CBSE NCERT Solutions For Class 11th Maths Chapter 1 : Set. NCERT Solutins For Class 11 Mathematics. Exercise 1.1, Exercise 1.2, Exercise 1.3, Exercise 1.4, Exercise 1.5, Exercise 1.6, (Miscellaneous Excercise) many more solutions
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Question 1:
Decide, among the following sets, which sets are subsets of one and another: A = {x: x ∈ R and x satisfy x^{2} – 8x + 12 = 0},
B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}. Answer
A = {x: x ∈ R and x satisfies x^{2} – 8x + 12 = 0}
2 and 6 are the only solutions of x^{2} – 8x + 12 = 0.
∴ A = {2, 6}
B = {2, 4, 6}, C = {2, 4, 6, 8 …}, D = {6}
∴ D ⊂ A ⊂ B ⊂ C
Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B, D ⊂ C
Question 2:
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
- If x ∈ A and A ∈ B, then x ∈ B
- If A ⊂ B and B ∈ C, then A ∈ C
- If A ⊂ B and B ⊂ C, then A ⊂ C
- If A ⊄ B and B ⊄ C, then A ⊄ C
- If x ∈ A and A ⊄ B, then x ∈ B
- If A ⊂ B and x ∉ B, then x ∉ A Answer
(i) False
Let A = {1, 2} and B = {1, {1, 2}, {3}}
Now,
∴ A ∈ B
However,
(ii) False
Let
As A ⊂ B
B ∈ C
However,
(iii) True
Let A ⊂ B and B ⊂ C.
Let x ∈ A
∴ A ⊂ C
(iv) False
Let
Accordingly, and .
However, A ⊂ C
(v) False
Let A = {3, 5, 7} and B = {3, 4, 6}
Now, 5 ∈ A and A ⊄ B
However, 5 ∉ B
(vi) True
Let A ⊂ B and x ∉ B.
To show: x ∉ A
If possible, suppose x ∈ A.
Then, x ∈ B, which is a contradiction as x ∉ B
∴x ∉ A
Question 3:
Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C. Answer
Let, A, B and C be the sets such that and . To show: B = C
Let x ∈ B
Case I
x ∈ A Also, x ∈ B
∴
- x ∈ A and x ∈ C
- x ∈ C
- B ⊂ C
Similarly, we can show that C ⊂ B.
∴ B = C
Question 4:
Show that the following four conditions are equivalent:
(i) A ⊂ B (ii) A – B = Φ
(iii) A ∪ B = B (iv) A ∩ B = A Answer
First, we have to show that (i) ⇔ (ii). Let A ⊂ B
To show: A – B ≠ Φ
If possible, suppose A – B ≠ Φ
This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B.
- A – B = Φ
- A ⊂ B ⇒ A – B = Φ Let A – B = Φ
To show: A ⊂ B Let x ∈ A
Clearly, x ∈ B because if x ∉ B, then A – B ≠ Φ
- A – B = Φ ⇒ A ⊂ B
∴ (i) ⇔ (ii) Let A ⊂ B
To show:
Clearly,
Let
Case I: x ∈ A
∴
Case II: x ∈ B
Then,
Conversely, let
Let x ∈ A
∴ A ⊂ B
Hence, (i) ⇔ (iii)
Now, we have to show that (i) ⇔ (iv).
Let A ⊂ B
Clearly
Let x ∈ A
We have to show that
As A ⊂ B, x ∈ B
∴
∴
Hence, A = A ∩ B
Conversely, suppose A ∩ B = A
Let x ∈ A
⇒
- x ∈ A and x ∈ B
- x ∈ B
∴ A ⊂ B
Hence, (i) ⇔ (iv).
Question 5:
Show that if A ⊂ B, then C – B ⊂ C – A.
Answer
Let A ⊂ B
To show: C – B ⊂ C – A
Let x ∈ C – B
- x ∈ C and x ∉ B
- x ∈ C and x ∉ A [A ⊂ B]
- x ∈ C – A
∴ C – B ⊂ C – A
Question 6:
Assume that P (A) = P (B). Show that A = B.
Answer
Let P(A) = P(B)
To show: A = B
Let x ∈ A
A ∈ P(A) = P(B)
- x ∈ C, for some C ∈ P(B) Now, C ⊂ B
- x ∈ B
- A ⊂ B
Similarly, B ⊂ A
∴ A = B
Question 7:
Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.
Answer
False
Let A = {0, 1} and B = {1, 2}
∴ A ∪ B = {0, 1, 2}
P(A) = {Φ, {0}, {1}, {0, 1}}
P(B) = {Φ, {1}, {2}, {1, 2}}
P(A ∪ B) = {Φ, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2}, {0, 1, 2}}
P(A) ∪ P(B) = {Φ, {0}, {1}, {0, 1}, {2}, {1, 2}}
∴ P(A) ∪ P(B) ≠ P(A ∪ B)
Question 8:
Show that for any sets A and B,
A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)
Answer
To show: A = (A ∩ B) ∪ (A – B)
Let x ∈ A
We have to show that x ∈ (A ∩ B) ∪ (A – B)
Case I
x ∈ A ∩ B
Then, x ∈ (A ∩ B) ⊂ (A ∪ B) ∪ (A – B)
Case II
x ∉ A ∩ B
⇒ x ∉ A or x ∉ B
- x ∉ B [x ∉ A]
- x ∉ A – B ⊂ (A ∪ B) ∪ (A – B)
- A ⊂ (A ∩ B) ∪ (A – B) … (1) It is clear that
A ∩ B ⊂ A and (A – B) ⊂ A
- (A ∩ B) ∪ (A – B) ⊂ A … (2) From (1) and (2), we obtain A = (A ∩ B) ∪ (A – B)
To prove: A ∪ (B – A) ⊂ A ∪ B
Let x ∈ A ∪ (B – A)
- x ∈ A or x ∈ (B – A)
- x ∈ A or (x ∈ B and x ∉ A)
- (x ∈ A or x ∈ B) and (x ∈ A or x ∉ A)
- x ∈ (A ∪ B)
∴ A ∪ (B – A) ⊂ (A ∪ B) … (3)
Next, we show that (A ∪ B) ⊂ A ∪ (B – A).
Let y ∈ A ∪ B
- y ∈ A or y ∈ B
- (y ∈ A or y ∈ B) and (y ∈ A or y ∉ A)
- y ∈ A or (y ∈ B and y ∉ A)
- y ∈ A ∪ (B – A)
∴ A ∪ B ⊂ A ∪ (B – A) … (4)
Hence, from (3) and (4), we obtain A ∪ (B – A) = A ∪B.
Question 9:
Using properties of sets show that
(i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.
Answer
(i) To show: A ∪ (A ∩ B) = A We know that
A ⊂ A
A ∩ B ⊂ A
- A ∪ (A ∩ B) ⊂ A … (1) Also, A ⊂ A ∪ (A ∩ B) … (2)
- From (1) and (2), A ∪ (A ∩ B) = A (ii) To show: A ∩ (A ∪ B) = A
A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B) = A ∪ (A ∩ B)
= A {from (1)}
Question 10:
Show that A ∩ B = A ∩ C need not imply B = C.
Answer
Let A = {0, 1}, B = {0, 2, 3}, and C = {0, 4, 5}
Accordingly, A ∩ B = {0} and A ∩ C = {0}
Here, A ∩ B = A ∩ C = {0}
However, B ≠ C [2 ∈ B and 2 ∉ C]
Question 11:
Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.
(Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use distributive law) Answer
Let A and B be two sets such that A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X. To show: A = B
It can be seen that
A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X]
- (A ∩ B) ∪ (A ∩ X) [Distributive law]
- (A ∩ B) ∪ Φ [A ∩ X = Φ]
- A ∩ B … (1)
Now, B = B ∩ (B ∪ X)
- B ∩ (A ∪ X) [A ∪ X = B ∪ X]
- (B ∩ A) ∪ (B ∩ X) [Distributive law]
- (B ∩ A) ∪ Φ [B ∩ X = Φ]
- B ∩ A
- A ∩ B … (2)
Hence, from (1) and (2), we obtain A = B.
Question 12:
Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = Φ.
Answer
Let A = {0, 1}, B = {1, 2}, and C = {2, 0}.
Accordingly, A ∩ B = {1}, B ∩ C = {2}, and A ∩ C = {0}.
∴ A ∩ B, B ∩ C, and A ∩ C are non-empty. However, A ∩ B ∩ C = Φ
Question 13:
In a survey of 600 students in a school, 150 students were found to be taking tea and 225 taking coffee, 100 were taking both tea and coffee. Find how many students were taking neither tea nor coffee?
Answer
Let U be the set of all students who took part in the survey. Let T be the set of students taking tea.
Let C be the set of students taking coffee.
Accordingly, n(U) = 600, n(T) = 150, n(C) = 225, n(T ∩ C) = 100
To find: Number of student taking neither tea nor coffee i.e., we have to find n(T’ ∩ C’). n(T’ ∩ C’) = n(T ∪ C)’
- n(U) – n(T ∪ C)
- n(U) – [n(T) + n(C) – n(T ∩ C)]
- 600 – [150 + 225 – 100]
- 600 – 275
- 325
Hence, 325 students were taking neither tea nor coffee.
Question 14:
In a group of students 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?
Answer
Let U be the set of all students in the group.
Let E be the set of all students who know English. Let H be the set of all students who know Hindi.
∴ H ∪ E = U
Accordingly, n(H) = 100 and n(E) = 50
= 25
n(U) = n(H) + – n(H ∩ E)
- 100 + 50 – 25
- 125
Hence, there are 125 students in the group.
Question 15:
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I,11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
- the number of people who read at least one of the newspapers.
- the number of people who read exactly one newspaper. Answer
Let A be the set of people who read newspaper H. Let B be the set of people who read newspaper T. Let C be the set of people who read newspaper I. Accordingly, n(A) = 25, n(B) = 26, and n(C) = 26
n(A ∩ C) = 9, n(A ∩ B) = 11, and n(B ∩ C) = 8 n(A ∩ B ∩ C) = 3
Let U be the set of people who took part in the survey.
(i) Accordingly,
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)
- 25 + 26 + 26 – 11 – 8 – 9 + 3
- 52
Hence, 52 people read at least one of the newspapers.
(ii) Let a be the number of people who read newspapers H and T only.
Let b denote the number of people who read newspapers I and H only. Let c denote the number of people who read newspapers T and I only. Let d denote the number of people who read all three newspapers.
Accordingly, d = n(A ∩ B ∩ C) = 3 Now, n(A ∩ B) = a + d
n(B ∩ C) = c + d n(C ∩ A) = b + d
∴ a + d + c + d + b + d = 11 + 8 + 9 = 28
⇒ a + b + c + d = 28 – 2d = 28 – 6 = 22
Hence, (52 – 22) = 30 people read exactly one newspaper.
Question 16:
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.
Answer
Let A, B, and C be the set of people who like product A, product B, and product C respectively.
Accordingly, n(A) = 21, n(B) = 26, n(C) = 29, n(A ∩ B) = 14, n(C ∩ A) = 12, n(B ∩ C) = 14, n(A ∩ B ∩ C) = 8
The Venn diagram for the given problem can be drawn as
Class XI Chapter 1 – Sets Maths
It can be seen that number of people who like product C only is {29 – (4 + 8 + 6)} = 11
NCERT Solutions for Class XI Maths: Chapter 1 – Set
Exercise 1.1 Solutions
Exercise 1.2 Solutions
Exercise 1.3 Solutions
Exercise 1.4 Solutions
Exercise 1.5 Solutions
Exercise 1.6 Solutions
Miscellaneous Solutions
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