The last exercise contains questions covering the entire topics present in the chapter. The Miscellaneous Exercise of NCERT Solutions for Class 11 Maths Chapter 1- Sets is based on the following topics:

- Sets and their Representations
- The Empty Set, Finite and Infinite Sets, Equal Sets
- Subsets, Power Set, Universal Set
- Venn Diagrams
- Operations on Sets
- The complement of a Set
- Practical Problems on Union and Intersection of Two Sets

The solutions provided here are based on the guidelines prescribed by the CBSE board. The NCERT Solutions for class 11 can help the students in preparing well for the Class 11 exams.

### Download PDF of NCERT Solutions for Class 11 Maths Chapter 1- Sets Miscellaneous Exercise

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### Solutions for Class 11 Maths Chapter 1- Sets Miscellaneous

**1. Decide, among the following sets, which sets are subsets of one and another: **

**A= {x: x âˆˆ R and x satisfy x ^{2} â€“ 8x + 12 = 0}, **

**B = {2, 4, 6},**

**C = {2, 4, 6, 8â€¦}, **

**D = {6}.**

**Solution:**

According to the question,

We have,

A = {x: x âˆˆ R and x satisfies x^{2} â€“ 8x + 12 =0}

2 and 6 are the only solutions of x^{2} â€“ 8x + 12 = 0.

Hence, A = {2, 6}

B = {2, 4, 6}, C = {2, 4, 6, 8 â€¦}, D = {6}

Hence, D âŠ‚ A âŠ‚ B âŠ‚ C

Hence, A âŠ‚ B, A âŠ‚ C, B âŠ‚ C, D âŠ‚ A, D âŠ‚ B, D âŠ‚ C

**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.**

**(i) If x âˆˆ A and A âˆˆ B, then x âˆˆ B**

**(ii) If A âŠ‚ B and B âˆˆ C, then A âˆˆ C**

**(iii) If A âŠ‚ B and B âŠ‚ C, then A âŠ‚ C**

**(iv) If A âŠ„ B and B âŠ„ C, then A âŠ„ C**

**(v) If x âˆˆ A and A âŠ„ B, then x âˆˆ B**

**(vi) If A âŠ‚ B and x âˆ‰ B, then x âˆ‰ A**

**Solution:**

(i) False

According to the question,

A = {1, 2} and B = {1, {1, 2}, {3}}

Now, we have,

2 âˆˆ {1, 2} and {1, 2} âˆˆ {1, {1, 2}, {3}}

Hence, we get,

A âˆˆ B

We also know,

{2} âˆ‰ {1, {1, 2}, {3}}

(ii) False

According to the question

Let us assume that,

A {2}

B = {0, 2}

And, C = {1, {0, 2}, 3}

From the question,

A âŠ‚ B

Hence,

B âˆˆ C

But, we know,

A âˆ‰ C

(iii) True

According to the question

A âŠ‚ B and B âŠ‚ C

Let us assume that,

x âˆˆ A

Then, we have,

x âˆˆ B

And,

x âˆˆ C

Therefore,

A âŠ‚ C

(iv) False

According to the question

AÂ âŠ„Â B

Also,

BÂ âŠ„Â C

Let us assume that,

A = {1, 2}

B = {0, 6, 8}

And,

C = {0, 1, 2, 6, 9}

âˆ´Â A âŠ‚ C

(v) False

According to the question,

x âˆˆ A

Also,

AÂ âŠ„Â B

Let us assume that,

A = {3, 5, 7}

Also,

B = {3, 4, 6}

We know that,

AÂ âŠ„Â B

âˆ´ 5 âˆ‰ B

(vi) True

According to the question,

AÂ âŠ‚Â B

Also,

xÂ âˆ‰Â B

Let us assume that,

x âˆˆ A,

We have,

x âˆˆ B,

From the question,

We have, xÂ âˆ‰Â B

âˆ´Â xÂ âˆ‰Â A

**3. Let A, B and C be the sets such that A âˆª B = A âˆª C and A âˆ© B = A âˆ© C. show that B = C.**

**Solution:**

According to the question,

A âˆª B = A âˆª C

And,

A âˆ© B = A âˆ© C

To show,

B = C

Let us assume,

x âˆˆ B

So,

x âˆˆ A âˆª B

x âˆˆ A âˆª C

Hence,

x âˆˆ A or x âˆˆ C

When x âˆˆ A, then,

x âˆˆ B

âˆ´ x âˆˆ A âˆ© B

As, A âˆ© BÂ = A âˆ© C

So,Â x âˆˆ A âˆ© C

âˆ´Â x âˆˆ A or x âˆˆ C

x âˆˆ C

âˆ´ B âŠ‚Â C

Similarly, it can be shown that CÂ âŠ‚Â B

Hence, B = C

**4. Show that the following four conditions are equivalent:**

**(i) A âŠ‚ B (ii) A â€“ B = Î¦**

**(iii) A âˆª B = B (iv) A âˆ© B = A**

**Solution:**

According to the question,

To prove, (i)Â â¬ŒÂ (ii)

Here, (i) = AÂ âŠ‚Â B and (ii) = A â€“ B â‰ Â Ï•

Let us assume that AÂ âŠ‚Â B

To prove, A â€“ B â‰ Â Ï•

Let A â€“ B â‰ Â Ï•

Hence, there exists XÂ âˆˆÂ A, X â‰ B, but since AâŠ‚Â B, it is not possible

âˆ´Â A â€“ B =Â Ï•

And AâŠ‚Â B â‡’ A â€“ B â‰ Â Ï•

Let us assume thatÂ A â€“ B â‰ Â Ï•

To prove: AÂ âŠ‚Â B

Let XâˆˆÂ A

So, XÂ âˆˆÂ B (if XÂ âˆ‰Â B, then A â€“ B â‰ Â Ï•)

Hence, A â€“ B =Â Ï•Â => AÂ âŠ‚Â B

âˆ´(i)Â â¬ŒÂ (ii)

Let us assume thatÂ AÂ âŠ‚Â B

To prove, AÂ âˆªÂ B = B

â‡’Â BÂ âŠ‚Â AÂ âˆªÂ B

Let us assume that, xÂ âˆˆÂ AâˆªÂ B

â‡’Â XÂ âˆˆÂ A or XÂ âˆˆÂ B

Taking Case I: XÂ âˆˆÂ B

AÂ âˆªÂ B = B

Taking Case II: XÂ âˆˆÂ A

â‡’Â XÂ âˆˆÂ B (AÂ âŠ‚Â B)

â‡’Â AÂ âˆªÂ BÂ âŠ‚Â B

Let AÂ âˆªÂ B = B

Let us assume that XÂ âˆˆÂ A

â‡’Â XÂ âˆˆÂ AÂ âˆªÂ B (AÂ âŠ‚Â AÂ âˆªÂ B)

â‡’Â XÂ âˆˆÂ B (AÂ âˆªÂ B = B)

âˆ´AâŠ‚Â B

Hence, (i)Â â¬ŒÂ (iii)

To prove (i)Â â¬ŒÂ (iv)

Let us assume that AÂ âŠ‚Â B

AÂ âˆ©Â BÂ âŠ‚Â A

Let XÂ âˆˆÂ A

To prove, XÂ âˆˆÂ Aâˆ©Â B

Since, AÂ âŠ‚Â B and XÂ âˆˆÂ B

Hence, XÂ âˆˆÂ A âˆ©Â B

â‡’Â AÂ âŠ‚Â AÂ âˆ©Â B

â‡’Â A = AÂ âˆ©Â B

Let us assume that AÂ âˆ©Â B = A

Let XÂ âˆˆÂ A

â‡’Â XÂ âˆˆÂ AÂ âˆ©Â B

â‡’Â XÂ âˆˆÂ B and XÂ âˆˆÂ A

â‡’Â AÂ âŠ‚Â B

âˆ´Â (i)Â â¬ŒÂ (iv)

âˆ´ (i) â¬ŒÂ (ii) â¬ŒÂ (iii) â¬ŒÂ (iv)

Hence, proved

**5. Show that if AÂ âŠ‚Â B, then C â€“ BÂ âŠ‚Â C â€“ A.**

**Solution:**

To show,

C â€“ BÂ âŠ‚Â C â€“ A

According to the question,

Let us assume that x is any element such that XÂ âˆˆÂ C â€“ B

âˆ´ xÂ âˆˆÂ C and x âˆ‰ B

Since, AÂ âŠ‚Â B, we have,

âˆ´ xÂ âˆˆÂ C and x âˆ‰ A

So, xÂ âˆˆÂ C â€“ A

âˆ´Â C â€“ BÂ âŠ‚Â C â€“ A

Hence, Proved.

**6. Assume that P (A) = P (B). Show that A = B**

**Solution:**

To show,

A = B

According to the question,

P (A) = P (B)

Let x be any element of set A,

xÂ âˆˆÂ A

Since, P (A) is the power set of set A, it has all the subsets of set A.

AÂ âˆˆÂ P (A) = P (B)

Let C be an element of set B

For any C âˆˆÂ P (B),

We have, x âˆˆÂ C

CÂ âŠ‚Â B

âˆ´ xÂ âˆˆÂ B

âˆ´Â AÂ âŠ‚Â B

Similarly, we have:

BÂ âŠ‚Â A

SO, we get,

IfÂ AÂ âŠ‚Â B andÂ B âŠ‚ A

âˆ´Â A = B

**7. Is it true that for any sets A and B, P (A)Â âˆªÂ P (B) = P (AÂ âˆªÂ B)? Justify your answer.**

**Solution:**

It is not true that for any sets A and B, P (A)Â âˆªÂ P (B) = P (AÂ âˆªÂ B)

Justification:

Let us assume,

A = {0, 1}

And, B = {1, 2}

âˆ´ AÂ âˆªÂ BÂ = {0, 1, 2}

According to the question,

We have,

P (A) = {Ï•, {0}, {1}, {0, 1}}

P (B) = {Ï•, {1}, {2}, {1, 2}}

âˆ´ P (A âˆª B) = {Ï•, {0}, {1}, {2}, {0, 1}, {1, 2}, {0, 2}, {0, 1, 2}}

Also,

P (A) âˆª P (B) = {Ï•, {0}, {1}, {2}, {0, 1}, {1, 2}}

âˆ´Â P (A) âˆª P (B â‰ P (A âˆª B)

Hence, the given statement is false

**8. Show that for any sets A and B,
A = (AÂ âˆ©Â B)Â âˆªÂ (A â€“ B) and AÂ âˆªÂ (B â€“ A) = (AÂ âˆªÂ B)**

**Solution:**

To Prove,

A = (AÂ âˆ©Â B)Â âˆªÂ (A â€“ B)

Proof: Let xÂ âˆˆÂ A

To show,

XÂ âˆˆÂ (AÂ âˆ©Â B)Â âˆªÂ (A â€“ B)

In Case I,

XÂ âˆˆÂ (A âˆ©Â B)

â‡’Â XÂ âˆˆÂ (AÂ âˆ©Â B)Â âŠ‚Â (AÂ âˆªÂ B)Â âˆªÂ (A â€“ B)

In Case II,

XÂ âˆ‰AÂ âˆ©Â B

â‡’Â XÂ âˆ‰Â B or XÂ âˆ‰Â A

â‡’Â XÂ âˆ‰Â B (XÂ âˆ‰Â A)

â‡’Â XÂ âˆ‰Â A â€“ BÂ âŠ‚Â (AÂ âˆªÂ B)Â âˆªÂ (A â€“ B)

âˆ´AÂ âŠ‚Â (AÂ âˆ©Â B)Â âˆªÂ (A â€“ B) (i)

It can be concluded that, AÂ âˆ©Â BÂ âŠ‚Â A and (A â€“ B)Â âŠ‚Â A

Thus, (AÂ âˆ©Â B)Â âˆªÂ (A â€“ B)Â âŠ‚Â A (ii)

Equating (i) and (ii),

A = (AÂ âˆ©Â B)Â âˆªÂ (A â€“ B)

We also have to show,

AÂ âˆªÂ (B â€“ A)Â âŠ‚Â AÂ âˆªÂ B

Let us assume,

XÂ âˆˆÂ AÂ âˆªÂ (B â€“ A)

XÂ âˆˆÂ A or XÂ âˆˆÂ (B â€“ A)

â‡’Â XÂ âˆˆÂ A or (XÂ âˆˆÂ B and XÂ âˆ‰A)

â‡’Â (XÂ âˆˆÂ A or XÂ âˆˆÂ B) and (XÂ âˆˆÂ A and XÂ âˆ‰A)

â‡’Â XÂ âˆˆÂ (BÂ âˆªA)

âˆ´Â AÂ âˆªÂ (B â€“ A)Â âŠ‚Â (AÂ âˆªÂ B) (iii)

According to the question,

To prove:

(AÂ âˆªÂ B)Â âŠ‚Â AÂ âˆª (B â€“ A)

Let yÂ âˆˆÂ AâˆªB

Y âˆˆÂ A or yÂ âˆˆÂ B

(yÂ âˆˆÂ A or yÂ âˆˆÂ B) and (XÂ âˆˆÂ A and XÂ âˆ‰A)

â‡’Â yÂ âˆˆÂ A or (yÂ âˆˆÂ B and yÂ âˆ‰A)

â‡’Â yÂ âˆˆÂ AÂ âˆªÂ (B â€“ A)

Thus, AÂ âˆªÂ BÂ âŠ‚Â AÂ âˆªÂ (B â€“ A) (iv)

âˆ´ From equations (iii) and (iv), we get:

AÂ âˆªÂ (B â€“ A) = AÂ âˆªÂ B

**9. Using properties of sets, show that:
(i) AÂ âˆªÂ (AÂ âˆ©Â B) = A
(ii) AÂ âˆ©Â (AÂ âˆªÂ B) = A.**

**Solution:**

(i) To show: A âˆª (A âˆ© B) = A

We know that,

A âŠ‚Â A

A âˆ© B âŠ‚Â A

âˆ´Â A âˆª (A âˆ© B) âŠ‚Â AÂ (i)

Also, according to the question,

We have:

AâŠ‚Â A âˆª (A âˆ© B)Â (ii)

Hence, from equation (i) and (ii)

We have:

A âˆª (A âˆ© B)Â = A

(ii) To show,

A âˆ© (A âˆª B) = A

A âˆ© (A âˆª B) = (A âˆ© A) âˆª (A âˆ© B)

= A âˆª (A âˆ© B)

= A

**10. Show that AÂ âˆ©Â B = AÂ âˆ©Â C need not imply B = C.**

**Solution:**

Let us assume,

A = {0, 1}

B = {0, 2, 3}

And, C = {0, 4, 5}

According to the question,

A âˆ© B = {0}

And,

A âˆ© C = {0}

âˆ´ A âˆ© B = A âˆ© C = {0}

But,

2Â âˆˆ B and 2 âˆ‰ C

Therefore, B â‰ C

**11. Let A and B be sets. If AÂ âˆ©Â X = BÂ âˆ©Â X = Ï• and AÂ âˆªÂ X = BÂ âˆªÂ X for some set X, show that A = B.
(Hints A = AÂ âˆ©Â (AÂ âˆªÂ X) , B = BÂ âˆ©Â (BÂ âˆªÂ X) and use Distributive law)**

**Solution:**

According to the question,

Let A and B be two sets such that A âˆ© X = B âˆ© X = Ï• and A âˆª X = B âˆª X for some set X.

To show, A = B

Proof:

A = A âˆ© (A âˆª X) = A âˆ© (B âˆª X) [A âˆª X = B âˆª X]

= (A âˆ© B) âˆª (A âˆ© X) [Distributive law]

= (A âˆ© B) âˆª Î¦ [A âˆ© X = Î¦]

= A âˆ© B (i)

Now, B = B âˆ© (B âˆª X)

= B âˆ© (A âˆª X) [A âˆª X = B âˆª X]

= (B âˆ© A) âˆª (B âˆ© X) â€¦ [Distributive law]

= (B âˆ© A) âˆª Î¦ [B âˆ© X = Î¦]

= A âˆ© B (i)

Hence, from equations (i) and (ii), we obtain A = B.

**12. Find sets A, B and C such that AÂ âˆ©Â B, BÂ âˆ©Â C and AÂ âˆ©Â C are non-empty sets and AÂ âˆ©Â BÂ âˆ©Â C = Î¦.
Solution:**

Let us assume, A {0, 1}

B = {1, 2}

And, C = {2, 0}

According to the question,

A âˆ© B = {1}

B âˆ© C = {2}

And,

A âˆ© C = {0}

âˆ´ A âˆ© B, B âˆ© C and A âˆ© C are not empty sets

Hence, we get,

A âˆ© B âˆ© C = Î¦

**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.**

**Solution:**

Let us assume that,

U = the set of all students who took part in the survey

T = the set of students taking tea

C = the set of the students taking coffee

Total number of students in a school, n (U) = 600

Number of students taking tea, n (T) = 150

Number of students taking coffee, n (C) = 225

Also,Â n (T âˆ© C) = 100

Now, we have to find that number of students taking neither coffee nor tea i.e. n (T âˆ© Câ€™)

âˆ´ According to the question,

n ( T âˆ© Câ€™ )= n( T âˆ© C )â€™

= n (U) â€“ n (T âˆ© C)

= n (U) â€“ [n (T) + n(C) â€“ n (T âˆ© C)]

= 600 â€“ [150 + 225 â€“ 100]

= 600 â€“ 275

= 325

âˆ´Â Number of students taking neither coffee nor tea = 325 students

**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?**

**Solution:**

Let us assume that,

U = the set of all students in the group

E = the set of students who know English

H = the set of the students who know Hindi

âˆ´ H âˆª E = U

Given that,

Number of students who know Hindi n (H) = 100

Number of students who knew English, n (E) = 50

Number of students who know both, n (H âˆ© E)Â = 25

We have to find the total number of students in the group i.e. n (U)

âˆ´ According to the question,

n (U) = n(H) + n(E) â€“ n(H âˆ© E)

= 100 + 50 â€“ 25

= 125

âˆ´Â Total number of students in the group = 125 students

**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: **

(i) The number of people who read at least one of the newspapers.

(ii) The number of people who read exactly one newspaper.

**Solution:**

(i)Â Let us assume that,

A = the set of people who read newspaper H

B = the set of people who read newspaper T

C = the set of people who read newspaper I

According to the question,

Number of people who read newspaper H, n (A) = 25

Number of people who read newspaper T, n (B) = 26

Number of people who read the newspaper I, n (C) = 26

Number of people who read both newspaper H and I, n (A âˆ© C) = 9

Number of people who read both newspaper H and T,Â n (A âˆ© B) = 11

Number of people who read both newspaper T and I, n (B âˆ© C) = 8

And,Â Number of people who read all three newspaper H, T and I, n (A âˆ© B âˆ© C) = 3

Now, we have to find the number of people who read at least one of the newspaper

âˆ´, we get.

= 25 + 26 + 26 â€“ 11 â€“ 8 â€“ 9 + 3

= 80 â€“ 28

= 52

âˆ´Â There are a total of 52 students who read at least one newspaper.

(ii)Â Let us assume that,

a = the number of people who read newspapers H and T only

b = the number of people who read newspapers I and H only

c = the number of people who read newspapers T and I only

d = the number of people who read all three newspapers

According to the question,

D = n(A âˆ© B âˆ© C) = 3

Now, we have:

n(A âˆ© B) = a + d

n(B âˆ© C) = c + d

And,

n(C âˆ© A) = b + d

âˆ´Â a + d + c +d + b + d = 11 + 8 + 9

a + b + c + d = 28 â€“ 2d

= 28 â€“ 6

= 22

âˆ´Â Number of people read exactly one newspaper = 52 â€“ 22

= 30 people

**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.**

**Solution:**

Let A, B and C = the set of people who like product A, product B and product C respectively.

Now, according to the question,

Number of students who like product A, n (A) = 21

Number of students who like product B, n (B) = 26

Number of students who like product C, n (C) = 29

Number of students who like both products A and B, n (A âˆ© B) = 14

Number of students who like both products A and C, n(C âˆ© A) = 12

Number of students who like both product C and B, n (B âˆ© C) = 14

Number of students who like all three product, n (A âˆ© B âˆ© C) = 8

From the Venn diagram, we get,

Number of students who only like product C = {29 â€“ (4 + 8 + 6)}

= {29 â€“ 18}

= 11 students

### Access other exercise solutions of Class 11 Maths Chapter 1- Sets

Exercise 1.1 Solutions 6 Questions

Exercise 1.2 Solutions 6 Questions

Exercise 1.3 Solutions 9 Questions

Exercise 1.4 Solutions 12 Questions

Exercise 1.5 Solutions 7 Questions

Exercise 1.6 Solutions 8 Questions