## CBSE Class 11 NCERT Exercise 1.3

Q1: 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}

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

Q2. Examine whether the following statements are true or false:
(i) { a, b } ⊄ { b, c, a }

Since elements of the set {a, b} are also present in the set {b, c, a},
∴ {a, b} ⊄ { b, c, a } is FALSE.

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

Vowels in English alphabets are a, e, i, o, u
{a, e} ⊂ { x : x is a vowel in the English alphabet } is TRUE.

(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }

(iv) { a } ⊂ { a, b, c }

(v) { a } ∈ { a, b, c }

Answer: False,  ∵ a  ∈ { a, b, c } and not { a }.

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

{ x : x is an even natural number less than 6}  = {2,4}
and { x : x is a natural number which divides 36} = {1,2,3,4,6,9,12,18,36}.
∴ the given statement is TRUE.

Q3. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are 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) ∅ ∈ A
(x) ∅ ⊂ A
(xi) {∅} ⊂ A

(i) {3, 4} ⊂ A - FALSE,  ∵ {3,4}  ∈ A

(ii) {3, 4} ∈ A  - TRUE  ∵ {3, 4} is an elament of set a

(iii) {{3, 4}} ⊂ A  - TRUE

(iv) 1 ∈ A  - TRUE

(v) 1 ⊂ A  - FALSE  ∵ 1 ∈ A

(vi) {1, 2, 5} ⊂ A  - TRUE

(vii) {1, 2, 5} ∈ A  - FALSE

(viii) {1, 2, 3} ⊂ A  - FALSE  ∵ 3 ∉ A

(ix) ∅ ∈ A - FALSE  ∵ ∅ ⊂ A

(x) ∅ ⊂ A - TRUE

(xi) {∅} ⊂ A - FALSE ∵ ∅ ⊂ A

Q4. Write down all the subsets of the following sets
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) ∅

(i) Subset of {a} are ∅, {a}
(ii) Subset of {a, b} are ∅, {a}, {b}, {a, b}
(iii) Subset of {1, 2, 3} are ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
(iv) Subset of ∅ is ∅ only as there is no other elment in the set.

Q5: How many elements has P(A), if A = ∅?

Answer: If A = ∅ then by the definition of power set,
P(A) = P(∅) = {∅}
= a set containing 1 element.

Q6: 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}

(i) {x : x ∈ R, – 4 < x ≤ 6}
The interval is given by (-4, 6]

(ii) {x : x ∈ R, – 12 < x < –10}
The interval is (-12, -10)

(iii) {x : x ∈ R, 0 ≤ x < 7}
The interval is given by [0,7)

(iv) {x : x ∈ R, 3 ≤ x ≤ 4}
The interval is [3,4]

Q7: Write the following intervals in set-builder form :
(i) (– 3, 0)
(ii) [6 , 12]
(iii) (6, 12]
(iv) [–23, 5)

(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 }

Q8: What universal set(s) would you propose for each of the following :
(i) The set of right triangles.

Answer: The universal set is set of triangles having angle as right angle.

(ii) The set of isosceles triangles.

Answer: The universal set is taken as set of all triangles which contain set of isoceles triangles.

Q9: 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) ∅
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}

Answer: Since universal set is the set which consists of all the elements of the small sets.
Since option (iii) {0,1,2,3,4,5,6,7,8,9,10} contains all the elements of sets A, B and C, it can be considered as universal set.  