• NCERT Solutions
    • NCERT Library
  • RD Sharma
    • RD Sharma Class 12 Solutions
    • RD Sharma Class 11 Solutions Free PDF Download
    • RD Sharma Class 10 Solutions
    • RD Sharma Class 9 Solutions
    • RD Sharma Class 8 Solutions
    • RD Sharma Class 7 Solutions
    • RD Sharma Class 6 Solutions
  • Class 12
    • Class 12 Science
      • NCERT Solutions for Class 12 Maths
      • NCERT Solutions for Class 12 Physics
      • NCERT Solutions for Class 12 Chemistry
      • NCERT Solutions for Class 12 Biology
      • NCERT Solutions for Class 12 Economics
      • NCERT Solutions for Class 12 Computer Science (Python)
      • NCERT Solutions for Class 12 Computer Science (C++)
      • NCERT Solutions for Class 12 English
      • NCERT Solutions for Class 12 Hindi
    • Class 12 Commerce
      • NCERT Solutions for Class 12 Maths
      • NCERT Solutions for Class 12 Business Studies
      • NCERT Solutions for Class 12 Accountancy
      • NCERT Solutions for Class 12 Micro Economics
      • NCERT Solutions for Class 12 Macro Economics
      • NCERT Solutions for Class 12 Entrepreneurship
    • Class 12 Humanities
      • NCERT Solutions for Class 12 History
      • NCERT Solutions for Class 12 Political Science
      • NCERT Solutions for Class 12 Economics
      • NCERT Solutions for Class 12 Sociology
      • NCERT Solutions for Class 12 Psychology
  • Class 11
    • Class 11 Science
      • NCERT Solutions for Class 11 Maths
      • NCERT Solutions for Class 11 Physics
      • NCERT Solutions for Class 11 Chemistry
      • NCERT Solutions for Class 11 Biology
      • NCERT Solutions for Class 11 Economics
      • NCERT Solutions for Class 11 Computer Science (Python)
      • NCERT Solutions for Class 11 English
      • NCERT Solutions for Class 11 Hindi
    • Class 11 Commerce
      • NCERT Solutions for Class 11 Maths
      • NCERT Solutions for Class 11 Business Studies
      • NCERT Solutions for Class 11 Accountancy
      • NCERT Solutions for Class 11 Economics
      • NCERT Solutions for Class 11 Entrepreneurship
    • Class 11 Humanities
      • NCERT Solutions for Class 11 Psychology
      • NCERT Solutions for Class 11 Political Science
      • NCERT Solutions for Class 11 Economics
      • NCERT Solutions for Class 11 Indian Economic Development
  • Class 10
    • NCERT Solutions for Class 10 Maths
    • NCERT Solutions for Class 10 Science
    • NCERT Solutions for Class 10 Social Science
    • NCERT Solutions for Class 10 English
    • NCERT Solutions For Class 10 Hindi Sanchayan
    • NCERT Solutions For Class 10 Hindi Sparsh
    • NCERT Solutions For Class 10 Hindi Kshitiz
    • NCERT Solutions For Class 10 Hindi Kritika
    • NCERT Solutions for Class 10 Sanskrit
    • NCERT Solutions for Class 10 Foundation of Information Technology
  • Class 9
    • NCERT Solutions for Class 9 Maths
    • NCERT Solutions for Class 9 Science
    • NCERT Solutions for Class 9 Social Science
    • NCERT Solutions for Class 9 English
    • NCERT Solutions for Class 9 Hindi
    • NCERT Solutions for Class 9 Sanskrit
    • NCERT Solutions for Class 9 Foundation of IT
  • CBSE Sample Papers
    • Previous Year Question Papers
    • CBSE Topper Answer Sheet
    • CBSE Sample Papers for Class 12
    • CBSE Sample Papers for Class 11
    • CBSE Sample Papers for Class 10
    • CBSE Sample Papers for Class 9
    • CBSE Sample Papers Class 8
    • CBSE Sample Papers Class 7
    • CBSE Sample Papers Class 6
  • Textbook Solutions
    • Lakhmir Singh
    • Lakhmir Singh Class 10 Physics
    • Lakhmir Singh Class 10 Chemistry
    • Lakhmir Singh Class 10 Biology
    • Lakhmir Singh Class 9 Physics
    • Lakhmir Singh Class 9 Chemistry
    • PS Verma and VK Agarwal Biology Class 9 Solutions
    • Lakhmir Singh Science Class 8 Solutions

Learn CBSE

NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12

Introduction to Set Theory – Basics, Definition, Representation of Sets

December 1, 2020 by Veerendra

Set Theory is a branch of mathematics and is a collection of objects known as numbers or elements of the set. Set theory is a vital topic and lays stronger basics for the rest of the Mathematics. You can learn about the axioms that are essential for learning the concepts of mathematics that are built with it. For instance, Element a belongs to Set A can be denoted by a ∈ A and a ∉ A represents the element a doesn’t belong to Set A.

{ 3, 4, 5} is an Example of Set. In this article of Introduction to Set Theory, you will find Representation of Sets in different forms such as Statement Form, Roster Form, and Set Builder Form, Types of Sets, Cardinal Number of a Set, Subsets, Operations on Sets, etc.

  • Representation of a Set
  • Types of Sets
  • Finite Sets and Infinite Sets
  • Power Set
  • Problems on Union of Sets
  • Problems on Intersection of Sets
  • Difference of two Sets
  • Complement of a Set
  • Problems on Complement of a Set
  • Problems on Operation on Sets
  • Word Problems on Sets
  • Venn Diagrams in Different Situations
  • Relationship in Sets using Venn Diagram
  • Union of Sets using Venn Diagram
  • Intersection of Sets using Venn Diagram
  • Disjoint of Sets using Venn Diagram
  • Difference of Sets using Venn Diagram
  • Examples on Venn Diagram

Set Definition

Set can be defined as a collection of elements enclosed within curly brackets. In other words, we can describe the Set as a Collection of Distinct Objects or Elements. These Elements of the Set can be organized into smaller sets and they are called the Subsets. Order isn’t that important in Sets and { 1, 2, 4} is the same as { 4,2, 1}.

Examples of Sets

  • Odd Numbers less than 20, i.e., 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
  • Prime Factors of 15 are 3, 5
  • Types of Triangles depending on Sides: Equilateral, Isosceles, Scalene
  • Top two surgeons in India
  • 10 Famous Engineers of the Society.

Among the Examples listed the first three are well-defined collections of elements whereas the rest aren’t.

Important Sets used in Mathematics

N: Set of all natural numbers = {1, 2, 3, 4, …..}

Q: Set of all rational numbers

R: Set of all real numbers

W: Set of all whole numbers

Z: Set of all integers = {….., -3, -2, -1, 0, 1, 2, 3, …..}

Z+: Set of all positive integers

Representation of a Set

Set can be denoted using three common forms. They are given along the following lines by taking enough examples

  • Statement Form
  • Roaster Form or Tabular Form
  • Set Builder Form

Statement Form: In this representation, elements of the set are given with a well-defined description. You can see the following examples for an idea

Example:

Consonants of the Alphabet

Set of Natural Numbers less than 20 and more than 5.

Roaster Form or Tabular Form: In Roaster Form, elements of the set are enclosed within a pair of brackets and separated by commas.

Example:

N is a set of Natural Numbers less than 7 { 1, 2, 3, 4, 5, 6}

Set of Vowels in Alphabet = { a, e, i, o, u}

Set Builder Form: In this representation, Set is given by a Property that the members need to satisfy.

{x: x is an odd number divisible by 3 and less than 10}

{x: x is a whole number less than 5}

Size of a Set

At times, we are curious to know the number of elements in the set. This is called cardinality or size of the set. In general, the Cardinality of the Set A is given by |A| and can be either finite or infinite.

Types of Sets

Sets are classified into many kinds. Some of them Finite Set, Infinite Set,  Subset, Proper Set, Universal Set, Empty Set, Singleton Set, etc.

Finite Set: A Set containing a finite number of elements is called Finite Set. Empty Sets come under the Category of Finite Sets. If at all the Finite Set is Non-Empty then they are called Non- Empty Finite Sets.

Example: A = {x: x is the first month in a year}; Set A will have 31 elements.

Infinite Set: In Contrast to the finite set if the set has infinite elements then it is called Infinite Set.

Example: A = {x : x is an integer}; There are infinite integers. Hence, A is an infinite set.

Power Set: Power Set of A is the set that contains all the subsets of Set A. It is represented as P(A).

Example:  If set A = {-5,7,6}, then power set of A will be:

P(A)={ϕ, {-5}, {7}, {6}, {-5,7}, {7,6}, {6,-5}, {-5,7,6}}

Sub Set: If Set A contains the elements that are in Set B as well then Set A is said to be the Subset of Set B.

Example:

If set A = {-5,7,6}, then Sub Set of A will be:

P(A)={ϕ, {-5}, {7}, {6}, {-5,7}, {7,6}, {6,-5}, {-5,7,6}}

Universal Set:

This is the base for all the other sets formed. Based on the Context universal set is decided and it can be either finite or infinite. All the other Sets are Subsets of Universal Set and is given by U.

Example: Set of Natural Numbers is a Universal Set of Integers, Real Numbers.

Empty Set: 

There will be no elements in the set and is represented by the symbol ϕ or {}. The other names of Empty Set are Null Set or Void Set.

Example: S = { x | x ∈ N and 9 < x < 10 } = ∅

Singleton Set:

If a Set contains only one element then it is called a Singleton Set.

Example: A = {x : x is an odd prime number}

Operations on Sets

Consider Two different sets A and B, they are several operations that are frequently used

Union: Union Operation is given by the symbol U. Set A U B denotes the union between Sets A and B. It is read as A union B or Union of A and B. It is defined as the Set that contains all the elements belonging to either of the Sets.

Intersection: Intersection Operation is represented by the symbol ∩. Set A ∩ B is read as A Intersection B or Intersection of A and B. A ∩ B is defined in general as a set that contains all the elements that belong to both A and B.

Complement: Usually, the Complement of Set A is represented as Ac or A‘ or ~A. The Complement of Set A contains all the elements that are not in Set A.

Power Set: The power set is the set of all possible subsets of S. It is denoted by P(S). Remember that Empty Set and the Set itself also comes under the Power Set. The Cardinality of the Power Set is 2n in which n is the number of elements of the set.

Cartesian Product: Consider A and B to be Two Sets. The Cartesian Product of the two sets is given by AxB i.e. the set containing all the ordered pairs (a, b) where a belong to Set A, b belongs to Set B.

Representation of Cartesian Product A × B = {(a, b) | a ∈ A ∧ b ∈ B}.

The cardinality of AxB is N*M where N is the cardinality of A and M is the Cardinality of B. Remember that AxB is not the same as BxA.

Filed Under: Mathematics

LearnCBSE Sample Papers
  • Words by Length
  • NEET MCQ
  • Factoring Calculator
  • Rational Numbers
  • CGPA Calculator
  • TOP Universities in India
  • TOP Engineering Colleges in India
  • TOP Pharmacy Colleges in India
  • Coding for Kids
  • Math Riddles for Kids with Answers
  • General Knowledge for Kids
  • General Knowledge
  • Scholarships for Students
  • NSP - National Scholarip Portal
  • Class 12 Maths NCERT Solutions
  • Class 11 Maths NCERT Solutions
  • NCERT Solutions for Class 10 Maths
  • NCERT Solutions for Class 9 Maths
  • NCERT Solutions for Class 8 Maths
  • NCERT Solutions for Class 7 Maths
  • NCERT Solutions for Class 6 Maths
  • NCERT Solutions for Class 6 Science
  • NCERT Solutions for Class 7 Science
  • NCERT Solutions for Class 8 Science
  • NCERT Solutions for Class 9 Science
  • NCERT Solutions for Class 10 Science
  • NCERT Solutions for Class 11 Physics
  • NCERT Solutions for Class 11 Chemistry
  • NCERT Solutions for Class 12 Physics
  • NCERT Solutions for Class 12 Chemistry
  • NCERT Solutions for Class 10 Science Chapter 1
  • NCERT Solutions for Class 10 Science Chapter 2
  • Metals and Nonmetals Class 10
  • carbon and its compounds class 10
  • Periodic Classification of Elements Class 10
  • Life Process Class 10
  • NCERT Solutions for Class 10 Science Chapter 7
  • NCERT Solutions for Class 10 Science Chapter 8
  • NCERT Solutions for Class 10 Science Chapter 9
  • NCERT Solutions for Class 10 Science Chapter 10
  • NCERT Solutions for Class 10 Science Chapter 11
  • NCERT Solutions for Class 10 Science Chapter 12
  • NCERT Solutions for Class 10 Science Chapter 13
  • NCERT Solutions for Class 10 Science Chapter 14
  • NCERT Solutions for Class 10 Science Chapter 15
  • NCERT Solutions for Class 10 Science Chapter 16

Free Resources

RD Sharma Class 12 Solutions RD Sharma Class 11
RD Sharma Class 10 RD Sharma Class 9
RD Sharma Class 8 RD Sharma Class 7
CBSE Previous Year Question Papers Class 12 CBSE Previous Year Question Papers Class 10
NCERT Books Maths Formulas
CBSE Sample Papers Vedic Maths
NCERT Library

 

NCERT Solutions

NCERT Solutions for Class 10
NCERT Solutions for Class 9
NCERT Solutions for Class 8
NCERT Solutions for Class 7
NCERT Solutions for Class 6
NCERT Solutions for Class 5
NCERT Solutions for Class 4
NCERT Solutions for Class 3
NCERT Solutions for Class 2
NCERT Solutions for Class 1

Quick Resources

English Grammar Hindi Grammar
Textbook Solutions Maths NCERT Solutions
Science NCERT Solutions Social Science NCERT Solutions
English Solutions Hindi NCERT Solutions
NCERT Exemplar Problems Engineering Entrance Exams
Like us on Facebook Follow us on Twitter
Watch Youtube Videos NCERT Solutions App