NCERT Solutions Class 11 Mathematics Sets 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

Answer:
(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} = {3, 6, 9 …}
      B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5, 6}
     A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}
    A ∪ B = {x: x = 1, 2, 4, 5 or a multiple of 3}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}
       B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}
       A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
       A∪ B = {x: x ∈ N and 1 < x < 10}
(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 ?

Answer:
Here, A = {a, b} and B = {a, b, c}
Yes, A ⊂ B.
Now, A ∪ B = {a, b, c} = B

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

Answer:
If A and B are two sets such that A ⊂ B, then A ∪ B = B
Example:
Let A = {a, b} and B = {a, b, c}
So, A ⊂ B.
Now, 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

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

Question 5:
Find the intersection of each pair of sets of question 1 above.

Answer:
(i) X = {1, 3, 5}, Y = {1, 2, 3}
    X ∩ Y = {1, 3}
(ii) A = {a, e, i, o, u}, B = {a, b, c}
      A ∩ B = {a}
(iii) A = {x: x is a natural number and multiple of 3} = (3, 6, 9 …}
      B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5}
So, A ∩ B = {3}
(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}
       B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}
Now, A ∩ B = φ
(v) A = {1, 2, 3}, B = φ. So, 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)

Answer:
Given, A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}
(i) A ∩ B = {7, 9, 11}
(ii) B ∩ C = {11, 13}
(iii) A ∩ C ∩ D = { A ∩ C} ∩ D = {11} ∩ {15, 17} = φ
(iv) A ∩ C = {11}
(v) B ∩ D = φ
(vi) A ∩ (B U C) = (A ∩ B) U (A ∩ C) = {7, 9, 11} U {11} = {7, 9, 11}
(vii) A ∩ D = φ
(viii) A ∩ (B U D) = (A ∩ B) U (A ∩ D) = {7, 9, 11} U Φ = {7, 9, 11}
(ix) (A ∩ B) ∩ (B C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}
(x) (A U D) ∩ (B U C) = {3, 5, 7, 9, 11, 15, 17) ∩ {7, 9, 11, 13, 15} = {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

Answer:
Given,
A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}
B ={x: x is an even natural number} = {2, 4, 6, 8 …}
C = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}
D = {x: x is a prime number} = {2, 3, 5, 7 …}
(i) A ∩B = {x: x is a even natural number} = B
(ii) A ∩ C = {x: x is an odd natural number} = C
(iii) A ∩ D = {x: x is a prime number} = D
(iv) B ∩ C = φ
(v) B ∩ D = {2}
(vi) C ∩ D = {x: x is odd prime number}

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}

Answer:
(i) {1, 2, 3, 4}
{x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}
Now, {1, 2, 3, 4} ∩ {4, 5, 6} = {4}
Therefore, this pair of sets is not disjoint.
(ii) {a, e, i, o, u} ∩ (c, d, e, f} = {e}
Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.
(iii) {x: x is an even integer} ∩ {x: x is an odd integer} = φ
Therefore, this pair of sets is disjoint.