B$.
What is a proper subset?
A proper subset is a subset that is not equal to the original set.
What is the formula for number of subsets?
2n
What is the number of proper subsets?
2n−1
Is the empty set a subset of every set?
Yes, the empty set is a subset of every set.
Why is Exercise 1.3 important?
It is essential for understanding set relationships and solving advanced problems.
Are NCERT solutions enough for preparation?
Yes, they are sufficient for CBSE exams and helpful for competitive exams.
Important Chapter Links
Sets Exercise 1.3 of Class 11 NCERT Mathematics covers important concepts related to subsets and proper subsets. In this exercise, students learn how to determine whether a set is a subset of another, count the total number of subsets, and understand properties of subsets. The formula for the number of subsets plays a key role in solving problems efficiently. This exercise builds a strong base for set operations and Venn diagrams and is highly useful for CBSE exams and competitive exams like JEE, NDA, IMUCET, and Merchant Navy entrance exams.
Exercise-wise NCERT Solutions
- Basic definition of sets
- Writing sets in roster and set-builder form
- Types of sets
- Finite and infinite sets
Exercise 1.3
- Subsets and proper subsets
- Number of subsets
- Set operations (union, intersection, complement)
- Advanced problems on set operations and Venn diagrams
- Mixed problems covering all concepts of the chapter