Sets Class 11 Exercise 1.3 Solutions

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 }

Solution. (i) ⊂    (ii) ⊄

(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}

Solution.  ⊂ 

(iv) {x : x is a circle in the plane} . . .{x : x is a circle in the same plane with radius 1 unit}

Solution. ⊄ 

(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}

Solution. ⊄ 

(vi) {x : x is an equilateral triangle in a plane} . . . {x : x is a triangle in the same plane}

Solution. ⊂ 

(vii) {x : x is an even natural number} . . . {x : x is an integer}

Solution. ⊂ 

2. Examine whether the following statements are true or false:

(i) { a, b } ⊄ { b, c, a }

True

(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}

True

(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }

False 

(iv) { a } ⊂ { a, b, c }

True 

(v) { a } ∈ { a, b, c }

True 

(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36}

True

3. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

(i) {3, 4} ⊂ A

Solution. Incorrect, since {3, 4 }∈A

 (ii) {3, 4} ∈ A 

Solution. Correct 

(iii) {{3, 4}} ⊂ A

Solution. correct 

(iv) 1 ∈ A 

Solution. correct 

(v) 1 ⊂ A 

Solution. Incorrect since 1∈A (i.e.,1 is an element of A) and {1}⊂ A .

(vi) {1, 2, 5} ⊂ A

Solution. correct 

(vii) {1, 2, 5} ∈ A

Solution. Incorrect since {1,2,3} is not an element of A but 1, 2 and 5∈ A.

 (viii) {1, 2, 3} ⊂ A

Solution. Incorrect because 3 is not the element of A

 (ix) φ ∈ A

Solution. Incorrect since φ is the subset of all set not an element of any set.

(x) φ ⊂ A 

Solution. Correct  

(xi) {φ} ⊂ A

Solution. Incorrect since φ is a set on its own so φ ⊂ A.

4. Write down all the subsets of the following sets

(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ

Solution.

(i) φ,  {a} 

(ii) φ, {a}, {b}, {a,b}

(iii) φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 

(number of subsets=2ⁿ)

(iv) φ

5. Write the following as intervals :

(i) {x : x ∈ R, – 4 < x ≤ 6} 

Solution. x∈(4,6]

(ii) {x : x ∈ R, – 12 < x < –10}

Solution. x∈ (-12,-10)

(iii) {x : x ∈ R, 0 ≤ x < 7} 

Solution. x∈ [0,7)

(iv) {x : x ∈ R, 3 ≤ x ≤ 4}

Solution. x∈ [3, 4]

6. Write the following intervals in set-builder form :

(i) (– 3, 0) 

Solution. {x : x ∈ R, – 3 < x < 0} 

(ii) [6 , 12] 

Solution. {x : x ∈ R,  6 ≤ x ≤ 12} 

(iii) (6, 12] 

Solution. {x : x ∈ R,  6 < x ≤ 12} 

(iv) [–23, 5)

Solution. {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.  (ii) The set of isosceles triangles

Solution. The universal set for these two sets is the set of triangles

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}

Solution. (iii) is the universal set for the sets A, B and C.