Exercise 1.4
Q1. 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} = {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}
Q2. 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.
As a ∈ B and b ∈ B
A ∪ B = {a, b, c} = B
{Rule: if A ∪ B = B then A ⊂ B Or if A ∪ B = A then B ⊂ A }
Q3. 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
Then A ∪ B = B
Illustration by example:
Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}
Here A ⊂ B Because All elements of A 1, 2, 3 ∈ B
[B also contains 1, 2, 3]
Now, A ∪ B = {1, 2, 3, 4, 5} = B
Therefore, A ∪ B = B
Q4. 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 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}
Q5. Find the intersection of each pair of sets of question 1 above.
(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}
(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}
∴ A ∩ B = {3}
(iv) A = {x: x is a natural number and 1 < x ≤ 6}
Or A = {2, 3, 4, 5, 6}
B = {x: x is a natural number and 6 < x < 10}
Or B = {7, 8, 9}
A ∩ B = Φ
(v) A = {1, 2, 3}, B = Φ.
So, A ∩ B = Φ
Q6. 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:
(i) A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13},
Therefore, A ∩ B = {7, 9, 11}
Solution:
(ii) B = {7, 9, 11, 13}, and C = {11, 13, 15}
Therefore, B ∩ C = {11, 13}
Solution:
(iii) A = { 3, 5, 7, 9, 11 }, C = {11, 13, 15} and D = {15, 17};
A ∩ C ∩ D = { A ∩ C} ∩ D = {11} ∩ {15, 17} = Φ
Solution:
(iv) A = { 3, 5, 7, 9, 11 }, C = {11, 13, 15}
Therefore, A ∩ C = {11}
Solution:
(v) B = {7, 9, 11, 13}, and D = {15, 17};
Therefore, B ∩ D = Φ
Solution:
(vi) A ∩ (B C) = (A ∩ B) (A ∩ C)
= {7, 9, 11} {11} = {7, 9, 11}
Solution:
(vii) A ∩ D = Φ
Solution:
(viii) A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, and D = {15, 17};
We know; A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D)
= {7, 9, 11} Φ = {7, 9, 11}
Solution:
(ix) A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, and C = {11, 13, 15}
(A ∩ B) = {7, 9, 11}
(B ∪ C) = {7, 9, 11, 13, 15}
(A ∩ B) ∩ (B ∪ C)
= {7, 9, 11} ∩ {7, 9, 11, 13, 15}
= {7, 9, 11}
Solution:
(x) Given A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17};
(A ∪ D) ∩ (B ∪ C)
= {3, 5, 7, 9, 11, 15, 17) ∩ {7, 9, 11, 13, 15}
= {7, 9, 11, 15}
Q7. 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 7:
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 ...}
Solution:
(i) A ∩ B = {2, 4, 6, 8 .....}
A ∩ B = {x : x is a even natural number}
A ∩ B = B
Solution:
(ii) A ∩ C = {1, 3, 5, 7, 9 ...}
A ∩ C = {x : x is an odd natural number}
A ∩ C = C
Solution:
(iii) A ∩ D = {2, 3, 5, 7 ...}
A ∩ D = {x : x is a prime number}
A ∩ D = D
Solution:
(iv) B ∩ C = Φ
Solution:
(v) B ∩ D = {2}
Solution:
(vi) C ∩ D = {3, 5, 7, 11 ....}
C ∩ D = {x : x is odd prime number}
Q8. 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}
{Note : Disjoint means no elements matched in each others, in other words intersection of two sets give null set then it said to be disjoint.}
Solution:
(i) Let A = {1, 2, 3, 4} and
B = {x: x is a natural number and 4 ≤ x ≤ 6}
Or B = {4, 5, 6}
Now, A ∩ B
= {1, 2, 3, 4} ∩ {4, 5, 6}
= {4}
Therefore, this pair of sets is not disjoint.
Solution:
(ii) Let X = {a, e, i, o, u} and
Y = (c, d, e, f}
Now, X ∩ Y
= {a, e, i, o, u} ∩ (c, d, e, f}
= {e}
Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.
Solution:
(iii) Let A = {x : x is an even integer}
Or A = {2, 4, 6, 8 ... } and
B = {x : x is an odd integer}
Or B = {1, 3, 5, 7 ... }
Now, A ∩ B
= {2, 4, 6, 8 ... } ∩ {1, 3, 5, 7 ... }
= Φ
Therefore,
{x : x is an even integer} ∩ {x : x is an odd integer}
= Φ
Therefore, A ∩ B gives null set so this pair of sets is disjoint.
Q9. 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:
(i) A – B = {3, 6, 9, 15, 18, 21}
(ii) A – C = {3, 9, 15, 18, 21}
(iii) A – D = {3, 6, 9, 12, 18, 21}
(iv) B – A = {4, 8, 16, 20}
(v) C – A = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 20}
(vii) B – C = {20}
(viii) B – D = {4, 8, 12, 16}
(ix) C – B = {2, 6, 10, 14}
(x) D – B = {5, 10, 15}
(xi) C – D = {2, 4, 6, 8, 12, 14, 16}
(xii) D – C = {5, 15, 20}
Q10. If X = { a, b, c, d } and Y = { f, b, d, g}, find
(i) X – Y
(ii) Y – X
(iii) X ∩ Y
Solution:
(i) X – Y = { a, c }
(ii) Y – X = {f, g}
(iii) X ∩ Y = {b, d}
Q11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?
Solution:
R = {x : x is set of real numbers}
Q = {x : x is set of rational numbers}
We know that,
Real numbers = Rational Numbers + Irrational numbers
Real numbers - Rational Numbers = Irrational numbers
R - Q = I
Therefore, R – Q is a set of irrational numbers.
Q12. 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.
Solution:
(i) False, Because { 2, 3, 4, 5 } ∩ { 3, 6} = {3}
(ii) { a, e, i, o, u } and { a, b, c, d } are disjoint sets.
Solution:
(ii) False, Because { a, e, i, o, u } ∩ { a, b, c, d } = {a}
(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
Solution:
(iii) True, Because { 2, 6, 10, 14 } ∩ { 3, 7, 11, 15} = Φ
(iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.
Solution:
(iv) True, Because { 2, 6, 10 } ∩ { 3, 7, 11} = Φ
