NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 11 - Sets

NCERT Solutions for Miscellaneous Exercise Chapter 1 Class 11 - Sets

Komal MiglaniUpdated on 05 May 2025, 04:27 PM IST

Have you ever tried to sort your wardrobe by colours or organise your playlist by mood? This is what the concept of sets teaches us in mathematics! It provides a systematic way to group and classify objects based on specific properties. From data science to probability, sets are used in every aspect of modern mathematics. The concepts like set notation, types of sets, Venn diagrams, union, intersection, complements, etc., are all going to be applied in this exercise.

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  1. Class 11 Maths Chapter 1 Sets Miscellaneous Exercise Solutions - Download PDF
  2. NCERT Solutions Class 11 Maths Chapter 1 Miscellaneous Exercise
  3. Topics covered in Chapter 1 Sets Miscellaneous Exercise
  4. NCERT Solutions of Class 11 Subject Wise
  5. Subject-Wise NCERT Exemplar Solutions

The NCERT Solutions for Chapter 1 Miscellaneous Exercise will help you break down each question in a simple and logical way. These NCERT solutions will improve your problem-solving speed and prepare you well for upcoming tests. Students can also access the NCERT notes for this chapter for feasible learning.


Class 11 Maths Chapter 1 Sets Miscellaneous Exercise Solutions - Download PDF

Download PDF


NCERT Solutions Class 11 Maths Chapter 1 Miscellaneous Exercise

Question 1: Decide, among the following sets, which sets are subsets of one another:

A = { x : x $\in$ R and x satisfy $x^{2}$ – 8x + 12 = 0 }, B = { 2, 4, 6 },

C = { 2, 4, 6, 8, . . . }, D = { 6 }.

Answer:

Solution of this equation are $x^{2}$ – 8x + 12 = 0

( x - 2 ) ( x - 6 ) = 0

X = 2,6

$\therefore$ A = { 2,6 }

B = { 2, 4, 6 }

C = { 2, 4, 6, 8, . . . }

D = { 6 }

From the sets given above, we can conclude that A $\subset$ B, A $\subset$ C, D $\subset$ A, D $\subset$ B, D $\subset$ C, B $\subset$ C

Hence, we can say that D $\subset$ A $\subset$ B $\subset$ C

Question 2: (i) 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 $\in$ A and A $\in$ B , then x $\in$ B

Answer:

The given statement is false,

example: Let A = { 2,4 }

B = { 1,{2,4},5}

x be 2.

then, 2 $\in$ { 2,4 } = x $\in$ A and { 2,4 } $\in$ { 1,{2,4},5} = A $\in$ B

But 2 $\notin$ { 1,{2,4},5} i.e. x $\notin$ B

Question 2: (ii) 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 A $\subset$ B and B $\in$ C , then A $\in$ C

Answer:

The given statement is false,

Let , A = {1}

B = { 1,2,3}

C = {0,{1,2,3},4}

Here, {1} $\subset$ { 1,2,3} = A $\subset$ B and { 1,2,3} $\in$ {0,{1,2,3},4} = B $\in$ C

But, {1} $\notin$ {0,{1,2,3},4} = A $\notin$ C

Question 2: (iii) 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 A $\subset$ B and B $\subset$ C , then A $\subset$C

Answer:

Let A ⊂ B and B ⊂ C

There is an element x such that

Let, x $\in$ A

$\Rightarrow$ x $\in$ B ( Because A $\subset$ B )

$\Rightarrow$ x $\in$ C ( Because B $\subset$ C )

Hence, the statement is true that A $\subset$ C

Question 2: (iv) 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 A $\not\subset$ B and B $\not\subset$ C , then A $\not\subset$ C

Answer:

The given statement is false

Let , A = {1,2}

B = {3,4,5 }

C = { 1,2,6,7,8}

Here, {1,2} $\not\subset$ {3,4,5 } = A $\not\subset$ B and {3,4,5 } $\not\subset$ { 1,2,6,7,8} = B $\not\subset$ C

But , {1,2} $\subset$ { 1,2,6,7,8} = A $\subset$ C

Question 2: (v) 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 $\in$ A and A $\not\subset$ B , then x $\in$ B

Answer:

The given statement is false,

Let x be 2

A = { 1,2,3}

B = { 4,5,6,7}

Here, 2 $\in$ { 1,2,3} = x $\in$ A and { 1,2,3} $\not \subset$ { 4,5,6,7} = A $\not \subset$B

But, 2$\notin$ { 4,5,6,7} implies x $\notin$ B

Question 2: (vi) 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 A $\subset$ B and x $\notin$ B , then x $\notin$ A

Answer:

The given statement is true,

Let, A $\subset$ B and x $\notin$ B

Suppose, x $\in$ A

Then, x $\in$ B , which is contradiction to x $\notin$ B

Hence, x $\notin$ A.

Question 3: Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C. Show that B = C.

Answer:

Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C

To prove: B = C.

A $\cup$ B = A + B - A $\cap$ B = A $\cup$ C = A + C - A $\cap$ C

A + B - A $\cap$ B = A + C - A $\cap$ C

B - A $\cap$ B = C - A $\cap$ C ( since A $\cap$ B = A $\cap$ C )

B = C

Hence proved that B = C.

Question 4: Show that the following four conditions are equivalent :

(i) A $\subset$ B(ii) A – B = $\phi$ (iii) A $\cup$ B = B (iv) A $\cap$ B = A

Answer:

First, we need to show A$\subset$ B $\Leftrightarrow$ A – B = $\phi$

Let A $\subset$ B

To prove: A – B = $\phi$

Suppose A – B $\neq$ $\phi$

this means, x $\in$ A and x $\not =$ B , which is not possible as A $\subset$ B .

SO, A – B = $\phi$.

Hence, A $\subset$ B $\implies$ A – B = $\phi$.

Now, let A – B = $\phi$

To prove: A $\subset$ B

Suppose, x $\in$ A

A – B = $\phi$ so x $\in$ B

Since, x $\in$ A and x $\in$ B and A – B = $\phi$ so A $\subset$ B

Hence, A$\subset$ B $\Leftrightarrow$ A – B = $\phi$.


Let A$\subset$ B

To prove : A $\cup$ B = B

We can say B $\subset$ A $\cup$ B

Suppose, x $\in$ A $\cup$ B

means x $\in$ A or x $\in$ B

If x $\in$ A

since A$\subset$ B so x $\in$ B

Hence, A $\cup$ B = B

and If x $\in$ B then also A $\cup$ B = B.


Now, let A $\cup$ B = B
To prove : A$\subset$ B

Suppose : x $\in$A

A $\subset$ A $\cup$ B so x $\in$ A $\cup$ B

A $\cup$ B = B so x $\in$ B

Hence,A$\subset$ B

ALSO, A$\subset$ B $\Leftrightarrow$ A $\cup$ B = B


NOW, we need to show A $\subset$ B $\Leftrightarrow$ A $\cap$ B = A

Let A $\subset$ B

To prove : A $\cap$ B = A

Suppose : x $\in$ A

We know A $\cap$ B $\subset$ A

x $\in$ A $\cap$ B Also ,A $\subset$ A $\cap$ B

Hence, A $\cap$ B = A


Let A $\cap$ B = A

To prove: A $\subset$ B

Suppose : x $\in$ A

x $\in$ A $\cap$ B ( replacing A by A $\cap$ B )

x $\in$ A and x $\in$ B

$\therefore$ A $\subset$ B

A $\subset$ B $\Leftrightarrow$ A $\cap$ B = A

Question 5: Show that if A $\subset$B, then C – B $\subset$ C – A.

Answer:

Given , A $\subset$ B

To prove: C – B $\subset$ C – A

Let, x $\in$ C - B means x$\in$ C but x$\notin$B

A $\subset$ B so x$\in$ C but x$\notin$A i.e. x $\in$ C - A

Hence, C – B $\subset$ C – A

Question 6: Show that for any sets A and B,

A = ( A $\cap$ B ) $\cup$ ( A – B ) and A $\cup$ ( B – A ) = ( A $\cup$ B )

Answer:

A = ( A $\cap$ B ) $\cup$ ( A – B )

L.H.S = A = Red coloured area

R.H.S = ( A $\cap$ B ) $\cup$ ( A – B )

( A $\cap$ B ) = green coloured

( A – B ) = yellow coloured

( A $\cap$ B ) $\cup$ ( A – B ) = coloured part

Hence, L.H.S = R.H.S = Coloured part


A $\cup$ ( B – A ) = ( A $\cup$ B )

A = sky blue coloured

( B – A )=pink coloured

L.H.S = A $\cup$ ( B – A ) = sky blue coloured + pink coloured

R.H.S = ( A $\cup$ B ) = brown coloured part

L.H.S = R.H.S = Coloured part

Question 7: (i) Using properties of sets, show that

A $\cup$ ( A $\cap$ B ) = A

Answer:

(i) A $\cup$ ( A $\cap$ B ) = A

We know that A $\subset$ A

and A $\cap$ B $\subset$ A

A $\cup$ ( A $\cap$ B ) $\subset$ A

and also , A $\subset$ A $\cup$ ( A $\cap$ B )

Hence, A $\cup$ ( A $\cap$ B ) = A

Question 7: (ii) Using properties of sets, show that

A $\cap$ ( A $\cup$ B ) = A

Answer:

This can be solved as follows

(ii) A $\cap$ ( A $\cup$ B ) = A

A $\cap$ ( A $\cup$ B ) = (A $\cap$ A) $\cup$ ( A $\cap$ B )

A $\cap$ ( A $\cup$ B ) = A $\cup$ ( A $\cap$ B ) { A $\cup$ ( A $\cap$ B ) = A proved in 7(i)}

A $\cap$ ( A $\cup$ B ) = A

Question 8: Show that A $\cap$ B = A $\cap$ C need not imply B = C.

Answer:

Let, A = {0,1,2}

B = {1,2,3}

C = {1,2,3,4,5}

Given, A $\cap$ B = A $\cap$ C

L.H.S : A $\cap$ B = {1,2}

R.H.S : A $\cap$ C = {1,2}

and here {1,2,3} $\not =$ {1,2,3,4,5} = B $\not =$ C.

Hence, A $\cap$ B = A $\cap$ C need not imply B = C.

Question 9: Let A and B be sets. If A $\cap$ X $=$B $\cap$ X $=$ $\phi$and A $\cup$ X $=$ B $\cup$ X for some set X, show that A $=$B.

Answer:

Given, A $\cap$ X $=$B $\cap$ X $=$ $\phi$ and A $\cup$ X $=$ B $\cup$ X

To prove: A = B

A = A $\cap$(A$\cup$X) (A $\cap$ X $=$B $\cap$ X)

= A $\cap$(B$\cup$X)

= (A$\cap$B) $\cup$ (A$\cap$X)

= (A$\cap$B) $\cup$ $\phi$ (A $\cap$ X $=$ $\phi$)

= (A$\cap$B)

B = B $\cap$(B$\cup$X) (A $\cap$ X $=$B $\cap$ X)

= B $\cap$(A$\cup$X)

= (B$\cap$A) $\cup$ (B$\cap$X)

= (B$\cap$A) $\cup$ $\phi$ (B $\cap$ X $=$ $\phi$)

= (B$\cap$A)

We know that (A$\cap$B) = (B$\cap$A) = A = B

Hence, A = B

Question 10: Find sets A, B and C such that A $\cap$ B, B $\cap$ C and A $\cap$ C are non-empty sets and A $\cap$ B $\cap$ C $=$ $\phi$

Answer:

Given, A $\cap$ B, B $\cap$ C and A $\cap$ C are non-empty sets

To prove : A $\cap$ B $\cap$ C $=$ $\phi$

Let A = {1,2}

B = {1,3}

C = {3,2}

Here, A $\cap$ B = {1}

B $\cap$ C = {3}

A $\cap$ C = {2}

and A $\cap$ B $\cap$ C $=$ $\phi$

Also read

Topics covered in Chapter 1 Sets Miscellaneous Exercise

1. Sets and set notation- A well-defined group of unique items is known as a set. They are expressed as $A=\{1,2,3\}$.

2. Venn diagrams- Venn diagrams are graphical representations of the relationships between various sets. They are drawn to understand union, intersection and complement.

3. Union and intersection of sets- The union of sets $A$ and $B$ is a set of all elements present in either $A$ or $B$ (or both). Only the components that are shared by both sets are in the intersection.

4. Set difference- The difference of two sets $A-B$ is the set of elements that are in $A$ but not in $B$.

5. Disjoint sets
When two sets have no elements in common, they are said to be disjoint; this indicates that the empty set is where they meet.

Also read

NCERT Solutions of Class 11 Subject Wise

Students can also access the NCERT solutions for other subjects and make their learning feasible.

Subject-Wise NCERT Exemplar Solutions

Use the links provided in the table below to get your hands on the NCERT exemplar solutions available for all the subjects.

Frequently Asked Questions (FAQs)

Q: How many exercises are there in CBSE Class 11 Maths chapter 1 ?
A:

There are a total of seven exercises including miscellaneous exercise in the CBSE Class 11 Maths chapter 1.

Q: What is the use of Venn's diagram ?
A:

Venn's diagram is useful in solving problems related to the union of sets, the intersection of sets. It is also useful in the chapters like relations and functions, probability.

Q: Find all the subsets of the set { –1, 0} ?
A:

φ, {–1}, {0}, {–1, 0} are the subsets of the given set.

Q: Find the power set of set A = {1,0} ?
A:

Power set of set A = { φ, {1}, {0}, {1, 0} }

Q: Does miscellaneous exercise is important for the CBSE exam ?
A:

As most of the questions in the CBSE exam are not asked from the miscellaneous exercise, it is not as important for CBSE exams but it is very important for the competitive exams.

Q: How many questions are given in the miscellaneous exercise sets ?
A:

There is a total of 16 questions and 7 solved examples given in the miscellaneous exercise sets.

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