Sets Class 11th Notes - Free NCERT Class 11 Maths Chapter 1 Notes - Download PDF

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# Sets Class 11th Notes - Free NCERT Class 11 Maths Chapter 1 Notes - Download PDF

Edited By Ramraj Saini | Updated on Mar 22, 2022 04:23 PM IST

Class 11 Math chapter 1 notes are regarding Sets. In chapter 1 we will be going through the set concepts in Sets Class 11 notes. This Class 11 Maths chapter 1 notes contains the following topics: sets, representation of sets, types of sets, subset, power set, Venn diagrams, operations of sets, Laws of algebra.

NCERT Class 11 Math chapter 1 notes also contain important formulas. NCERT Class 11 Math chapter 1 contains systematic explanations of topics using examples and exercises. NCERT Notes for Class 11 Math chapter 1 include FAQ’s or frequently asked questions about the chapter. In CBSE Class 11 Maths chapter 1 notes, topics are explained step by step. These concepts can also be downloaded from Class 11 Maths chapter 1 notes pdf download, Class 11 notes Sets, Class 11 Sets notes pdf download.

Also, students can refer,

## Sets:

Set is a well-defined collection of elements. These elements may include numbers, symbols, variables, etc.

We generally represent them using capital letter(A,B,C,D,Z,N etc)

Elements in the sets are represented by small letters(a,b,c,d,e)

Elements of sets are represented using closed brackets { } .

Examples of sets:

• i) set of first n natural numbers: N=1,2,3,4,5,.................,n
• ii) set of rational numbers: R.
• iii) set of whole numbers: W.
• iv) set of positive integers: Z+
• v) set of negative integers: Z-

## Representation of Sets:

We have 2 methods to represent a set:

Roaster form or tabular form:

In this form, elements are separated with commas and within braces { }.

Eg: set of natural numbers below 10: { 1,2,3,4,5,6,7,8,9}

Set-builder form: In this form, each element contains a common character that any other element outside the set does not contain.
Eg: F={x: x is a natural number that divides 24}

## Types of Sets:

Empty set: A set that does not contain any element and is empty such sets are called empty (or) null (or) void sets.

Represented as: { } or ∅

Eg: A={x:1<x,x is a natural number}

So A is an empty set as we don't have any natural element that is less than 1.

Finite set: Sets that contain a fixed or finite number of elements are called finite sets.

Represented by: A= { , , , , , }

Eg: A={natural numbers <20}

A= {1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19}

NOTE: Empty set is also a finate set.

Infinite sets: Sets that are not finite or the sets that are infinite are called infinite sets.

Represented by: {, , , , , , , , ……………………………}

Eg: A={n natural numbers}

A={1,2,3,4,5,6,7,8,9,..................................................}

NOTE: All finite sets can be represented in set-builder form but infinite sets cannot be represented in set-builder form because they have no fixed pattern.

Equal sets: Sets where two sets have exactly the same elements.

Represented with the symbol: “ = “

Examples:

i) Let A, B be two sets.

A= set of natural even elements less than 10

A={2,4,6,8}

B= set of multiples of 2 less than 10 that are natural

B={2,4,6,8}

Here A=B. Since every element of A is also an element of B and vice versa.

ii) A={1,2,3,4}

B={1,3,2,1,4,1}

Then also A=B, the set elements do not change even if repeated.

The order of the elements also doesn’t matter.

## Subset:

Any Set that has every element of another set is called a subset.

Representation: A⊂B if a ∈A ⇒ a ∈B

The above representation means that A should have every element of set B, but B need not have every element of A.

NOTE: Every set is a subset of itself

Eg: A = {1,3,5,7,9}

B={1,2,3,4,5,6,7,8,9}

Here A⊂B because every element of A is an element of B and vice versa doesn’t matter

Venn diagram of subset:

Proper set: If A⊂B and A≠B then such set A is called a proper subset of B.

B is called the superset of A.

Eg: A={1,2,3,4,5,6}

B={1,2,3,4,5,6,7,8}

Here A⊂B and A≠B.

So A is a proper subset of B.

B is a superset of A.

Intervals of subsets:

When a,b ∈ real numbers, a<b then,

Set of real numbers: {x: a<x<b} is open interval, represented as (a,b).

The set that includes end points also then such intervals are closed intervals.

{y:a≤y≤b}, closed interval represented by [a,b].

Eg: (p,q] ⇒ {x:p<x≤q}

Here in the set elements includes q and exclude p.

Pictorial representation of intervals:

Power set: It is a set that is a collection of all subsets including null or empty sets and itself denotes a power set.

Represented by p(A).

Eg: A={1,2,3,4}

Subsets: ∅,{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Total subsets:16

Points :

1. First is a null set

2. Next singleton sets(sets with one element}

3. Next with two elements

4. The set itself

Remember: {1,2}={2,1}

(order doesn’t matter so it should be written only once)

Formula to find a number of subsets:

m= number of elements in set A

Subsets: ∅,{1},{2},{3},{4},{1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4}, {2,3,4},{1,2,3,4}

Total subsets:16

Universal Set:

The set that includes all the elements is called a universal set.

Represented using U.

Venn Diagram:

{0,1,2,3,4,5,6,7,8,9} all belong to the universal set.

## Operation of Sets:

Union of sets: The set contains all the elements of A and B and repeating elements of both sets A, B is taken only once.

Represented by: ∪

Set builder form of union sets:

{y: y∈A or y∈B}

Venn diagram of union:

Eg: A={1,2,3,4,5} B={2,4,5,6,7,8}

A∪B={1,2,3,4,5,6,7,8}

Properties of union:

1. A∪B = B∪A is called commutative property.

2. A∪(B∪C) = (A∪B)∪C is associative property.

3. A∪∅ = A is identity law

4. A∪A is an idempotent law

5. U∪A is union law

The intersection of Sets:

This set contains the elements that are common in the sets.

Represented as: ∩

Set builder form of intersection:

{y:y∈A and y∈B}

Venn diagram of AB:

Eg: A={1,2,3,4,5,6} B=(2,4,6,8,10}

A∩B={2,4,6}

Properties of intersection:

a)A∩B=B∩A is called commutative property.

b) A∩(B∩C) = (A∩B)∩C is an associative property.

c) A∩∅=∅ is identity law

d) A∩A is idempotent law

e) U∩A is union law

f) A∩(B∪C) = (A∩B)∪(A∩C) is distribution property.

Difference of Sets:

In the set the elements belong to A but do not belong to B, such sets are called Differences of sets.

Represented as: A-B

Set builder form: {y:y∈A and y∉B}

Venn diagram of A-B:

Eg: A={1,2,3,4,5}

B={2,3,4,8,9}

A-B= {1,5}

Here even though we have {2,3,4} in A as they belong to B also do not include them in A-B.

Complement of set:

The elements other than A that is in the universe come under complement elements.

Represented by: A’

Set builder form: {y:y∈U and y∉A}

Venn diagram:

Eg: A={y:y∈U and not divisor of 24}

A’={1,2,3,4,6,8,12,24}

Properties of complement sets:

1. A∪A’ = U (complement laws)

2. A∩ A’ = ∅ (complement laws)

3. (A∪B)’ = A’∩B’ (Demorgan’s law)

4. (A∩B)’ = A’∪ B’ (Demorgan’s law)

5. (A’)’ = A (double complement)

6. U’ = ∅ and ∅’= U

## Theorems and Proofs:

Theorems 1:

n(A∪B) = n(A) + n (B) - n(A∩B)

Proof:

n ( A – B) + n ( A ∩ B ) + n ( B – A )

n(A) - n ( A ∩ B ) + n(B) - n( A ∩ B ) + n( A ∩ B )

n(A) + n(B) - n( A ∩ B )

Hence proved;

Theorem 2: n(A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )

Proof:

n ( A ∪ B ∪ C ) = n (A) + n ( B ∪ C ) – n [ A ∩ ( B ∪ C ) ]

n (A) + n ( B ) + n ( C ) – n ( B ∩ C ) – n [ A ∩ ( B ∪ C ) ]

n [ A ∩ ( B ∪ C ) ] = n ( A ∩ B ) + n ( A ∩ C ) – n [ ( A ∩ B ) ∩ (A ∩ C)] n ( A ∩ B ) + n ( A ∩ C ) – n (A ∩ B ∩ C)

Similarly for the other two:

Finally proved;

n ( A ∪ B ∪ C ) = n (A) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )

With this topic we conclude NCERT Class 11 chapter 1 notes.

The link for NCERT textbook pdf is given below:

URL: ncert.nic.in/ncerts/l/kemh101.pdf

## Significance of NCERT Class 11 Maths Chapter 1 Notes:

NCERT Class 11 Maths chapter 1 notes will be very much helpful for students to score maximum marks in their 11 board exams. In Sets Class 11 chapter 1 notes we have discussed many topics: sets, representation of sets, types of sets, subset, power set, Venn diagrams, operations of sets, Laws of algebra. NCERT Class 11 Mathematics chapter 1 is also very useful to cover major topics of Class 11 CBSE Mathematics Syllabus.

The CBSE Class 11 Maths chapter 1 will help students to understand the formulas, statements, and rules in detail. This pdf also contains previous year’s questions and NCERT textbook pdf. The next part includes FAQ’s or frequently asked questions along with topic-wise explanations. These topics can also be downloaded from Class 11 chapter 1 Sets pdf download.

### NCERT Class 11 Notes Chapter Wise.

 NCERT Class 11 Maths Chapter 1 Notes NCERT Class 11 Maths Chapter 2 Notes NCERT Class 11 Maths Chapter 3 Notes NCERT Class 11 Maths Chapter 4 Notes NCERT Class 11 Maths Chapter 5 Notes NCERT Class 11 Maths Chapter 6 Notes NCERT Class 11 Maths Chapter 7 Notes NCERT Class 11 Maths Chapter 8 Notes NCERT Class 11 Maths Chapter 9 Notes NCERT Class 11 Maths Chapter 10 Notes NCERT Class 11 Maths Chapter 11 Notes NCERT Class 11 Maths Chapter 12 Notes NCERT Class 11 Maths Chapter 13 Notes NCERT Class 11 Maths Chapter 14 Notes NCERT Class 11 Maths Chapter 15 Notes NCERT Class 11 Maths Chapter 16 Notes

### NCERT Books and Syllabus

 NCERT Book for Class 11 NCERT Syllabus for Class 11

1. Is {0} an empty set in Sets Class 11 notes?

No, {0} cannot be an empty set. Even though 0 is an element. It comes under a singleton set.

2. Who is the father of NCERT notes Class 11 Maths Chapter 1 Sets?

Georg Ferdinand Ludwig Philipp Cantor, a German mathematician is the father of Set theory.

3. What is the importance of ncert notes for Class 11 Maths chapter 1 in real life?

It is used to separate and arrange the things in groups and finding them out regarding the requirement is the use of sets in real life.

4. What are the symbols of sets in Class 11 Math chapter 1 notes?

{ }, A⊂B, A⊆B, A∈B, A∪B, A∩B, A⊇B, A⊃B, A-B, A’ and many more.

5. Give a few daily life examples where sets are used as per ncert Class 11 Maths chapter 1 notes?

Kitchen, house, cupboards, playlist, etc are examples of sets that we use in our daily life.

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Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available
##### Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages. Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available
##### Talent Agent

The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

3 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
##### Producer

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

2 Jobs Available
##### Fashion Blogger

Fashion bloggers use multiple social media platforms to recommend or share ideas related to fashion. A fashion blogger is a person who writes about fashion, publishes pictures of outfits, jewellery, accessories. Fashion blogger works as a model, journalist, and a stylist in the fashion industry. In current fashion times, these bloggers have crossed into becoming a star in fashion magazines, commercials, or campaigns.

2 Jobs Available
##### Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available
##### Fashion Journalist

Fashion journalism involves performing research and writing about the most recent fashion trends. Journalists obtain this knowledge by collaborating with stylists, conducting interviews with fashion designers, and attending fashion shows, photoshoots, and conferences. A fashion Journalist  job is to write copy for trade and advertisement journals, fashion magazines, newspapers, and online fashion forums about style and fashion.

2 Jobs Available
##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available
##### QA Manager

Quality Assurance Manager Job Description: A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes.

2 Jobs Available

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.

2 Jobs Available
##### Reliability Engineer

Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment.

2 Jobs Available
##### Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### IT Consultant

An IT Consultant is a professional who is also known as a technology consultant. He or she is required to provide consultation to industrial and commercial clients to resolve business and IT problems and acquire optimum growth. An IT consultant can find work by signing up with an IT consultancy firm, or he or she can work on their own as independent contractors and select the projects he or she wants to work on.

2 Jobs Available
##### Data Architect

A Data Architect role involves formulating the organisational data strategy. It involves data quality, flow of data within the organisation and security of data. The vision of Data Architect provides support to convert business requirements into technical requirements.

2 Jobs Available
##### Security Engineer

The Security Engineer is responsible for overseeing and controlling the various aspects of a company's computer security. Individuals in the security engineer jobs include developing and implementing policies and procedures to protect the data and information of the organisation. In this article, we will discuss the security engineer job description, security engineer skills, security engineer course, and security engineer career path.

2 Jobs Available
##### UX Architect

A UX Architect is someone who influences the design processes and its outcomes. He or she possesses a solid understanding of user research, information architecture, interaction design and content strategy.

2 Jobs Available