TOPICS
Exercise - 1.5

Question-1 :- Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find
(i) A'
(ii) B'
(iii) (A ∪ C)'
(iv) (A ∪ B)'
(v) (A')'
(vi) (B - C)'

Solution :-
  Given that U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6};

(i) A' = U - A
  A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4} 
  A' = {5, 6, 7, 8, 9}
(ii) B' = U - B
  B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8} 
  B' = {1, 3, 5, 7, 9}
(iii) (A ∪ C)' = U - (A ∪ C)
  A ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6} 
  A ∪ C = {1, 2, 3, 4, 5, 6}
  (A ∪ C)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}
  (A ∪ C)' = {7, 8, 9}
      
  
(iv) (A ∪ B)' = U - (A ∪ B)
  A ∪ B = {1, 2, 3, 4} ∪ {2, 4, 6, 8} 
  A ∪ B = {1, 2, 3, 4, 6, 8}
  (A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8}
  (A ∪ B)' = {5, 7, 9}
(v) (A')' = U - A'
  A' = U - A
  A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4} 
  A' = {5, 6, 7, 8, 9}
  (A')' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9}
  (A')' = {1, 2, 3, 4} = A
(vi) (B - C)' = U - (B - C)
  B - C = {2, 4, 6, 8} - {3, 4, 5, 6}
  B - C = {2, 8}
  (B - C)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 8}
  (B - C)' = {1, 3, 4, 5, 6, 7, 9}
    

Question-2 :- If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}

Solution :-
  Given that  U = {a, b, c, d, e, f, g, h}.

(i) A = {a, b, c}
  A' = U - A
  A' = {a, b, c, d, e, f, g, h} - {a, b, c}
  A' = {d, e, f, g, h}
(ii) B = {d, e, f, g}
  B' = U - B
  B' = {a, b, c, d, e, f, g, h} - {d, e, f, g}
  B' = {a, b, c, h} 
(iii) C = {a, c, e, g}
  C' = U - C
  C' = {a, b, c, d, e, f, g, h} - {a, c, e, g}
  C' = {b, d, f, h}
(iv) D = {f, g, h, a}
  D' = U - D
  D' = {a, b, c, d, e, f, g, h} - {f, g, h a}
  D' = {b, c, d, e}

Question-4 :- Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x: x is an even natural number}
(ii) {x: x is an odd natural number}
(iii) {x: x is a positive multiple of 3}
(iv) {x: x is a prime number}
(v) {x: x is a natural number divisible by 3 and 5}
(vi) {x: x is a perfect square}
(vii) {x: x is perfect cube}
(viii) {x: x + 5 = 8}
(ix) {x: 2x + 5 = 9}
(x) {x: x ≥ 7}
(xi) {x: x ∈ N and 2x + 1 > 10}

Solution :-
  Given that U = {x : x is set of natural numbers} = {1, 2, 3,....}

(i) A = {x: x is an even natural number} 
  A = {2, 4, 6,....}
  A' = U - A 
  A' = {1, 2, 3,....} - {2, 4, 6,....}
  A' = {1, 3, 5....}
  A' = {x : x is set of odd natural numbers, x ∈ N}
(ii) B = {x: x is an odd natural number} 
  B = {1, 3, 5,....}
  B' = U - B 
  B' = {1, 2, 3,....} - {1, 3, 5,....}
  B' = {2, 4, 6....}
  B' = {x : x is set of even natural numbers, x ∈ N}
(iii) C = {x: x is a positive multiple of 3} 
  C = {3, 6, 9,....}
  C' = U - C 
  C' = {1, 2, 3,....} - {3, 6, 9,....}
  C' = {2, 4, 5....}
  C' = {x : x is not a positive multiple of 3, x ∈ N}
(iv) D = {x: x is a prime number}
  D = {2, 3, 5,....}
  D' = U - D 
  D' = {1, 2, 3,....} - {2, 3, 5,....}
  D' = {1, 4, 6....}
  D' = {x : x is not a prime number, x ∈ N}      
     
(v) E = {x: x is a natural number divisible by 3 and 5}
  E = {15, 30, 45,....}
  E' = U - E 
  E' = {1, 2, 3,....} - {15, 30, 45,....}
  E' = {1, 3, 5....}
  E' = {x : x is not a natural number divisible by 3 and 5, x ∈ N} 
(vi) F = {x: x is a perfect square}
  F = {1, 4, 9,....}
  F' = U - F 
  F' = {1, 2, 3,....} - {1, 4, 9,....}
  F' = {2, 3, 5....}
  F' = {x : x is not a perfect square, x ∈ N} 
    
     
(vii) G = {x: x is perfect cube} 
  G = {1, 8, 27,....}
  G' = U - G 
  G' = {1, 2, 3,....} - {1, 8, 27,....}
  G' = {2, 3, 4....}
  G' = {x : x is not a perfect cube, x ∈ N} 
(viii) H = {x: x + 5 = 8}
  x + 5 = 8
  x = 8 - 5
  x = 3
  H = {3}
  H' = U - H 
  H' = {1, 2, 3,....} - {3}
  H' = {1, 2, 4....}
  H' = {x : x ≠ 3, x ∈ N} 
(ix) I = {x: 2x + 5 = 9} 
  2x + 5 = 9
  2x = 9 - 5
  2x = 4
  x = 4/2
  x = 2
  I = {2}
  I' = U - I
  I' = {1, 2, 3,....} - {2}
  I' = {1, 3, 4,....}
  I' = {x : x ≠ 2, x ∈ N} 
(x) J = {x: x ≥ 7}
  J = {7, 8, 9,....}
  J' = U - J
  J' = {1, 2, 3,....} - {7, 8, 9,....}
  J' = {1, 2, 3, 4, 5, 6}
  J' = {x : x < 7, x ∈ N} 
    
     
(xi) K = {x: x ∈ N and 2x + 1 > 10}
  K = {5, 6, 7,....}
  K' = U - K
  K' = {1, 2, 3,....} - {5, 6, 7,....}
  k' = {1, 2, 3, 4}
  K' = {x: x ∈ N and x ≤ 9/2} 
    

Question-4 :- If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) (A ∪ B)' = A'∩ B'
(ii) (A ∩ B)' = A' ∪ B'

Solution :-
  Given that  U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}.

(i) (A ∪ B)' = A'∩ B'
L.H.S 
  (A ∪ B)' = U - (A ∪ B)
  A ∪ B = {2, 4, 6, 8} ∪ {2, 3, 5, 7}
  A ∪ B = {2, 3, 4, 5, 6, 7, 8}
  (A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5, 6, 7, 8}
  (A ∪ B)' = {1, 9}
R.H.S
  A'∩ B' = (U - A) ∩ (U - B)
  A' = U - A
  A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
  A' = {1, 3, 5, 7, 9}
  B' = U - B
  B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7}
  B' = {1, 4, 6, 8, 9}
  A'∩ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
  A'∩ B' = {1, 9}
  L.H.S = R.H.S
(ii) (A ∩ B)' = A' ∪ B' 
L.H.S 
  (A ∩ B)' = U - (A ∩ B)
  A ∩ B = {2, 4, 6, 8} ∩ {2, 3, 5, 7}
  A ∩ B = {2}
  (A ∩ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2}
  (A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9}
R.H.S
  A'∪ B' = (U - A) ∪ (U - B)
  A' = U - A
  A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
  A' = {1, 3, 5, 7, 9}
  B' = U - B
  B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7}
  B' = {1, 4, 6, 8, 9}
  A'∪ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
  A'∪ B' = {1, 3, 4, 5, 6, 7, 8, 9}
  L.H.S = R.H.S

Question-5 :- Draw appropriate Venn diagram for each of the following:
(i) (A ∪ B)'
(ii) A'∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'

Solution :-
(i) (A ∪ B)' 
sets
(ii) A'∩ B'
sets
(iii) (A ∩ B)' 
sets
(iv) A' ∪ B'
sets
    

Question-6 :- Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is ?

Solution :-
  Given that  U = {x : x is a triangle}.
  A = {x : x  is a triangle and has at least one angle different from 60°}
  A' = U - A
  A' = {x : x is a triangle} - {x : x  is a triangle and has at least one angle different from 60°}
  A' = {x : x is a triangle and has all angles equal to 60°}
  A' = set of all equilateral triangles 

Question-7 :- Fill in the blanks to make each of the following a true statement:
(i) A ∪ A'= ...
(ii) Φ′∩ A = ...
(iii) A ∩ A' = ...
(iv) U'∩ A' = ...

Solution :-
(i) A ∪ A'= U 
(ii) Φ′∩ A = U ∩ A = A 
(iii) A ∩ A' = Φ 
(iv) U'∩ A' = Φ ∩ A = Φ
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