TOPICS
Exercise - 1.3

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} ⊂ {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 is 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 in 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} ∈ (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. 
   Each element of {a, b} is also an element of {b, c, a}. 

(ii) True. 
   a, e are two vowels of the English alphabet.
     
(iii) False. 
   2 ∈ {1, 2, 3}; however, 2 ∉ {1, 3, 5}
     
(iv) True. 
   Each element of {a} is also an element of {a, b, c}. 

(v)	False. 
   The elements of {a, b, c} are a, b, c. 
   Therefore, {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}

Question-3 :- 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) Φ ∈ A
(x) Φ ⊂ A
(xi) {Φ} ⊂ A

Solution :-
    Give that A = {1, 2, {3, 4}, 5}
(i) The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3 ∉ A. 
(ii) The statement {3, 4} ∈ A is correct because {3, 4} is an element of A. 
(iii) The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A. 
(iv) The statement 1 ∈ A is correct because 1 is an element of A. 
(v) The statement 1 ⊂ A is incorrect because an element of a set can never be a subset of itself. 
(vi) The statement {1, 2, 5} ⊂ A is correct because each element of {1, 2, 5} is also an element of A. 
(vii) The statement {1, 2, 5} ∈ A is incorrect because {1, 2, 5} is not an element of A. 
(viii) The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.
(ix) The statement Φ ∈ A is incorrect because Φ is not an element of A. 
(x) The statement Φ ⊂ A is correct because Φ is a subset of every set. 
(xi) The statement {Φ} ⊂ A is incorrect because Φ ∈ {Φ}; however, Φ ∈ A. 
    

Question-4 :- Write down all the subsets of the following sets:
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Φ

Solution :-
(i) The subsets of {a} = {Φ, {a}}. 
(ii) The subsets of {a, b} = {Φ, {a}, {b}, {a, b}}. 
(iii) The subsets of {1, 2, 3} = {Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}} 
(iv) The only subset of Φ = {Φ}. 

Question-5 :- How many elements has P(A), if A = Φ?

Solution :-
  We know that if A is a set with m elements i.e., n(A) = m, then n[P(A)] = 2ᵐ. 
  If A = Φ, then n(A) = 0.
∴ n[P(A)] = 2⁰ = 1
  Hence, P(A) has one element.
    

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) {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] 

Question-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} 

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. 
   Therefore, the set of triangles contain all types of triangles. 
   Hence, U = {x : x is a triangle in a plane.}

(ii) Isosceles triangle is a type of triangle. 
   Therefore, the set of triangles contain all types of triangles. 
   Hence, 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 universals 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 :-
    Given that sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}.
    So, U = {0, 1, 2, 3, 4, 5, 6, 8}

(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. 
   Therefore 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}.
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