Sets (Ex – 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.
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}. All elements of set A are present in set B.
∴ 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?
Solution.
Here A and B are two sets such that A ⊂ B.
Take A = {1, 2} and B = {1, 2, 3}.
A ∪ B = {1, 2, 3) = B.
Ex 1.4 Class 11 Maths Question 4.
If A = {11, 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 ∪ O
(v) A ∪ B ∪ C
(vi) A ∪ B ∪ D
(vii) B ∪ C ∪ D
Solution.
Here A = {11, 2, 3, 4}, B = {3, 4, 5, 6}, C={5, 6, 7, 8} and D = {7, 8, 9, 10}
Question 5.
Find the intersection of each pair of sets of .
(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) Here X = {1, 3, 5} and Y = {1, 2, 3}
∴ X ∩ Y= {1,3}
(ii) Here A = {a, e, i, o, u} and B = {a, b, c}
∴ A ∩ B = {a}
(iii) Here A = {x: x is a natural number and multiple of 3} = {3, 6, 9,12,….} and B = {x: x is a natural number less than 6}
= {1, 2, 3, 4, 5} ∴ A ∩ B = {3}
(iv) Here A = {x: x is a natural number and 1 < x < 6} ={2, 3, 4, 5, 6} and B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9} ∴ A ∩ B = φ
(v) Here A = {1, 2, 3) and B = φ
∴ A ∩ B = φ
Ex 1.4 Class 11 Maths 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.
Here A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}
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.
Here A = {x: x is a natural number}
= (1, 2, 3, 4, 5, …….}
B = {x: x is an even natural number}
= 12, 4, 6,………}
C = {x: x is an odd natural number}
= {1, 3, 5, 7,………}
and D = {x: x is a prime number}
= {2, 3, 5, 7,….}
(i) A ∩ B = {x: x is a natural number} ∩ {x: x is an even natural number}
= {x: x is an even natural number} = B.
(ii) A ∩ C = {x: x is a natural number} ∩ {x: x is an odd natural number}
= {x: x is an odd natural number} = C.
(iii) A ∩ D = {x: x is a natural number} ∩ {x: x is a prime number}
= {x: x is a prime number} = D.
(iv) B ∩ C = {x: x is an even natural number} ∩{x: x is an odd natural number} = φ .
(v) B ∩ D = [x: x is an even natural number} ∩ {x: x is a prime number} = {2}.
(vi) C ∩ D = {x: x is an odd natural number} ∩ {x: x is a prime number} = {x: x is an 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}
Solution.
(i) Let A = {1,2,3,4}
and B = {x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}
∴ A ∩ B = {1,2,3,4} n {4,5, 6} = {4}
Hence A and B are not disjoint sets.
(ii) Let A = {a, e, i, o, u} and B = {c, d, e, f}
∴ A ∩ B = {e}
Hence A and B are not disjoint sets.
(iii) Let A = {x : x is an even integer} and B = {x: x is an odd integer}
∴ A ∩ B = φ. Hence A and B are disjoint sets.
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} and 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.
Here 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} = {3, 6, 9,15,18, 21}
(ii) A – C = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12, 14, 16} = {3, 9, 15, 18, 21}
(iii) A – D = {3, 6, 9, 12, 15, 18, 21} – {5,10,15, 20} = {3, 6, 9, 12, 18, 21}
(iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21} = {4, 8,16, 20}
(v) C – A = {2,4, 6, 8, 10, 12, 14, 16} – {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}
(vii) B – C={4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {20}
(viii) B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20} = {4, 8, 12, 16}
(ix) C – B = {2,4, 6, 8, 10, 12, 14, 16} – {4, 8, 12, 16, 20} = {2, 6, 10, 14}
(x) D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20} = {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 10, 12, 14, 16} – {5, 10, 15, 20} = {2, 4, 6, 8, 12, 14, 16}
(xii) D – C={5, 10, 15, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {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.
Here X = {a, b, c, d} and Y = {f, b, d, g}
(i) X – Y = {a, b, c, d} – {f, b, d, g} = {a, c}
(ii) Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}
(iii) X ∩ Y = {a, b, c, d} ∩ {f, b, d, g} = {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.
We know that set of real numbers contain rational and irrational numbers. So R – Q = 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
Solution.
