EXERCISE 1.3

1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:(Ex 1.3 sets ncert maths solution class 11)

(i) {2, 3, 4} … {1, 2, 3, 4, 5}

(ii) {abc} … {bcd}

(iii) {xx is a student of Class XI of your school} … {xx student of your school}

(iv) {xx is a circle in the plane} … {xx is a circle in the same plane with radius 1 unit}

(v) {xx is a triangle in a plane}…{xx is a rectangle in the plane}

(vi) {xx is an equilateral triangle in a plane}… {xx is a triangle in the same plane}

(vii) {xx is an even natural number} … {xx is an integer}

Solution:  (i) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}

(ii) {abc} ⊄ {bcd}

(iii) {xx is a student of Class XI of your school} ⊂ {xx student of your school}

(iv) {xx is a circle in the plane} ⊄ {xx is a circle in the same plane with radius 1 unit}

(v) {xx is a triangle in a plane} ⊄ {xx is a rectangle in the plane}

(vi) {xx is an equilateral triangle in a plane} ⊂ {xx is a triangle in the same plane}

(vii) {xx is an even natural number} ⊂ {xx is an integer}

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

(i) {ab} ⊄ {bca}

(ii) {ae} ⊂ {xx is a vowel in the English alphabet}

(iii) {1, 2, 3} ⊂ {1, 3, 5}

(iv) {a} ⊂ {abc}

(v) {a} ∈ (abc)

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

Solution:  (i) False.

Here each element of {ab} is an element of {bca}.

(ii) True.

We know that ae are two vowels of the English alphabet.

(iii) False.

2 ∈ {1, 2, 3} where, 2∉ {1, 3, 5}

(iv) True.

Each element of {a} is also an element of {abc}.

(v) False.

Elements of {abc} are abc. Hence, {a} ⊂ {abc}

(vi) True.

{xx is an even natural number less than 6} = {2, 4}

{xx 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 is 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

Solution:  It is given that 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) Φ ∈ A is incorrect

Φ is not an element of A.

(x) Φ ⊂ A is correct

Φ is a subset of every set.

(xi) {Φ} ⊂ A is incorrect

Φ∈ {Φ}; where, Φ ∈ A.

4. Write down all the subsets of the following sets:

(i) {a}

(ii) {ab}

(iii) {1, 2, 3}

(iv) Φ

Solution: (i) Subsets of {a} are

Φ, {a}.

(ii) Subsets of {ab} are

Φ, {a}, {b}, {ab}.

(iii) Subsets of {1, 2, 3} are

Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}.

(iv) Only subset of Φ is Φ.

5. How many elements has P (A), if A = Φ?

Solution: If A is a set with m elements

(A) = m then [P (A)] = 2m

If A = Φ we get n (A) = 0

[P(A)] = 20 = 1

Therefore, P (A) has one element.

6. Write the following as intervals:

(i) {xx ∈ R, –4 < x ≤ 6}

(ii) {xx ∈ R, –12 < x < –10}

(iii) {xx ∈ R, 0 ≤ x < 7}

(iv) {xx ∈ R, 3 ≤ x ≤ 4}

Solution: (i) {x∈ R, –4 < x ≤ 6} = (–4, 6]

(ii) {x∈ R, –12 < x < –10} = (–12, –10)

(iii) {x∈ R, 0 ≤ x < 7} = [0, 7)

(iv) {x∈ R, 3 ≤ x ≤ 4} = [3, 4]

7. 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∈ R, –3 < x < 0}

(ii) [6, 12] = {x∈ R, 6 ≤ x ≤ 12}

(iii) (6, 12] ={x∈ R, 6 < x ≤ 12}

(iv) [–23, 5) = {x∈ R, –23 ≤ x < 5}

8. 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) Among the set of right triangles,

The universal set is the set of all triangles or the set of polygons.

(ii) Among the set of isosceles triangles,

The universal set is the set of all triangles or the set of polygons or the set of two-dimensional figures.

9. 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) 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.

Ex 1.2 sets ncert maths solution class 11