NCERT Solutions Class 11 Maths
The NCERT Solutions in English Language for Class 11 Mathematics Chapter – 1 Sets Exercise 1.3 has been provided here to help the students in solving the questions from this exercise.
Chapter 1 (Sets)
Chapter : 1 Sets
- NCERT Class 11 Maths Solution Ex – 1.1
- NCERT Class 11 Maths Solution Ex – 1.2
- NCERT Class 11 Maths Solution Ex – 1.4
- NCERT Class 11 Maths Solution Ex – 1.5
- NCERT Class 11 Maths Solution Ex – 1.6
| Exercise – 1.3 |
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}
2. Examine whether the following statements are true or false:
(i) {a, b} ⊄ {b, c, a}
Solution – (False) Here each element of {a, b} is an element of {b, c, a}.
(ii) {a, e} ⊂ {x: x is a vowel in the English alphabet}
Solution – (True) We know that a, e are two vowels of the English alphabet.
(iii) {1, 2, 3} ⊂ {1, 3, 5}
Solution – (False) 2 ∈ {1, 2, 3} where, 2∉ {1, 3, 5}
(iv) {a} ⊂ {a. b, c}
Solution – (True) Each element of {a} is also an element of {a, b, c}.
(v) {a} ∈ (a, b, c)
Solution – (False) Elements of {a, b, c} are a, b, c. Hence, {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 – (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}
3. Let A= {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
Solution – {3, 4} ⊂ A is incorrect. Here 3 ∈ {3, 4}, where 3 ∉ A.
(ii) {3, 4}}∈ A
Solution – {3, 4} ∈ A is correct. {3, 4} is an element of A.
(iii) {{3, 4}} ⊂ A
Solution – {{3, 4}} ⊂ A is correct. {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
(iv) 1 ∈ A
Solution – 1 ∈ A is correct. 1 is an element of A.
(v) 1 ⊂ A
Solution – 1 ⊂ A is incorrect. An element of a set can never be a subset of itself.
(vi) {1, 2, 5} ⊂ A
Solution – {1, 2, 5} ⊂ A is correct. Each element of {1, 2, 5} is also an element of A.
(vii) {1, 2, 5} ∈ A
Solution – {1, 2, 5} ∈ A is incorrect. { 1, 2, 5 } is not an element of A.
(viii) {1, 2, 3} ⊂ A
Solution – {1, 2, 3} ⊂ A is incorrect. 3 ∈ {1, 2, 3}; where, 3 ∉ A.
(ix) ∅ ∈ A
Solution – ∅ ∈ A is incorrect. ∅ is not an element of A.
(x) ∅ ⊂ A
Solution – ∅ ⊂ A is correct. ∅ is a subset of every set.
(xi) {∅} ⊂ A
Solution – {∅} ⊂ A is incorrect. {∅} is not present in A.
4. Write down all the subsets of the following sets:
(i) {a}
Solution – Subsets of {a} are ∅ and {a}.
(ii) {a, b}
Solution – Subsets of {a, b} are {a}, {b}, and {a, b}.
(iii) {1, 2, 3}
Solution – Subsets of {1, 2, 3} are ∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}.
(iv) ∅
Solution – Only subset of ∅ is ∅.
5. Write the following as intervals:
(i) {x : x ∈ R, – 4 < x ≤ 6}
Solution – {x : x ∈ R, – 4 < x ≤ 6} = (-4, 6]
(ii) {x : x ∈ R, – 12 < x < –10}
Solution – {x : x ∈ R, – 12 < x < –10} = (-12, -10)
(iii) {x : x ∈ R, 0 ≤ x < 7}
Solution – {x : x ∈ R, 0 ≤ x < 7} = [0, 7)
(iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Solution – {x : x ∈ R, 3 ≤ x ≤ 4} = [3, 4]
6. Write the following intervals in set-builder form:
(i) (–3, 0)
Solution – (–3, 0) = {x: x ∈ R, –3 < x < 0}
(ii) [6, 12]
Solution – [6, 12] = {x : x ∈ R, 6 ≤ x ≤ 12}
(iii) (6, 12]
Solution – (6, 12] = {x : x ∈ R, 6 < x ≤ 12}
(iv) [–23, 5)
Solution – [–23, 5) = {x : x ∈ R, -23 ≤ x < 5}
7. What universal set (s) would you propose for each of the following?
(i) The set of right triangles
Solution – Among the set of right triangles, the universal set is the set of triangles or the set of polygons.
(ii) The set of isosceles triangles
Solution – Among the set of isosceles triangles, the universal set is the set of triangles or the set of polygons or the set of two-dimensional figures.
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 universals set (s) for all the three sets A, B and C?
(i) {0, 1, 2, 3, 4, 5, 6}
Solution –
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) ∅
Solution – A ⊄ ∅, B ⊄ ∅, C ⊄ ∅
Hence, ∅ cannot be the universal set for the sets A, B, and C.
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Solution –
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) {1, 2, 3, 4, 5, 6, 7, 8}
Solution –
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.
