**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}

**Solution.**

**(i)** {2, 3, 4} ⊂ {11, 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}

**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} ∈ la, b, c}**(vi)**{x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}

**Solution.**

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

**Solution.**

**Question 4.** **Write down all the subsets of the following sets**

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

**Solution.**

**(i)** Number of elements in given set = 1

Number of subsets of given set = 2^{1} = 2

∴ Subsets of given set are φ , {a}.

**(ii)** Number of elements in given set = 2

Number of subsets of given set = 2^{12} = 4

∴ Subsets of given set are φ, {a}, {b}, {a, b}.

**(iii)** Number of elements in given set = 3

Number of subsets of given set = 2^{3} = 8

Subsets of given set are φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}.

**(iv)** Number of elements in given set = 0

Number of subsets of given set = 2^{0}= 1

∴ Subset of given set is φ.

**Question 5. How many elements has P(A), if A = φ?**

**Solution.**

Number of elements in set A = 0

Number of subset of set A = 2^{0} = 1

Hence, number of elements of P(A) is 1.

**Question 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)**Let A = {x: x ∈ R, -4 < x ≤ 6}

It can be written in the form of interval as (-4, 6)

**(ii)** Let A= {x: x ∈ R, -12 < x < -10}

It can be written in the form of interval as (-12, -10)

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

It can be written in the form of interval as (0, 7).

**(iv)** Let A = {x: x ∈ R, 3 ≤ x ≤ 4}

It can be written in the form of interval as (3,4).

**Question 7.** **Write the following intervals in set-builder form:**

**(i)**(-3,0)**(ii)**[6, 12]**(iii)**(6, 12]**(iv)**[-23, 5)

**Solution.**

**(i)** The interval (-3, 0) can be written in set-builder form as {x : x ∈ R,-3 < x < 0}.

**(ii)** The interval [6, 12] can be written in set-builder form as {x : x ∈ R, 6 ≤ x ≤ 12}.

**(iii)** The interval (6, 12] can be written in set-builder form as {x : x ∈ R, 6 < x ≤ 12}

**(iv)** The interval [-23,5) can be written in set-builder form as {x : x ∈ R, -23 ≤ x < 5}

**Question 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)** Right triangle is a type of triangle. So the set of triangles contain all types of triangles.

∴ U = {x : x is a triangle in a plane}

**(ii)** Isosceles triangle is a type of triangle. So the set of triangles contain all types of triangles.

∴ U = }x : x is a triangle in a plane}

**Question 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.**

**(i)** {0, 1, 2, 3, 4, 5, 6} is not a universal set for A, B, C because 8 ∈ C but 8 is not a member of {0, 1, 2, 3, 4, 5, 6}.

**(ii)** φ is a set which contains no element. So it is not a universal set for A, B, C.

**(iii)** {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is a universal set for A, B, C because all members of A, B, C are present in {0,1 , 2, 3, 4, 5, 6, 7, 8, 9, 10).

**(iv)** (1, 2, 3, 4, 5, 6, 7, 8) is not a universal set for A, B, C because 0 ∈ C but 0 is not a member of {1, 2, 3, 4, 5, 6, 7, 8)