Written by : Anmol Gupta
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NCERT Class 11 Maths Chapter 1 Sets Exercise 1.3 Solutions
Table of Contents
Solved Examples of Exercise 1.3
Example 9. Consider the sets φ, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ⊂ or ⊄ between each of the following pair of sets:
(i) φ . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C
Answer 9. (i) φ ⊂ B as φ is a subset of every set.
(ii) A ⊄ B as 3 ∈ A and 3 ∉ B
(iii) A ⊂ C as 1, 3 ∈ A also belongs to C
(iv) B ⊂ C as each element of B is also an element of C.
Example 10. Let A = {a, e, i, o, u} and B = {a, b, c, d}. Is A a subset of B ? Is B a subset of A?
Answer 10. A ⊄ B as e, i, o, u ∈ A but e, i, o, u ∉ B
B ⊄ A as b, c, d ∈ B but b, c, d ∉ A
Example 11. Let A, B and C be three sets. If A ∈ B and B ⊂ C, is it true that A ⊂ C?. If not, give an example
Answer 11. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ∈ B as A = {1} and B ⊂ C.
But A ⊄ C as 1 ∈ A and 1 ∉ C.
Exercise 1.3 solved questions
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}
Answer 1.
(i) { 2, 3, 4 } ⊂ { 1, 2, 3, 4, 5 } as every element of set { 2, 3, 4 } is in set { 1, 2, 3, 4, 5 }
(ii) { a, b, c } ⊄ { b, c, d } as a ∈ { a, b, c } but a ∉ {b, c, d}
(iii) {x : x is a student of Class XI of your school} ⊂ {x : x student of your school} as every student of Class XI of your school is a 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} as every circle in the plane is not a circle with radius 1 unit, the radius can also be between 0 and 1 and more than 1
(v) {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane} as triangle is not a rectangle
(vi) {x : x is an equilateral triangle in a plane} ⊂ {x : x is a triangle in the same plane} as every equilateral triangle in the plane is a triangle in the same plane
(viii) {x : x is an even natural number} ⊂ {x : x is an integer} as every even natural number is an integer
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}
Answer 2.
(i) False as every element of { a, b } is in { b, c, a }
(ii) True as elements of set {a, e} are in elements of set {a, e, i, o, u}
(iii) False as element 2 is not in set {1, 3, 5}
2 ∈ {1, 2, 3} but 2 ∉ {1, 3, 5}
(iv) True as element a is in set {a, b, c}
a ∈ { a, b, c }
(v) False as set {a} does not belong or is not in set {a, b, c}
{a} ∉ {a, b, c}
(vi) True
Even natural numbers less than 6 are 2, 4
2, 3, 4, 9, 12, 18, 36 are natural numbers that can divide 36
{2, 4} ⊂ {2, 3, 4, 9, 12, 18, 36}
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) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A
Answer 3.
(i) False as 3 ∉ A, 4 ∉ A
(ii) True as set {3, 4} is in set A
(iii) True as {3, 4} ∈ A
(iv) True as element 1 is in A
(v) False as 1 is an element not a set, only a set can be a subset of another set
(vi) True as 1 ∈ A, 2 ∈ A, 5 ∈ A
(vii) False as set {1, 2, 5} is not an element of set A
(viii) False as element 3 ∉ A
(ix) False as φ does not belongs to set A
(x) True as φ is subset of every set
(xi) False as φ does not belongs to set A
Question 4. Write down all the subsets of the following sets
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
Answer 4. (i) {a}, φ
(ii) {a}, {b}, {a, b} φ
(iii) {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1, 2, 3}, φ
(iv) φ
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}
Answer 5. (i) (-4,6]
(ii) (-12, -10)
(iii) [0, 7)
(iv) [3,4]
Question 6. Write the following intervals in set-builder form :
(i) (– 3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [–23, 5)
Answer 6.
(i) {x : x ∈ R, – 3 < x < 0}
(ii) {x : x ∈ R, 6 ≤ x ≤ 12}
(iii) {x : x ∈ R, 6 < x ≤ 12}
(iv) {x : x ∈ R, – 23 ≤ x < 5}
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
Answer 7. The set of all triangles for both (i) and (ii)
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) φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}
Answer 8. (iii)