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Exercise - 1.4

Question-1 :- Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = {a, e, i, o, u} B = {a, b, c}
(iii) A = {x: x is a natural number and multiple of 3} B = {x: x is a natural number less than 6}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} B = {x: x is a natural number and 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ

Solution :-
(i) X = {1, 3, 5} Y = {1, 2, 3}
    X ∪ Y = {1, 2, 3, 5}

(ii) A = {a, e, i, o, u} B = {a, b, c} 
    A ∪ B = {a, b, c, e, i, o, u}

(iii) A = {x: x is a natural number and multiple of 3} B = {x: x is a natural number less than 6}
    A = {3, 6, 9, 12,....} and B = {1, 2, 3, 4, 5} 
    A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}

(iv) A = {x: x is a natural number and 1 < x ≤ 6} B = {x: x is a natural number and 6 < x < 10}
    A = {2, 3, 4, 5, 6} and B = {7, 8, 9}
    A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

(v) A = {1, 2, 3}, B = Φ
   A ∪ B = {1, 2, 3}
    

Question-2 :- Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

Solution :-
  Here, A = {a, b} and B = {a, b, c}
  Yes, A ⊂ B.
  Hence, A ∪ B = {a, b, c} = B

Question-3 :- If A and B are two sets such that A ⊂ B, then what is A ∪ B?

Solution :-
 Given that A and B are two sets such that A ⊂ B
 Let A = {a, b} and B = {a, b, c} then A ∪ B = {a, b, c} = B

Question-4 :- If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
(i) A ∪ B
(ii) A ∪ C
(iii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪ D
(vii) B ∪ C ∪ D

Solution :-
   Given that A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}

(i) A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} 
  A ∪ B = {1, 2, 3, 4, 5, 6}

(ii) A ∪ C = {1, 2, 3, 4} ∪ {5, 6, 7, 8}
  A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(iii) B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7, 8}
  B ∪ C = {3, 4, 5, 6, 7, 8}

(iv) B ∪ D = {3, 4, 5, 6} ∪ {7, 8, 9, 10}
  B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}   
         
(v) A ∪ B ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {5, 6, 7, 8}
  A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}    
         
(vi) A ∪ B ∪ D = {1, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {7, 8, 9, 10}
  A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}      
         
(vii) B ∪ C ∪ D = {3, 4, 5, 6} ∪ {5, 6, 7, 8} ∪ {7, 8, 9, 10}
  B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
    

Question-5 :- Find the intersection of each pair of sets:
(i) X = {1, 3, 5}, Y = {1, 2, 3}
(ii) A = {a, e, i, o, u}, B = {a, b, c}
(iii) A = {x: x is a natural number and multiple of 3}, B = {x: x is a natural number less than 6}
(iv) A = {x: x is a natural number and 1 < x ≤ 6}, B = {x: x is a natural number and 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ

Solution :-
(i) X = {1, 3, 5}, Y = {1, 2, 3}
  X ∩ Y = {1, 3, 5} ∩ {1, 2, 3}
  X ∩ Y = {1, 3} 
       
(ii) A = {a, e, i, o, u}, B = {a, b, c} 
  A ∩ B = {a, e, i, o, u} ∩ {a, b, c} 
  A ∩ B = {a}

(iii) A = {x: x is a natural number and multiple of 3}, B = {x: x is a natural number less than 6} 
  A ∩ B = {3, 6, 9,.....} ∩ {1, 2, 3, 4, 5} 
  A ∩ B = {3}

(iv) A = {x: x is a natural number and 1 < x ≤ 6}, B = {x: x is a natural number and 6 < x < 10}
  A ∩ B = {2, 3, 4, 5, 6} ∩ {7, 8, 9} 
  A ∩ B = Φ

(v) A = {1, 2, 3}, B = Φ
  A ∩ B = {1, 2, 3} ∩ Φ
  A ∩ B = Φ

Question-6 :- If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find
(i) A ∩ B
(ii) B ∩ C
(iii) A ∩ C ∩ D
(iv) A ∩ C
(v) B ∩ D
(vi) A ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩ (B ∪ D)
(ix) (A ∩ B) ∩ (B ∪ C)
(x) (A ∪ D) ∩ (B ∪ C)

Solution :-
   Given that A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}.

(i) A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}
  A ∩ B = {7, 9, 11}

(ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15}
  B ∩ C = {7, 11}       
         
(iii) A ∩ C ∩ D = {3, 5, 7, 9, 11} ∩ {11, 13, 15} ∩ {15, 17}
  A ∩ C ∩ D = Φ
         
(iv) A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15}
  A ∩ C = {11}
               
(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17}
  B ∩ D = Φ
               
(vi) A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ [{7, 9, 11, 13} ∪ {11, 13, 15}]
  A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}  
  A ∩ (B ∪ C) = {7, 9, 11}   
          
(vii) A ∩ D = {3, 5, 7, 9, 11} ∩ {15, 17}
  A ∩ D = Φ

(viii) A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ [{7, 9, 11, 13} ∪ {15, 17}]
  A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17}
  A ∩ (B ∪ D) = {7, 9, 11} 

(ix) (A ∩ B) ∩ (B ∪ C) = [{3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}] ∩ [{7, 9, 11, 13} ∪ {11, 13, 15}] 
  (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15}   
  (A ∩ B) ∩ (B ∪ C) = {7, 9, 11}    
         
(x) (A ∪ D) ∩ (B ∪ C) = [{3, 5, 7, 9, 11} ∪ {15, 17}] ∩ [{7, 9, 11, 13} ∪ {11, 13, 15}]
  (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 13, 15}
  (A ∪ D) ∩ (B ∪ C) = {7, 9, 11, 15}
    

Question-7 :- If A = {x: x is a natural number}, B ={x: x is an even natural number}, C = {x: x is an odd natural number} and D = {x: x is a prime number}, find
(i) A ∩ B
(ii) A ∩ C
(iii) A ∩ D
(iv) B ∩ C
(v) B ∩ D
(vi) C ∩ D

Solution :-
  Given that :
  A = {x: x is a natural number} = {1, 2, 3,....}
  B ={x: x is an even natural number} = {2, 4, 6,....}
  C = {x: x is an odd natural number} = {1, 3, 5,....}
  D = {x: x is a prime number} = {2, 3, 5,....}

(i) A ∩ B = {1, 2, 3,....} ∩ {2, 4, 6,....}
  A ∩ B = {2, 4, 6,....} i.e.,
  A ∩ B = {x: x is an even natural number}

(ii) A ∩ C = {1, 2, 3,....} ∩ {1, 3, 5,....}
  A ∩ C = {1, 3, 5,....} i.e.,
  A ∩ C = {x: x is an odd natural number}

(iii) A ∩ D = {1, 2, 3,....} ∩ {2, 3, 5,....}
  A ∩ D = {2, 3, 5,....} i.e.,
  A ∩ D = {x: x is a prime number}

(iv) B ∩ C = {2, 4, 6,....} ∩ {1, 3, 5,....}
  B ∩ C = Φ

(v) B ∩ D = {2, 4, 6,....} ∩ {2, 3, 5,....}
  B ∩ D = {2}

(vi) C ∩ D = {1, 3, 5,....} ∩ {2, 3, 5,....}
  C ∩ D = {3, 5, 7,....} i.e.,
  C ∩ D = {x: x is an odd natural number ∉ 1}

Question-8 :- Which of the following pairs of sets are disjoint
(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6}
(ii) {a, e, i, o, u}and {c, d, e, f}
(iii) {x: x is an even integer} and {x: x is an odd integer}

Solution :-
(i)	A = {1, 2, 3, 4}, B = {x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6} 
  A ∩ B = {1, 2, 3, 4} ∩ {4, 5, 6}
  A ∩ B = {4}
  Therefore,{1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6} are not disjoint.

(ii) A = {a, e, i, o, u}, B = {c, d, e, f} 
  A ∩ B = {a, e, i, o, u} ∩ (c, d, e, f} 
  A ∩ B = {e} 
  Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.

(iii) A = {x: x is an even integer} ∩ {x: x is an odd integer} = Φ Therefore, this pair of sets is disjoint.
        {x: x is an even integer} ∩ {x: x is an odd integer} = Φ
    

Question-9 :- If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; find
(i) A – B,    (ii) A – C
(iii) A – D,    (iv) B – A
(v) C – A,    (vi) D – A
(vii) B – C,    (viii) B – D
(ix) C – B,     (x) D – B
(xi) C – D,     (xii) D – C

Solution :-
  Given that A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}; 

(i) A – B = {3, 6, 9, 12, 15, 18, 21} - {4, 8, 12, 16, 20}
  A – B = {3, 6, 9, 15, 18, 21} 

(ii) A – C = {3, 6, 9, 12, 15, 18, 21} - {2, 4, 6, 8, 10, 12, 14, 16}
  A – C = {3, 9, 15, 18, 21} 

(iii) A – D = {3, 6, 9, 12, 15, 18, 21} - {5, 10, 15, 20}
  A – D = {3, 6, 9, 12, 18, 21} 

(iv) B – A = {4, 8, 12, 16, 20} - {3, 6, 9, 12, 15, 18, 21}
  B – A = {4, 8, 16, 20}

(v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} - {3, 6, 9, 12, 15, 18, 21}
  C – A = {2, 4, 8, 10, 14, 16} 

(vi) D – A = {5, 10, 15, 20} - {3, 6, 9, 12, 15, 18, 21}
  D – A = {5, 10, 20}

(vii) B – C = {4, 8, 12, 16, 20} - {2, 4, 6, 8, 10, 12, 14, 16}
  B – C = {20}

(viii) B – D = {4, 8, 12, 16, 20} - {5, 10, 15, 20}
  B – D = {4, 8, 12, 16}

(ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} - {4, 8, 12, 16, 20}
  C – B = {2, 6, 10, 14} 

(x) D – B = {5, 10, 15, 20} - {4, 8, 12, 16, 20}
  D – B = {5, 10, 15}

(xi) C – D = {2, 4, 6, 8, 10, 12, 14, 16} - {5, 10, 15, 20}
  C – D = {2, 4, 6, 8, 12, 14, 16}

(xii) D – C = {5, 10, 15, 20} - {2, 4, 6, 8, 10, 12, 14, 16} 
  D – C = {5, 15, 20} 

Question-10 :- If X = {a, b, c, d} and Y = {f, b, d, g}, find
(i) X – Y
(ii) Y – X
(iii) X ∩ Y

Solution :-
  Given that X = {a, b, c, d} and Y = {f, b, d, g};

(i) X – Y = {a, b, c, d} - {f, b, d, g}
  X – Y = {a, c}

(ii) Y – X = {f, b, d, g} - {a, b, c, d}
  Y – X = {f, g}

(iii) X ∩ Y = {a, b, c, d} ∩ {f, b, d, g}
  X ∩ Y = {b, d}
    

Question-11 :- If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

Solution :-
  Given that :
  R: set of real numbers 
  Q: set of rational numbers
  Therefore, R – Q is a set of irrational numbers.

Question-12 :- State whether each of the following statement is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
(ii) {a, e, i, o, u } and {a, b, c, d} are disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.

Solution :-
(i) A = {2, 3, 4, 5}, B = {3, 6}
  A ∩ B = {3}
  A and B are not disjoint. Therefore statement is false. 

(ii) A = {a, e, i, o, u }, B = {a, b, c, d}
  A ∩ B = {a}
  A and B are not disjoint. Therefore statement is false. 

(iii) A = {2, 6, 10, 14}, B = {3, 7, 11, 15}
  A ∩ B = Φ
  A and B are disjoint. Therefore statement is true. 

(iv) A = {2, 6, 10}, B = {3, 7, 11}
  A ∩ B = Φ
  A and B are disjoint. Therefore statement is true.
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