NCERT Class 11 Maths Chapter 1 Sets Exercise 1.3 Solutions

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.

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

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.

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

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)