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}

2. Examine whether the following statements are true or false:
(i) { a, b } ⊄ { b, c, a } False
(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 } False
(vi) { x : x is an even natural number less than 6} ⊂ { x : x is a natural number which divides 36}
{2,4}⊂{1,2,3,4,6,9,12,18,36} True

3. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A   False   {3,4}∈A, 3∈{3,4} but 3∉A 

(ii) {3, 4} ∈ A   True   {3,4} is an element of A 

(iii) {{3, 4}} ⊂ A  True  {3,4}∈{{3,4}} and {3,4}∈A 

(iv) 1 ∈ A   True  1 is an element of A 

(v) 1 ⊂ A    False  1∈A and an element can never be s subset of itself 

(vi) {1, 2, 5} ⊂ A  True  elements of {1,2,5} is also an element of A 

(vii) {1, 2, 5}∈ A  False  {1,2,5} is not an element of A 

(viii) {1, 2, 3} ⊂ A   False    3∉A 

(ix) φ ∈ A   False   ϕ∉A 

(x) φ ⊂ A   True   ϕ is subset of every set 

(xi) {φ}⊂A  False   ϕ∉A

4. Write down all the subsets of the following sets
(i) {a}    ϕ, {a}
 
(ii) {a , b}     ϕ, {a}, {b}, {a,b}
 
(iii) {1, 2, 3}   ϕ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
 
(iv) φ   ϕ

5. How many elements has P(A), if A = φ?
We know that if A is set with m elements i.e. n(A)=m, then n[P(A)]=$\displaystyle \small 2^{m}$
if A=ϕ, then n(A)=0
∴ n[P(A)]=$\displaystyle \small 2^{0}=1$
Hence, P(A) has one element.

6. Write the following as intervals :
(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) = {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.
The universal set can be,
Set of triangles
Set of polygons
Set of 2D figures

(ii) The set of isosceles triangles.
The universal set can be,
Set of triangles
Set of polygons
Set of 2D 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}
A⊂ {0,1,2,3,4,5,6}
B⊂ {0,1,2,3,4,5,6}
C⊄ {0,1,2,3,4,5,6}
∴ {0,1,2,3,4,5,6} cannot be universal set for A,B,C

(ii) φ
A⊄ϕ, B⊄ϕ, C⊄ϕ
∴ ϕ cannot be universal set for A,B,C

(iii) {0,1,2,3,4,5,6,7,8,9,10}
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}
∴ {0,1,2,3,4,5,6} is the universal set for A,B,C

(iv) {1,2,3,4,5,6,7,8}
A⊂ {1,2,3,4,5,6,7,8}
B⊂ {1,2,3,4,5,6,7,8}
C⊄ {1,2,3,4,5,6,7,8}

∴ {1,2,3,4,5,6,7,8} cannot be universal set for A,B,C