NCERT Solutions for Class 11 Maths Chapter 1: Sets - Exercise 1.5

1. Let $U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$ and $C=\left\{ 3,4,5,6 \right\}$, find 

i. $A'$

Ans: Given that,

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$, $C=\left\{ 3,4,5,6 \right\}$

To find,

$A'$

The complement of set A is the set of all elements of U which are not the elements of A.

$A'=U-A$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 1,2,3,4 \right\}$

$=\left\{ 5,6,7,8,9 \right\}$

$\therefore A'=\left\{ 5,6,7,8,9 \right\}$

ii. $B'$

Ans: Given that,

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$, $C=\left\{ 3,4,5,6 \right\}$

To find,

$B'$

The complement of set A is the set of all elements of U which are not the elements of A.

$B'=U-B$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 2,4,6,8 \right\}$

$=\left\{ 1,3,5,7,9 \right\}$

$\therefore B'=\left\{ 1,3,5,7,9 \right\}$

iii. $\left( A\cup C \right)'$

Ans: Given that,

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$, $C=\left\{ 3,4,5,6 \right\}$

To find,

$\left( A\cup C \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

$A\cup C=\left\{ 1,2,3,4,5,6 \right\}$

$\left( A\cup C \right)'=U-\left( A\cup C \right)$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 1,2,3,4,5,6 \right\}$

$=\left\{ 7,8,9 \right\}$

$\therefore \left( A\cup C \right)'=\left\{ 7,8,9 \right\}$

iv. $\left( A\cup B \right)'$

Ans: Given that,

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$, $C=\left\{ 3,4,5,6 \right\}$

To find,

$\left( A\cup B \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

$A\cup B=\left\{ 1,2,3,4,6,8 \right\}$

$\left( A\cup B \right)'=U-A\cup B$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 1,2,3,4

,6,8 \right\}$

$=\left\{ 5,7,9 \right\}$

$\therefore \left( A\cup B \right)'=\left\{ 5,7,9 \right\}$

v. $\left( A' \right)'$

Ans: Given that,

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$, $C=\left\{ 3,4,5,6 \right\}$

To find,

$\left( A' \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

$\left( A' \right)'=A$

$=\left\{ 1,2,3,4 \right\}$

$\therefore \left( A' \right)'=\left\{ 1,2,3,4 \right\}$

vi. $\left( B-C \right)'$

Ans: Given that,

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 1,2,3,4 \right\},B=\left\{ 2,4,6,8 \right\}$, $C=\left\{ 3,4,5,6 \right\}$

To find,

$\left( B-C \right)'$

The complement of set A is the set of all elements of U which are not the elements of A.

$B-C=\left\{ 2,8 \right\}$

$\left( B-C \right)'=U-\left( B-C \right)$

$=\left\{ 1,2,3,4,5,6,7,8,9 \right\}-\left\{ 2,8 \right\}$

$=\left\{ 1,3,4,5,6,7,9 \right\}$

$\therefore \left( B-C \right)'=\left\{ 1,3,4,5,6,7,9 \right\}$

2. If $U=\left\{ a,b,c,d,e,f,g,h \right\}$, then find the complements of the following sets:

i. $A=\left\{ a,b,c \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$A=\left\{ a,b,c \right\}$

To find the complement of A

The complement of set A is the set of all elements of U which are not the elements of A.

$A'=U-A$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ a,b,c \right\}$

$=\left\{ d,e,f,g,h \right\}$

$\therefore $The complement of $A$ is $A'=\left\{ d,e,f,g,h \right\}$

ii. $B=\left\{ d,e,f,g \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$b=\left\{ d,e,f,g \right\}$

To find the complement of B

The complement of set A is the set of all elements of U which are not the elements of A.

$B'=U-B$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ d,e,f,g \right\}$

$=\left\{ a,b,c,h \right\}$

$\therefore $The complement of $B$ is $B'=\left\{ b,e,c,h \right\}$

iii. $C=\left\{ a,c,e,g \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$C=\left\{ a,c,e,g \right\}$

To find the complement of A

The complement of set A is the set of all elements of U which are not the elements of A.

$C'=U-C$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ a,c,e,g \right\}$

$=\left\{ b,d,f,h \right\}$

$\therefore $The complement of $C$ is $C'=\left\{ b,d,f,h \right\}$

iv. $D=\left\{ f,g,h,a \right\}$

Ans: Given that,

$U=\left\{ a,b,c,d,e,f,g,h \right\}$

$A=\left\{ f,g,h,a \right\}$

To find the complement of A

The complement of set A is the set of all elements of U which are not the elements of A.

$D'=U-D$

$=\left\{ a,b,c,d,e,f,g,h \right\}-\left\{ f,g,h,a \right\}$

$=\left\{ b,c,d,e \right\}$

$\therefore $The complement of $D$ is $D'=\left\{ b,c,d,e \right\}$

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:

i. $\left\{ x:x\text{ is an even natural number} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of even natural number

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is an even natural number} \right\}'=\left\{ x:x\text{ is an odd natural number} \right\}$

ii. $\left\{ x:x\text{ is an odd natural number} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of odd natural number

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is an odd natural number} \right\}'=\left\{ x:x\text{ is an even natural number} \right\}$

iii. $\left\{ x:x\text{ is a positive multiple of 3} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of positive multiples of $3$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a positive multiple of 3} \right\}'=\left\{ x:x\in N\text{ and x is not a positive multiple of 3} \right\}$

iv. $\left\{ x:x\text{ is a prime number} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of prime number

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a prime number} \right\}'=\left\{ x:x\text{ is a positive composite number and x=1} \right\}$

v. $\left\{ x:x\text{ is a natural number divisible by 3 and 5} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of natural number divisible by $3$ and $5$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a number divisible by 3 and 5} \right\}'=$

$\left\{ x:x\text{ is a natural number that is not divisible by 3 or 5} \right\}$

vi. $\left\{ x:x\text{ is a perfect square} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of perfect squares.

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a perfect squares} \right\}'=\left\{ x:x\in N\text{ and x is not a perfect square} \right\}$

vii. $\left\{ x:x\text{ is a perfect cube} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the set of perfect cube

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x\text{ is a perfect cube} \right\}'=\left\{ x:x\in N\text{ and x is not a perfect cube} \right\}$

viii. $\left\{ x:x+5=8 \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of $\left\{ x:x+5=8 \right\}$

$x+5=8$

$x=3$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore \left\{ x:x+5=8 \right\}'=\left\{ x:x\in N\text{ and x}\ne \text{3} \right\}$

ix. $\left\{ x:2x+5=9 \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the 

$\left\{ x:2x+5=9 \right\}$

The complement of set A is the set of all elements of U which are not the elements of A.

$2x+5=9$

$2x=4$

$x=2$

$\therefore \left\{ x:2x+5=9 \right\}'=\left\{ x:x\in N\text{ and x}\ne \text{2 } \right\}$

x. $\left\{ x:x\ge 7 \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of 

$\left\{ x:x\ge 7 \right\}$

The complement of set A is the set of all elements of U which are not the elements of A.

$\therefore {{\left\{ x:x\ge 7 \right\}}^{\prime }}=\left\{ x:x\in N\text{ and x}<\text{7} \right\}$

xi. $\left\{ x:x\in N\text{ and 2x+1 > 10} \right\}$

Ans: Given that,

The set of natural number is the universal set

To find the complement of the 

$\left\{ x:x\in N\text{ and 2x+1 > 10} \right\}$

The complement of set A is the set of all elements of U which are not the elements of A.

$2x+1>10$

$2x>9$

$x>\frac{9}{2}$

$\therefore \left\{ x:x\in N\text{ and 2x+1 > 10} \right\}'$$=\left\{ x:x\in N\text{ and x}\le \frac{9}{2} \right\}$

4. If $U=\left\{ 1,2,3,4,5,6,7,8,9 \right\},A=\left\{ 2,4,6,8 \right\}$ and $B=\left\{ 2,3,5,7 \right\}$. Verify that,

i. $\left( A\cup B \right)'=A'\cap B'$

Ans: Given that.

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\}$

$A=\left\{ 2,4,6,8 \right\}$

$B=\left\{ 2,3,5,7 \right\}$

To prove that $\left( A\cup B \right)'=A'\cap B'$

$A\cup B=\left\{ 2,4,6,8 \right\}\cup \left\{ 2,3,5,7 \right\}$

$=\left\{ 2,3,4,5,6,7,8 \right\}$

$\left( A\cup B \right)'=U=A\cup B$

$=\left\{ 1,9 \right\}$

$A'=U-A$

$=\left\{ 1,3,5,7,9 \right\}$

$B'=U-B$

$=\left\{ 1,4,6,8,9 \right\}$

$A'\cap B'=\left\{ 1,3,5,7,9 \right\}\cap \left\{ 1,4,6,8,9 \right\}$

$=\left\{ 1,9 \right\}$

Hence it has been proved that $\left( A\cup B \right)'=A'\cap B'$

ii. $\left( A\cap B \right)'=A'\cup B'$

Ans: Given that.

$U=\left\{ 1,2,3,4,5,6,7,8,9 \right\}$

$A=\left\{ 2,4,6,8 \right\}$

$B=\left\{ 2,3,5,7 \right\}$

To prove that $\left( A\cap B \right)'=A'\cup B'$

$A\cap B=\left\{ 2,4,6,8 \right\}\cap \left\{ 2,3,5,7 \right\}$

$=\left\{ 2 \right\}$

$\left( A\cap B \right)'=U-A\cap B$

$=\left\{ 1,3,4,5,6,7,8,9 \right\}$

$A'=U-A$

$=\left\{ 1,3,5,7,9 \right\}$

$B'=U-B$

$=\left\{ 1,4,6,8,9 \right\}$

$A'\cup B'=\left\{ 1,3,5,7,9 \right\}\cup \left\{ 1,4,6,8,9 \right\}$

$=\left\{ 1,3,4,5,6,7,8,9 \right\}$

Hence it has been proved that $\left( A\cap B \right)'=A'\cup B'$

5. Draw appropriate Venn diagrams for each of the following:

i. $\left( A\cup B \right)'$

Ans: To draw the Venn diagram for $\left( A\cup B \right)'$

(Image Will Be Updated Soon)

ii. $A'\cap B'$

Ans: To draw the Venn diagram for $A'\cap B'$

(Image Will Be Updated Soon)

iii. $\left( A\cap B \right)'$

Ans: To draw the Venn diagram for $\left( A\cap B \right)'$

(Image Will Be Updated Soon)

iv. $A'\cup B'$

Ans: To draw the Venn diagram for $A'\cup B'$

(Image Will Be Updated Soon)

6. Let $U$ be the set of all triangles in a plane. If $A$ is the set of all triangles with at least one angle different from $60{}^\circ $, what is $A'$?

Ans: Given that,

$U$ is the set of all triangles in the plane

$A=$Set of triangles different form $60{}^\circ $

To find $A'$

The complement of set A is the set of all elements of U which are not the elements of A.

$A'=U-A$

$=$Set of all equilateral triangles

$\therefore A'$ is the set of all equilateral triangles

7. Fill in the blanks to make each of the following a true statement:

i. $A\cup A'=...$

Ans: To fill the blanks given in the statement

The union of the set and its complement is the universal set

$\therefore A\cup A'=U$

ii. $\varnothing '\cap A=...$

Ans: To fill the blanks given in the statement

We know that,

$\varnothing '\cap A=U\cap A=A$

$\therefore \varnothing '\cap A=A$

iii. $A\cap A'=...$

Ans: To fill the blanks given in the statement

The intersection of the set and its complement is an empty set.

$\therefore A\cap A'=\varnothing $

iv. $U'\cap A=...$

Ans: To fill the blanks given in the statement

We know that,

$\varnothing \cap A=U'\cap A=\varnothing $

$\therefore U'\cap A=\varnothing $

Class 11 Maths NCERT Solutions Chapter 1 – An Overview of Sets

Sets are one of the most crucial as well as fundamental concepts of Mathematics and have found their application across all the branches of the subject. With the help of fundamentals of sets, the concept of functions and relations can be defined easily. Also, it has found its application in the study of geometry, sequences and several other concepts.

In this particular chapter, you will learn about new definitions of sets and will also become familiar with its various operations.

To elaborate, this chapter touches upon concepts as follows.

• Introduction to the concept and functionality of sets.

• Representation of sets.

• The empty set.

• Finite and Infinite sets.

• Equal sets.

• Subsets of a set of real numbers.

• Intervals as subsets of R.

• Power set as a universal set.

• Venn diagram.

• Operations on sets.

• Differences in sets.

• A complement of a set.

• Union and intersection of two sets.

A fair idea of these concepts will not only help you to apply them as and when required to solve sums but will also prove useful in higher classes. Also, make it a point to get all your doubts cleared in advance before you delve into related chapters in details.

To facilitate the same, you can use our study materials like NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5. It will help you to pick up shortcut techniques for solving sums and will also help you get your chapter-related doubts cleared effectively.

Our exercise-based study guides like Class 11 Maths Ex 1.5 Solutions provide a suitable explanation for each answer. It also helps to obtain a fair understanding of the step by step approach that goes into solving each question of this chapter. Now, take a look at the questions included in our NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 to have a proper understanding of how our study material can help you score better grades.

NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 – All Questions

Let us take an example of Ex. 1.5 Class 11 Maths and figure out how our updated NCERT Solutions for Class 11 Maths Chapter 1 benefits your revisions for exam preparation. It comprises a total of 7 questions, each of which is based on the essential concepts of Sets.

• Exercise 1.5 Class 11 Maths – Question 1

To begin with, the first question of this exercise provides you with four sets, namely U, A, B and C. The objective of the question is to test your knowledge of sets and its operation. For example, the first sub-part of the question requires you to find the complement of set A. Similarly; you are to provide the complement of the union of set A and C in the third subpart of this question. 

To solve these questions effectively, you would first need to learn how to find the complement of a given set. Also, an understanding of how to find the union of two separate sets will give you an upper hand for solving such questions accurately in exams. With a quick reference of our Class 11 Maths NCERT Solutions Chapter 1, you will learn not just how to solve such sums but will pick up smart techniques of approaching them. 

Additionally, our NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 provide essential insight into the steps that should be applied to solve sums like section (iv.) of question 1. The said question requires you to find the complement of a given set (A).

While going through our solutions, you will find out the step by step method used to find the complement of a given set (A’). It will further show how to find out the complement of the new set (A’)’.

Such useful insight will allow you to solve similar problems in this exercise with more confidence, which will directly benefit your exam preparation. Regardless, make it a point to go for multiple revisions of this chapter so that the various methods and laws that come in handy while solving its sums are instilled in your mind.

Subsequently, include our Class 11 Maths Exercise 1.5 Solutions into your revision plan to take your exam preparation to the next level. Such inclusion will prove immensely beneficial and will help you gain the confidence you need to ace your upcoming exams.

• Class 11 Maths NCERT Solutions Chapter 1 Exercise 1.5 – Question 2

Based on a similar concept and approach as above, this particular sum in Ex 1.5 Class 11 Maths requires you to provide the complements of different sets respectively. A universal set has been provided in the same, which serves as a reference point for students. 

Solve these questions at a go, and then figure out if the answers you have achieved are correct or not. You can use our NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 as a guide and check the accuracy of your answers as well as the effectiveness of your method.

Nonetheless, it must be noted that these sums are quite easy to solve and require a simple approach. That being said, students are more likely to make careless mistakes while solving them quickly.

Consequently, to ensure that you do not commit minute errors while rushing through these sums, you need to adopt a few smart shortcut techniques. For instance, our Class 11 Maths Chapter 1 Exercise 1.5 Solutions offer students several easy techniques that help to solve these sums quite early, thus saving time.

Since our solutions are updated frequently, the tips and techniques we share with undergo revision to lend improved efficiency. It directly benefits the quality of exam preparation and helps you to upgrade your approach towards planning the same. 

Download our NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 from Vedantu, and go through all the latest tips and techniques that can help you take your learning experience to the next level.

• Class 11 Maths Exercise 1.5 Solutions – Question 3

Before discussing this particular question, it is essential to look into other vital aspects of Sets that will facilitate the process of solving the remaining questions of this exercise.

Through the course of this chapter, students will be introduced to several new methods and touches upon the different rules of sets in details. Possessing a basic idea of these laws simplifies the concepts of this chapter to a great extent. It also provides you with a fair understanding of the approach different sums require for arriving at their respective solutions faster and with zero mistakes.

Which is why, before moving ahead in this exercise, you must make it a point to go through these following in details –

1. Commutative law

2. Associative law

3. Law of identity

4. Idempotent law

5. Law of U

6. Distributive law

7. De Morgan’s law

8. Complements law

9. Law of double complementation

10. Laws of empty set and universal set

Once you gain a substantial idea of these laws of set operations, you can move onto solving the remaining questions of Class 11 Maths Chapter 1 Exercise 1.5 effectively. As for the third question of this exercise, it is quite similar to the previous two questions. However, to solve it correctly, you need to be equipped with the idea and concept of Sets in more depth. 

To elaborate, you are to use the Natural Set as your Universal Set and come up with the accurate complements for the respective sets.

Consequently, you need to be aware of a few things to ensure the accuracy of their solution. To begin with, you need to know what a Natural Set and a Universal Set are.

Similarly, you must also know about the following.

• Odd natural numbers.

• Even natural numbers.

• Positive multiples of three.

• Prime numbers.

• Divisibility of natural numbers.

• Characteristics of a perfect square.

• Characteristics of a perfect cube.

Equip yourself with the basic idea of these concepts and then proceed to approach 11 problems that comprise this question. Arrive at the accurate answers for this unique question by referring to our NCERT Solutions for Class 11 Maths Chapter 1 that we have made available online. You can access the free PDF downloads of such solutions for free and go through them as per your convenience.

• Ex. 1.5 Class 11 Maths – Question 4 

To solve this particular question from NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 effectively, you need to have a fair understanding of the concepts that were touched upon in the previous exercises. For instance, refresh your knowledge of how to form the union of one set with the other. Also, while at it, make sure to go through the steps that have to be applied to create an intersection set.

These basics will help you solve this particular sum with more confidence and ease. To elaborate, this question requires you to verify that the complement of the union of two sets (A and B) is equivalent to the intersection of the union of the complement of each set (A’ and B’).

When it comes to sums based on proving, the right approach is essential to arrive at accurate answers. Make it a point to be familiar with the accurate approach and know the step by step method of solving similar problems.

Regardless of your ability to answer this type of questions; you can follow the approach that our subject experts have adopted to solve this said question. By doing so, you will gain a significant idea of the steps that go into solving such sums, and will further be able to pick up effective shortcut techniques.

As qualified teaching professionals cross-check them, you need not worry about the accuracy of the methods we have shared in our Ex. 1.5 Class 11 NCERT Solutions. With the incorporation of shortcut techniques we have shared with you through our solutions, you can improve your problem-solving skills significantly.

Also, a familiarity with the use of such techniques and a working knowledge of the same will help you solve similar problems in exams quite efficiently. Over time, it will sharpen your skills to solve problems with a few steps and will also come in handy in your CBSE Class 12 Board Examination.

Avail more of such revision-friendly tips and tricks from us to take your exam preparation a level up and make the entire revision process smooth and convenient. Just visit our online platform and choose from a world of learning options like never before.

• Class 11 Maths Chapter 1 Exercise 1.5 – Question 5

How proficient are you when it comes to answering questions based on the Venn diagram? It is recommended that you get your basics cleared when it comes to Venn diagram before beginning with this question. It requires you to draw an accurate representation of the relation between the union and intersection of different sets.

For example, the first sum of this question requires you to draw a Venn diagram showing a complement of the union of set A and B.  Subsequently, in the next one, you will have to show the intersection of set A’ and B’ in the form of a Venn diagram.

Also, you need to refresh your memory of three basics concepts to solve these and other similar questions accurately –

• The concept of the Venn diagram.

• The meaning of intersection and complement of sets.

• The point of difference between intersection and complement of sets.

Figure out why you need to be familiar with these three by following the answer format of our expert-approved solutions. With a quick look, you will be able to figure out how these come in to play while solving related questions.

You can use our NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 as a learning guide and cross-check your methods as well as answers with a quick reference. To make it effective immediately, download our latest chapter-wise solutions from our official website and begin your revision for complete preparation. Also, by including our study guide into your revision plan, you will be able to keep track of your improvement. It will further help you systematically plan your exam revision and make changes to it as and when deemed necessary.

• Ex 1.5 Class 11 NCERT Solutions – Question 6

Word problems are always fun to solve. They do not just gauge your ability to understand the question but also test how well you approach them. The next question on sets maintains the same spirit and is represented as an interesting word problem.

Nonetheless, it must be noted that to solve this problem; you need to have a clear idea of the fundamentals of both Sets and Triangles. Only if you have mastered the two; you will be able to solve problems like this one in no time.

Also, if you need to figure out the accurate answer for this particular problem and more, you can download our NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.5. It will provide you with all the answers and effective shortcut techniques that can be used effectively to solve not just this sum but other similar problems as well.

• Class 11 Maths Exercise 1.5 – Question 7

The highlight of this question is that it is represented in the format of fill in the blanks comprising seven sub-parts. You are to provide a suitable solution in the blanks provided to justify the given statements.

To effectively solve them, you need to be familiar with the functioning of –

• Universal sets.

• Complement sets.

• Null sets.

Additionally, you should also have a fair idea of how two sets can be related or their function can be restricted. Since it is a unique way of approaching this type of question, gain a better insight into how to solve it with the help of Ex 1.5 Class 11 NCERT Solutions. A quick look at the same will help you to solve it more effectively. Also, the scope of making any errors will considerably reduce.

Nonetheless, you must dedicate ample time and attention to the solution of this chapter to be able to retain it for a long time. Also, with regular revision, you will be able to boost your confidence level and perform well in exams.

• Class 11 Maths Exercise 1.5 – Other Exercise of Sets

There are a total of 6 exercises of this chapter namely –

1. Exercise 1.1

2. Exercise 1.2

3. Exercise 1.3

4. Exercise 1.4

5. Exercise 1.5

6. Exercise 1.6

Other than these, one miscellaneous exercise is also included that touches upon all the topics of sets and consists of different types of questions.

The following will offer you a fair idea of all the exercises mentioned above. It will further tell you how our NCERT Solutions for Class 11 Maths Chapter 1 proves beneficial in solving each one of them.

• Exercise 1.1 of Sets 

In this exercise, you will be solving a total of 6 questions; each of which tends to test your basic knowledge of Sets.

For example, the first question comprising nine problems requires you to figure out which of the given collections qualify as a set. Once identified, you will further be required to justify your answer for the same.

Though it may be easy to identify which of the collections form a set, students often falter at providing a suitable justification. You can avoid such unwarranted problem by simply following our latest NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5. Our solution will not only help you to justify your answer but will also enable you to solve similar problems with minimum assistance.

Besides, the other question of this exercise requires you to have a substantial idea of the following –

1. The symbols of set theory.

2. The roster format.

3. The set builder format.

4. Elements of sets.

With a clear idea of these pointers, you will be able to solve the remaining questions quite effectively. Also, take note of the key pointers that are shared in our CBSE NCERT Books free PDF downloads. They are highlighted in blue and represented in boxes to make it easier for you to spot them.

Having a fair hold of them will help you solve the sums quite easily. For example, one of the key pointers preceding the first exercise states: “The order in which the elements are listed in the roster form is immaterial.” While going through the same, you will also learn that all elements in the roster form are considered to be distinct.

In case you want to have more of such learning tips and shortcut techniques of solving problems, you can refer to our study materials available online. Subject experts have formulated our study solutions like Class 11 Maths NCERT Solutions Chapter 1 Ex 1.5. Their years of experience and thorough knowledge have helped us to come up with brilliant shortcut techniques and prompt learning tips. All of these facilitate an effective learning process and allow you to prepare for your upcoming exams better.

• Exercise 1.2 of Sets

When you move onto the next question of this exercise, you will find that it consists of 6 questions, all of which are entirely based on concepts of finite, infinite and empty sets.

To elaborate, the first one of the lot requires you to state which one of the given sets can be deemed as null sets. The following two questions also direct you to pick up the finite and infinite set from the sets provided.

However, as you move on to the fourth question, you will notice that you need to state if the two specific sets are similar to one another. Though question 5 and 6 are also on the same line and requires a similar approach, you have to offer suitable reasons for your choice of answer.

Here are two potent tips for you to solve these and other similar questions easily –

1. Go through the solved examples that precede the exercise.

2. Practice our compact NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5.

A perfect blend of both will come a long way and allow you to polish your understanding of the chapter significantly. Similarly, it will offer a functional idea of which approach to adopt for solving these types of questions.

By incorporating this blend into your study regime, you will benefit your exam revision and will improve your scope of performing exceptionally well in your upcoming exam. To facilitate the same at the earliest, access our study solutions now and take your exam preparation to the next level.

• Exercise 1.3 of Sets

How well-acquainted are you are with concepts like – subsets, power sets and universal sets? In case you are not well-versed with them, it is recommended that you improve your knowledge about them before moving on to this exercise. As all nine questions of this exercise are largely based on these concepts mentioned above, brushing up your memory of the same will prove immensely helpful.

That being said; let us take a quick look at how the questions are based on these concepts. For example, the first question tests your knowledge of subsets. First, you have to identify if the two given set of elements are subsets or not. Then, based on your identification, you have to fill in the accurate symbol for each case.

The following question in NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 is also based on the same concepts. Here, you need to figure out if the following statements are true or false. Subsequently, you need to provide a suitable explanation for your choice of answer.

While you are at your revision, pay close attention to provide subsets of given sets and the steps to compute the number of elements in a given set. All these techniques and information will allow you to solve the remaining exercises quite efficiently.

Regardless of your comfort level with these concepts of sets, you can access the latest PDF free downloads of our study guides for further assistance. As they are formulated by subject experts, you would not have to guess the accuracy of our solutions, and the methods applied. 

Also, you can access all the accurate answers and accompanying explanations in our chapter guides like Class 11 Maths Exercise 1.5 study solutions. Take a cue from the question-answering approach we have shared in our study materials to improve on your question-answering skills.

• Exercise 1.4 of Sets

The exercise carries a total of 12 questions, and each of them is designed to test your familiarity with the operations of sets.

To refresh your memory, these are the operations that you need to know in details to offer the accurate step by step solution for each question –

i. Union of Sets – In this case, the union of two sets (A and B) would consist of all the elements of both sets. The union between the two sets is denoted by the symbol ‘U’.

Also, the operation of the union has these following properties –

• As per Commutative law, A ∪ B = B ∪ A ()

• As per Associative law, (A ∪ B) ∪ C = A ∪ ( B ∪ C) 

• As per the Law of identity element, φ is the identity of ∪ A ∪ φ = A 

• As per Idempotent law A ∪ A = A 

• As per Law of U (v) U ∪ A = U

ii. Intersections of Sets – In this case, the intersection of two sets (A and B) would consist of those elements that are common to both sets. An intersection between the two sets is denoted by the symbol ‘∩’

Also, operation of the intersection has these following properties –

• As per Commutative law, A ∩ B = B ∩ A.

• As per Associative law, (A ∩ B) ∩ C = A ∩ (B ∩ C).

• As per Law of φ and U, φ ∩ A = φ, U ∩ A = A (U).

• As per Idempotent law, A ∩ A = A 

• As per Distributive law, A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )

ii. The difference of Sets – In this case, the set elements which belong to A do not belong to B. The same would be denoted as A – B.

With the help of this information, you would be able to solve 12 questions of NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 quickly and accurately. Also, a close reference to the solved exercises provided at the beginning of this exercise will help you understand how these laws are applied in each case. Simultaneously, follow the step by step problem-solving methods shown in our NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 to understand the application of these laws better.

• Exercise 1.6 of Sets

How good are you at solving word problems? Put your skills to test by solving the practical problems on the union and intersection of two sets. The section preceding the exercise will provide you with ample information on how to solve word problems based on sets. You will also gain potential insight into why a particular step was applied for solving a problem.

Subsequently, try the eight-word problems of this exercise to figure out how well you have understood the requisite approach for solving similar problems. If you still think you would require a little more assistance to solve them properly, you can take a quick look at our Ex 1.5 Class 11 NCERT Solutions online. Doing so, you will not only figure the accurate answer for each question but will also be able to pick up the best techniques of solving them.

Nonetheless, to be able to attempt these questions in the first place, you need to be thorough with the topics covered in previous exercises. A firm understanding of those will pave your way for solving it correctly.

Hence, make it a point to practice the chapter and our study guides like Exercise 1.5 Class 11 Maths Solutions frequently and give your exam preparation the much-required boost.

• Miscellaneous Exercise of Sets

Once you have solved all the exercises of this chapter, you can easily move on to the miscellaneous exercises. The thing about this particular exercise is that it is based on all the important concepts of this chapter. It is why you must be thorough with them for successfully solving all the 14 questions of NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 correctly.

To get the hang of it, you can go through the solved miscellaneous examples that are provided in the chapter and pick up the necessary pointers from them. Once you have glanced through them, you can move on to solving the chapter-based questions with confidence.

Regardless, you need to know that the problems mostly require you –

• To identify subsets.

• To determine the accuracy of the statement and to provide a suitable explanation for the same.

• To prove the equivalence between two conditions.

Other than these, you would also be tested on your previous knowledge of solving practical problems based on sets. Additionally, a firm idea of the operation of sets and their properties will help you solve most of the sums with greater ease. 

Once you are done solving the miscellaneous exercise, you can use our updated learning tools like NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 to cross-check your answers. It will also serve as a potent tool for your last-minute exam revision and make the learning process smooth. It will further directly help you plan your revision systematically and allow you to obtain higher scores in exams.

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