Punjab State Board PSEB 11th Class Maths Book Solutions Chapter 1 Sets Ex 1.4 Textbook Exercise Questions and Answers.
PSEB Solutions for Class 11 Maths Chapter 1 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}
Answer.
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}
Answer.
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}
Answer.
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 = {1, 2, 3, 4, 5, 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}
Answer.
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
Answer.
A = {1, 2, 3}, B = Φ
A ∪ B = {1, 2, 3}.
Question 2.
Let A = {a, 6}, B = {a, b, c}. Is Ac B? What is A ∪ B?
Answer.
Here, A = {a, b} and B = {a, b, c}
Yes, A ⊂ B and A ∪ B = {a, b, c}.
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.
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.
(i) A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6}
= {1, 2, 3, 4, 5, 6}.
(ii) A ∪ C = {1, 2, 3, 4} ∪ {5, 6, 7, 8}
={1, 2, 3, 4, 5, 6, 7, 8}.
(iii) B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7, 8}
={3, 4, 5, 6, 7, 8}.
(iv) B ∪ D = {3, 4, 5, 6} ∪ {7, 8, 9, 10}
= {3, 4, 5, 6, 7, 8, 9, 10}.
(v) A ∪ B ∪ C = {l, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {5, 6, 7, 8}
= {1, 2, 3, 4, 5, 6} ∪ {5, 6, 7, 8}
= {1, 2, 3, 4, 5, 6, 7, 8}.
(vi) A ∪ B ∪ D = {1, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {7,8, 9, 10}
= {1, 2, 3, 4, 5, 6} ∪ {7, 8, 9, 10}
= {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}
= {3, 4, 5, 6, 7, 8} ∪ {7, 8, 9, 10}
= {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}
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}
Answer.
A = {a, e, i, o, u}, B = {a, b, c}
A ∩ B = {o}
(iii) A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
Answer.
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}
B = {x : x is a natural number and 6 < x < 10}
Answer.
(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 = Φ
(v) A = {1, 2, 3},
B = Φ
Answer.
A = {1, 2, 3}, B= 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)
Answer.
(i) A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}
= {7, 9, 11}.
(ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15}
= {11, 13}.
(iii) A ∩ C ∩ D = {3, 5, 7, 9, 11} ∩ {11, 13, 15} ∩ {15, 17}
= {11} ∩ {15, 17}
= Φ.
(iv) A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} =
{11}
(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17}
= Φ
(vi) A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {11, 13, 15})
= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}
= {7, 9, 11}.
(vii) A ∩ D = {3, 5, 7, 9, 11} ∩ {15, 17}
= Φ
(viii) A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ ({7, 9, 11, 13} ∪ {15, 17}
= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17}
= {7, 9, 11}
(ix) (A ∩ B) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}
= {7, 9, 11}
B ∪ C = {7, 9, 11, 13} ∪ {11, 13, 15}
= {7, 9, 11, 13, 15}
∴ (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15}
= {7, 9, 11}
(x) (A ∩ D) = {3, 5, 7, 9, 11} ∩ {15, 17}
= {3, 5, 7, 9, 11, 15 ,17}
B ∪ C = {7, 9,11, 13, 15} [From part (ix)]
∴ (A ∪ D) ∩ (B ∪ 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.
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 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}.
Answer.
{1, 2, 3, 4} and {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}and {c, d, e, f}
Answer.
{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} and {x : x is an odd integer}
Answer.
{x : x is an even integer} ∩ {x : x is an odd integer} = Φ.
Therefore, this pair of sets is disjoint.
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
Answer.
(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, 15, 18, 21}.
(iii) A – D = {3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20}
= {3, 6, 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
Answer.
X – Y = {a, b, c, d} – {f, b, d, g} = {a, c}.
(ii)Y – X
Answer.
Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}.
(iii) X ∩ Y
Answer.
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?
Answer.
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.
Answer.
False
As 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}
⇒ {2, 3, 4, 5} ∩ {3, 6} = {3}
(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
Answer.
False
As a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d}
⇒ {a, e, i, o, u } ∩ {a, b, c, d} = {a}
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
Answer.
True
As {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ
(iv) {2, 6, 10} and {3, 7,11} are disjoint sets.
Answer.
True
As {2, 6, 10} ∩ {3, 7, 11} = Φ