# 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) { 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}**

**(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}**

**Solution: ****(i) False.**

Here each element of {*a*, *b*} is an element of {*b*, *c*, *a*}.

**(ii) True.**

We know that *a*, *e* 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 {*a*, *b*, *c*}.

**(v) False.**

Elements of {*a*, *b*, *c*} are *a*, *b*, *c*. Hence, {*a*} ⊂ {*a*, *b*, *c*}

**(vi) 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 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) { a, b}**

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

**(iv) Φ**

**Solution: (i) **Subsets of {*a*} are

Φ, {*a*}.

(ii) Subsets of {*a*, *b*} are

Φ, {*a*}, {*b*}, {*a*, *b*}.

(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

*n *(A) = *m* then *n *[P (A)] = 2^{m}

If A = Φ we get* n *(A) = 0

*n *[P(A)] = 2^{0} = 1

Therefore, P (A) has one element.

**6. 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}**

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

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

**(iii)** {*x*: *x *∈ R, 0 ≤ *x* < 7} = [0, 7)

**(iv) **{*x*: *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*: *x *∈ R, –3 < *x* < 0}

**(ii) **[6, 12] = {*x*: *x *∈ R, 6 ≤ *x* ≤ 12}

**(iii) **(6, 12] ={*x*: *x *∈ R, 6 < *x* ≤ 12}

**(iv)** [–23, 5) = {*x*: *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.