This document provides an introduction to set theory. It begins with definitions of fundamental set concepts like elements, membership, representation of sets in roster and set-builder forms, empty and singleton sets, finite and infinite sets, equal and equivalent sets. It then discusses types of sets such as subsets and proper subsets, the power set of a set, and universal sets. Examples are provided to illustrate each concept. The document also introduces Venn diagrams to represent relationships between sets.
SETS, RELATIONS AND FUNCTIONS - JEE Main 2014Ednexa
A set is a well-defined collection of objects where it is clear whether an element belongs to the set or not. Sets can be represented in roster form by listing elements or in set-builder form using properties to define elements. There are different types of sets including the empty set, singleton sets, finite and infinite sets, and equal and subset relationships between sets. Cartesian products and relations are ways to represent relationships between sets.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced. Sets can be represented using a roster method by listing elements or a set-builder form using a common property. Sets are classified as finite or infinite based on the number of elements, and other set types like the empty set and singleton set are defined. Equal, equivalent, and disjoint sets are also defined.
The document discusses sets and set notation. It begins by defining a set as a collection of elements and noting that sets are fundamental data structures in mathematics. It then provides examples of set notation using curly braces and discusses properties of sets such as that a set is defined by its elements regardless of ordering or repetition. The document also introduces concepts like subsets, membership using the Greek letter epsilon, the empty set, cardinality, and power sets.
This document is a workbook from Esperanza National High School covering sets and number sense for 7th grade mathematics. It includes lessons on defining and describing sets using roster and rule methods, set operations like union, intersection, difference and complement, and problems involving Venn diagrams. It also covers absolute value on the number line. The workbook contains examples and exercises for students to practice these set theory and number sense concepts.
This document defines and explains key concepts in sets and set theory, including:
- Defining sets and listing elements using roster and set-builder notation
- Important mathematical sets like natural numbers, integers, and real numbers
- Describing relationships between sets such as subsets, equality, and Venn diagrams
- Calculating cardinality to determine the number of elements in a set
- Forming Cartesian products to combine elements from multiple sets into ordered pairs
Discrete mathematics Ch1 sets Theory_Dr.Khaled.Bakro د. خالد بكروDr. Khaled Bakro
This document provides an overview of discrete mathematics and sets theory. It outlines the main topics covered in discrete mathematics including propositional logic, set theory, simple algorithms, functions, sequences, relations, counting methods, introduction to number theory, graph theory, and trees. It then defines what a discrete mathematics is and contrasts discrete vs continuous mathematics. The remainder of the document defines fundamental concepts in sets theory such as subsets, supersets, set operations, Venn diagrams, cardinality, and power sets. It also discusses ways to represent sets using arrays, linked lists, and bit strings.
This document provides a lesson on the complement of a set. It begins with an example problem about student populations to introduce the concept. The lesson then defines the complement of a set A as the set of all elements in the universal set U that are not in A. It explains how to find the complement using a Venn diagram and the formula that the cardinality of the complement is equal to the total elements of U minus the elements of A. Several examples are provided to illustrate computing and representing complements of sets using Venn diagrams. The lesson concludes by solving the initial problem about student selection using the complement concept.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
SETS, RELATIONS AND FUNCTIONS - JEE Main 2014Ednexa
A set is a well-defined collection of objects where it is clear whether an element belongs to the set or not. Sets can be represented in roster form by listing elements or in set-builder form using properties to define elements. There are different types of sets including the empty set, singleton sets, finite and infinite sets, and equal and subset relationships between sets. Cartesian products and relations are ways to represent relationships between sets.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced. Sets can be represented using a roster method by listing elements or a set-builder form using a common property. Sets are classified as finite or infinite based on the number of elements, and other set types like the empty set and singleton set are defined. Equal, equivalent, and disjoint sets are also defined.
The document discusses sets and set notation. It begins by defining a set as a collection of elements and noting that sets are fundamental data structures in mathematics. It then provides examples of set notation using curly braces and discusses properties of sets such as that a set is defined by its elements regardless of ordering or repetition. The document also introduces concepts like subsets, membership using the Greek letter epsilon, the empty set, cardinality, and power sets.
This document is a workbook from Esperanza National High School covering sets and number sense for 7th grade mathematics. It includes lessons on defining and describing sets using roster and rule methods, set operations like union, intersection, difference and complement, and problems involving Venn diagrams. It also covers absolute value on the number line. The workbook contains examples and exercises for students to practice these set theory and number sense concepts.
This document defines and explains key concepts in sets and set theory, including:
- Defining sets and listing elements using roster and set-builder notation
- Important mathematical sets like natural numbers, integers, and real numbers
- Describing relationships between sets such as subsets, equality, and Venn diagrams
- Calculating cardinality to determine the number of elements in a set
- Forming Cartesian products to combine elements from multiple sets into ordered pairs
Discrete mathematics Ch1 sets Theory_Dr.Khaled.Bakro د. خالد بكروDr. Khaled Bakro
This document provides an overview of discrete mathematics and sets theory. It outlines the main topics covered in discrete mathematics including propositional logic, set theory, simple algorithms, functions, sequences, relations, counting methods, introduction to number theory, graph theory, and trees. It then defines what a discrete mathematics is and contrasts discrete vs continuous mathematics. The remainder of the document defines fundamental concepts in sets theory such as subsets, supersets, set operations, Venn diagrams, cardinality, and power sets. It also discusses ways to represent sets using arrays, linked lists, and bit strings.
This document provides a lesson on the complement of a set. It begins with an example problem about student populations to introduce the concept. The lesson then defines the complement of a set A as the set of all elements in the universal set U that are not in A. It explains how to find the complement using a Venn diagram and the formula that the cardinality of the complement is equal to the total elements of U minus the elements of A. Several examples are provided to illustrate computing and representing complements of sets using Venn diagrams. The lesson concludes by solving the initial problem about student selection using the complement concept.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
This document contains a mathematics test for 7th grade students on the topic of sets. The test has 10 questions and provides 3 different types of questions for each number - Type A questions are worth 80 points, Type B questions are worth 90 points, and Type C questions are worth 100 points. Students must choose one type of question for each number and show their work. The questions cover topics like examples of sets in daily life, set notation, Venn diagrams, subsets, and relationships between sets. Students are given 60 minutes to complete the test and must sign the answer sheet along with their teacher and parent.
The answer for:
1)Give me a group of girls whose height is > than 156 cm is E,F,G.
2) The answers for Piano and Guitar question is:
n(U) =8,
n(A)=3,
n(B)=4
(A n B) = 1
( A U B)= 6
(A U B)' = 2
Only Piano ( A - B)=2
Only guitar(B-A) =3
Sets [Algebra] in an easier and interesting way to learn! Specially suited for young children and for those who find Sets difficult to grasp.
Content-
Venn diagram,
Set builder(Rule method),
List method(Roster method),
Universal set,
Union of sets,
Intersection of set
This document provides a teaching guide for a 7th grade math lesson on sets. It introduces concepts like well-defined sets, subsets, universal sets, and the null set. Students will use Venn diagrams to represent sets and subsets. The lesson defines terms like union and intersection of sets and teaches students to perform set operations and represent unions and intersections using Venn diagrams.
This document provides an overview of mathematical functions and relations through a series of lessons:
1. It defines key concepts like domains, ranges, and intervals used to describe functions and relations. Functions are defined as relations where no two ordered pairs have the same first element.
2. One-to-one functions are introduced, which satisfy both vertical and horizontal line tests. Only one-to-one functions can have inverse functions.
3. The process for finding the inverse of a function is described. The inverse is formed by swapping the inputs and outputs of the original function and solving for the new output. The domain of the original becomes the range of the inverse, and vice versa.
4
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Q1-Q2)LiGhT ArOhL
The document provides information about sets and set operations including:
1) It defines the complement of a set as the elements in the universal set that are not in the given set.
2) It provides examples of finding the complement of sets and using Venn diagrams to represent complements.
3) It solves a word problem about selecting a student who is not a sophomore by finding the complement of the set of sophomores.
This document provides an introduction to the basics of set theory. It defines a set as a collection of objects and explains that sets play an important role in mathematics. The key concepts covered include set notation using curly brackets, the membership symbol to indicate if an element belongs to a set, comparing sets for equality or determining if one is a subset of another, calculating cardinality to determine the number of elements in a set, and performing the set operations of union and intersection. Examples using playing cards help illustrate these set theory concepts.
Grade 7 Learning Module in Math (Quarter 1 to 4)R Borres
Here are the answers:
(a) A B is shown in Set 2. It contains all elements that belong to A or B or both.
(b) A B is shown in Set 3. It contains elements that belong to both A and B.
2. Given sets P = {1, 3, 5, 7, 9} and Q = {2, 4, 6, 8}, find P Q and P Q.
3. Draw a Venn diagram to represent the following sets:
A = {x | x is a prime number less than 10}
B = {x | x is an even number less than 10}
The document provides notes on set theory concepts including:
1. Sets are well-defined collections of objects called elements or members. Examples of sets include collections of boys in a class or numbers within a specified range.
2. For a collection to be considered a set, the objects must be well-defined so it is clear which objects are members.
3. There are different ways to represent sets including roster form listing elements, statement form describing characteristics of elements, and rule form using logical statements.
This document provides an introduction to sets and the real number system. It defines key concepts such as elements, membership, cardinality, types of sets including finite, infinite, empty, subsets, proper subsets, improper subsets, universal sets, power sets, and set relations like equal, equivalent, joint, and disjoint sets. Examples are provided to illustrate each concept. The last section provides a quiz to test understanding of basic set concepts covered.
This document introduces sets and their representations. It discusses:
1) Georg Cantor developed the theory of sets in the late 19th century while working on trigonometric series. Sets are now fundamental in mathematics.
2) A set is a well-defined collection of objects where we can determine if an object belongs to the set or not. Sets are represented using roster form (listing elements between braces) or set-builder form (using properties of elements).
3) The empty set, denoted {}, is the set with no elements. It is different from non-existence of a set.
This document provides an overview of key concepts in set theory, including:
1) Sets can be represented in roster or set-builder form. Common sets used in mathematics include the natural numbers, integers, rational numbers, and real numbers.
2) The empty set contains no elements. Finite sets have a definite number of elements, while infinite sets have an unlimited number of elements.
3) Two sets are equal if they contain exactly the same elements. A set is a subset of another set if all its elements are also elements of the other set.
4) The power set of a set contains all possible subsets of that set.
This document introduces the concept of sets. It defines a set as a well-defined collection of distinct objects, called elements or members. There are two main ways to represent a set: a roster form that lists the elements between curly brackets, and a set-builder form that uses a description to define the set of elements that satisfy a given property. The document discusses different types of sets such as empty, singleton, finite, and infinite sets. It also introduces set operations and relations like equivalent sets, equal sets, and subsets.
This document provides information about sets, relations, and functions in mathematics. It begins by giving examples of sets and non-sets to illustrate what makes a collection a well-defined set. It then defines various set concepts like finite and infinite sets, the empty set, singleton sets, equal sets, subsets, unions and intersections of sets. It introduces the concept of relations and functions, defining a function as a special type of relation. It concludes by stating the objectives of learning about sets, relations and functions.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced, such as using capital letters to represent sets and lowercase letters for elements. Two methods for representing sets are described: roster form and set-builder form. The document then classifies sets as finite or infinite, empty, singleton, equal, equivalent, and disjoint sets. Examples are provided to illustrate each concept.
CBSE Class 10 Mathematics Set theory Topic
Set theory Topics discussed in this document:
Introduction
Sets and their representations
Two methods of representing a set:
Roster form
Set-builder form
The empty set
Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
Intervals as subsets of R
Power set
Universal Set
Venn Diagrams
More Topics under Class 10th Set theory (CBSE):
Introduction to sets
Operations on sets
Visit Edvie.com for more topics
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
This document provides an introduction to the concept of sets in mathematics. It defines what a set is and explains that a set is a well-defined collection of distinct objects. It discusses different ways of representing sets, such as using a roster or set-builder notation. It also covers various types of sets like finite, infinite, empty, singleton, equivalent and equal sets. The document explains set operations like union, intersection and complement. It introduces Venn diagrams to visually represent relationships between sets. Finally, it discusses algebraic properties that set operations satisfy, such as commutative, associative and distributive laws.
The document defines sets and provides examples of different types of sets. It discusses how sets can be defined using roster notation by listing elements within braces or using set-builder notation stating properties elements must have. Some key types of sets mentioned include the empty set, natural numbers, integers, rational numbers, and real numbers. Operations on sets like union, intersection, and difference are introduced along with examples. Subsets, Venn diagrams, and the power set are also covered.
This document contains a mathematics test for 7th grade students on the topic of sets. The test has 10 questions and provides 3 different types of questions for each number - Type A questions are worth 80 points, Type B questions are worth 90 points, and Type C questions are worth 100 points. Students must choose one type of question for each number and show their work. The questions cover topics like examples of sets in daily life, set notation, Venn diagrams, subsets, and relationships between sets. Students are given 60 minutes to complete the test and must sign the answer sheet along with their teacher and parent.
The answer for:
1)Give me a group of girls whose height is > than 156 cm is E,F,G.
2) The answers for Piano and Guitar question is:
n(U) =8,
n(A)=3,
n(B)=4
(A n B) = 1
( A U B)= 6
(A U B)' = 2
Only Piano ( A - B)=2
Only guitar(B-A) =3
Sets [Algebra] in an easier and interesting way to learn! Specially suited for young children and for those who find Sets difficult to grasp.
Content-
Venn diagram,
Set builder(Rule method),
List method(Roster method),
Universal set,
Union of sets,
Intersection of set
This document provides a teaching guide for a 7th grade math lesson on sets. It introduces concepts like well-defined sets, subsets, universal sets, and the null set. Students will use Venn diagrams to represent sets and subsets. The lesson defines terms like union and intersection of sets and teaches students to perform set operations and represent unions and intersections using Venn diagrams.
This document provides an overview of mathematical functions and relations through a series of lessons:
1. It defines key concepts like domains, ranges, and intervals used to describe functions and relations. Functions are defined as relations where no two ordered pairs have the same first element.
2. One-to-one functions are introduced, which satisfy both vertical and horizontal line tests. Only one-to-one functions can have inverse functions.
3. The process for finding the inverse of a function is described. The inverse is formed by swapping the inputs and outputs of the original function and solving for the new output. The domain of the original becomes the range of the inverse, and vice versa.
4
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Q1-Q2)LiGhT ArOhL
The document provides information about sets and set operations including:
1) It defines the complement of a set as the elements in the universal set that are not in the given set.
2) It provides examples of finding the complement of sets and using Venn diagrams to represent complements.
3) It solves a word problem about selecting a student who is not a sophomore by finding the complement of the set of sophomores.
This document provides an introduction to the basics of set theory. It defines a set as a collection of objects and explains that sets play an important role in mathematics. The key concepts covered include set notation using curly brackets, the membership symbol to indicate if an element belongs to a set, comparing sets for equality or determining if one is a subset of another, calculating cardinality to determine the number of elements in a set, and performing the set operations of union and intersection. Examples using playing cards help illustrate these set theory concepts.
Grade 7 Learning Module in Math (Quarter 1 to 4)R Borres
Here are the answers:
(a) A B is shown in Set 2. It contains all elements that belong to A or B or both.
(b) A B is shown in Set 3. It contains elements that belong to both A and B.
2. Given sets P = {1, 3, 5, 7, 9} and Q = {2, 4, 6, 8}, find P Q and P Q.
3. Draw a Venn diagram to represent the following sets:
A = {x | x is a prime number less than 10}
B = {x | x is an even number less than 10}
The document provides notes on set theory concepts including:
1. Sets are well-defined collections of objects called elements or members. Examples of sets include collections of boys in a class or numbers within a specified range.
2. For a collection to be considered a set, the objects must be well-defined so it is clear which objects are members.
3. There are different ways to represent sets including roster form listing elements, statement form describing characteristics of elements, and rule form using logical statements.
This document provides an introduction to sets and the real number system. It defines key concepts such as elements, membership, cardinality, types of sets including finite, infinite, empty, subsets, proper subsets, improper subsets, universal sets, power sets, and set relations like equal, equivalent, joint, and disjoint sets. Examples are provided to illustrate each concept. The last section provides a quiz to test understanding of basic set concepts covered.
This document introduces sets and their representations. It discusses:
1) Georg Cantor developed the theory of sets in the late 19th century while working on trigonometric series. Sets are now fundamental in mathematics.
2) A set is a well-defined collection of objects where we can determine if an object belongs to the set or not. Sets are represented using roster form (listing elements between braces) or set-builder form (using properties of elements).
3) The empty set, denoted {}, is the set with no elements. It is different from non-existence of a set.
This document provides an overview of key concepts in set theory, including:
1) Sets can be represented in roster or set-builder form. Common sets used in mathematics include the natural numbers, integers, rational numbers, and real numbers.
2) The empty set contains no elements. Finite sets have a definite number of elements, while infinite sets have an unlimited number of elements.
3) Two sets are equal if they contain exactly the same elements. A set is a subset of another set if all its elements are also elements of the other set.
4) The power set of a set contains all possible subsets of that set.
This document introduces the concept of sets. It defines a set as a well-defined collection of distinct objects, called elements or members. There are two main ways to represent a set: a roster form that lists the elements between curly brackets, and a set-builder form that uses a description to define the set of elements that satisfy a given property. The document discusses different types of sets such as empty, singleton, finite, and infinite sets. It also introduces set operations and relations like equivalent sets, equal sets, and subsets.
This document provides information about sets, relations, and functions in mathematics. It begins by giving examples of sets and non-sets to illustrate what makes a collection a well-defined set. It then defines various set concepts like finite and infinite sets, the empty set, singleton sets, equal sets, subsets, unions and intersections of sets. It introduces the concept of relations and functions, defining a function as a special type of relation. It concludes by stating the objectives of learning about sets, relations and functions.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced, such as using capital letters to represent sets and lowercase letters for elements. Two methods for representing sets are described: roster form and set-builder form. The document then classifies sets as finite or infinite, empty, singleton, equal, equivalent, and disjoint sets. Examples are provided to illustrate each concept.
CBSE Class 10 Mathematics Set theory Topic
Set theory Topics discussed in this document:
Introduction
Sets and their representations
Two methods of representing a set:
Roster form
Set-builder form
The empty set
Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
Intervals as subsets of R
Power set
Universal Set
Venn Diagrams
More Topics under Class 10th Set theory (CBSE):
Introduction to sets
Operations on sets
Visit Edvie.com for more topics
The document defines key concepts in set theory including:
1. A set is a well-defined collection of distinct objects called elements. Georg Cantor is credited with creating set theory.
2. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties the elements share.
3. Operations on sets include intersection, union, difference, and complement. Intersection is the set of common elements, union is all elements in either set, difference is elements only in the first set, and complement is elements not in the set.
This document provides an introduction to the concept of sets in mathematics. It defines what a set is and explains that a set is a well-defined collection of distinct objects. It discusses different ways of representing sets, such as using a roster or set-builder notation. It also covers various types of sets like finite, infinite, empty, singleton, equivalent and equal sets. The document explains set operations like union, intersection and complement. It introduces Venn diagrams to visually represent relationships between sets. Finally, it discusses algebraic properties that set operations satisfy, such as commutative, associative and distributive laws.
The document defines sets and provides examples of different types of sets. It discusses how sets can be defined using roster notation by listing elements within braces or using set-builder notation stating properties elements must have. Some key types of sets mentioned include the empty set, natural numbers, integers, rational numbers, and real numbers. Operations on sets like union, intersection, and difference are introduced along with examples. Subsets, Venn diagrams, and the power set are also covered.
- A set is a well-defined collection of objects. The empty set is a set with no elements. A set is finite if it has a definite number of elements, otherwise it is infinite.
- Two sets are equal if they have exactly the same elements. A set A is a subset of set B if every element of A is also an element of B.
- The power set of a set A is the collection of all subsets of A, including the empty set and A itself. It is denoted by P(A).
The document defines different types of sets and methods of representing sets. It discusses empty sets, singleton sets, finite and infinite sets. It also defines equivalent sets as sets with the same number of elements, and equal sets as sets containing the same elements. Disjoint sets are defined as sets that do not share any common elements. Examples are provided to illustrate these key set concepts and relationships between sets.
The document provides information about sets and set operations including:
1) It defines the complement of a set as the elements in the universal set that are not in the given set.
2) It provides examples of finding the complement of sets and using Venn diagrams to represent complements.
3) It solves a word problem about selecting a student who is not a sophomore by finding the complement of the set of sophomores.
The document provides information about sets and operations on sets such as union, intersection, and complement. It includes examples and exercises involving defining sets based on given criteria, finding the elements and cardinality of unions, intersections, and complements of sets, and using Venn diagrams to represent relationships between sets. The key concepts covered are defining sets, unions and intersections of sets, complements of sets, and using Venn diagrams to illustrate set relationships.
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
The document defines set concepts and their properties. It explains that a set is a collection of distinct objects, called elements. Properties discussed include how sets are denoted, elements belong or don't belong to sets, order doesn't matter, and counting each element once. It also covers set theory, Venn diagrams, ways to represent sets, types of sets (empty, finite, infinite), equal sets, subsets, cardinality, and operations like intersection, union, difference, and complement of sets. Examples are provided to illustrate each concept.
Moazzzim Sir (25.07.23)CSE 1201, Week#3, Lecture#7.pptxKhalidSyfullah6
This document provides an overview of key concepts in set theory including:
- The definition of a set as an unordered collection of distinct elements
- Common ways to describe and represent sets such as listing elements, set-builder notation, and Venn diagrams
- Important set terminology including subset, proper subset, set equality, cardinality (size of a set), finite vs infinite sets, power set, and Cartesian product
The document uses examples and explanations to illustrate each concept over 34 pages. It appears to be lecture material introducing students to the basic foundations of set theory.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
The document provides an introduction to set theory. It defines what a set is and discusses different ways of representing sets using roster form and set-builder form. It also defines types of sets such as the empty set, singleton set, finite sets, and equivalent sets. Subsets are introduced, including proper and improper subsets. Important subsets of the real numbers like the natural numbers, integers, rational numbers, and irrational numbers are identified. Intervals are also discussed as subsets of the real line.
1. The lesson plan is for a mathematics class on irrational numbers for 1X standard students.
2. It includes learning outcomes like identifying irrational numbers, discussing problems involving irrational numbers, and observing aspects of irrational numbers.
3. The teaching involves activities like discussing pythagoras' theorem and using examples to show that some lengths cannot be expressed as rational numbers, such as the diagonal of a square with sides of length 1.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times denominators are the same.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
1. The document outlines a lesson plan on teaching division of irrationals to 9th grade students.
2. It includes learning objectives like recalling products of irrationals, recognizing division of irrational numbers, and identifying concepts of irrational numbers.
3. The lesson involves group discussions, individual work, and explanations from the teacher using examples and a chart on dividing irrational numbers.
The area of a triangle within a rectangle is half the area of the rectangle if they share the same base. If the triangle is required to be isosceles, the point defining the triangle's third vertex must be at the midpoint of the rectangle's other side. If the triangle must also be a right triangle, the third vertex point must be at the corner of the rectangle opposite the shared base.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Assessment and Planning in Educational technology.pptxKavitha Krishnan
In an education system, it is understood that assessment is only for the students, but on the other hand, the Assessment of teachers is also an important aspect of the education system that ensures teachers are providing high-quality instruction to students. The assessment process can be used to provide feedback and support for professional development, to inform decisions about teacher retention or promotion, or to evaluate teacher effectiveness for accountability purposes.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
3. Preface
The book “ An Introduction to Set theory” is intended for the secondary students and teachers in
Kerala syllabus. In this book all the topic have been deal with in a simple and lucid manner. A
sufficiently large number of problems have been solved. By studying this book , the student is
expected to understand the concept of set and their representations, types of sets, operations on
set, practical situations. To do more problems involving the types of sets, operations of sets and
express set builder form to roster form and roster form to set builder form.
Suggestion for the further improvement of this book will be highly
appreciated.
Veena v.
4. CONTENTS
Title Page No:
Preface
Chapter 1. Set and their representations 1 – 4
Chapter 2. Types of sets 5 – 7
Chapter 3. Subsets 8 – 13
Chapter 4. Operations on sets 14 – 20
Chapter 5. Practical problems on union and
intersection of two sets 21 – 22
Summary 23
Reference
5. 1
CHAPTER 1
SETS AND THEIR REPRESENTATIONS
In these days of conflict between ancient and modern studies ; there must surely be something to
be said for a study which did not begin with Pythagoras and will not end with Einstein ; but is
the oldest and the youngest. - G . H . HARDY *
Introduction
The concept of set serves as a fundamental part of the present day of mathematics . Today this
concept is being used in almost every branch of mathematics . Sets are used to define the
concept of relations and functions. The study of geometry, sequences, probability ,etc. requires
the knowledge of sets.
The theory of sets was developed by German mathematician George Cantor ( 1845 – 1918 ). He
first encountered sets while working on ‘ problems on trigonometric series’.
In this chapter , we discuss some basic definitions and operations involving sets.
Definition 1
A set is a well – defined collection of objects. The following points may be noted :
I. Objects , elements and members of a set are synonymous terms.
II. Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
III. The elements of a set are represented by small letters a, b, c, x, y, z, etc.
6. If a is an element of a set A , we say that ‘a belongs to A’ the Greek symbol ϵ ( epsilon ) is
used to denote the phrase ‘ belongs to’ .Thus we write a∈A . If b is not an element of a set A,
we write b∉A and read ‘ b does not belongs to A’.
2
There are two methods of representing a set :
I. Roster or tabular form.
II. Set – builder form.
i. In roster form, all the elements of a set are listed, the elements are being separated by
commas and are enclosed within braces { }.For example , the set of all even positive
integers less than 7 is described in roster form as { 2,4,6 }
ii. In set – builder form, all the elements of a set possess a single common property which is
not possessed by any elements outside the set.
For example, in the set { a,e,i,o,u,s } all the elements possess a common property , namely each
of them is a vowel in the English alphabet. Denoting this set by V , we write,
V = { x: x is a vowel in English alphabet }.
We described the element of the set by using a symbol x ( any other symbol like the letters y,z,
etc. could be used ) which is followed by a colon ‘ :’ . After the sign of colon , we write the
characteristics property possessed by the elements of the set and then enclose the whole
description within braces.
The above description of the set V is read as ‘ the set of all x such that x is a vowel of the English
alphabet’. The braces stands for ‘ the set of all’ , the colon stands for ‘ such that’.
NOTE : It may be noted that while writing the set in roster form an element is not generally
repeated , i.e; all the elements are taken as distinct .
Example 1 : Write the solution set of the equation x2
+ x- 2 = 0 in roster form
Solution : The given equation can be written as,
7. ( x + 2 ) (x – 1 ) = 0 ; i.e; x = 1, -2 .
The roster form as { 1, -2 }.
3
Example 2 : Write the set { x : x is a positive integer and x2
< 40 } in the roster form .
Solution : The required numbers are 1,2,3,4,5,6. So , the given set in the roster form is
{1,2,3,4,5,6 }.
Example 3 : Write the set A = { 1,4,9,16,25,…} in set builder form.
Solution : We may write the set A as ,
A = { x : x is the square of a natural number }
Exercise
1. Which of the following are sets ? Justify your answer.
a) The collection of all the months of a year beginning with the letter J .
b) The collection of ten most talented writers of India.
c) A team of eleven best cricket bats man of the world.
d) The collection of all boy’s in your class.
e) The collection of all even integers.
2. Let A = { 1,2,3,4,5,6 }. Insert the appropriate symbol ∈ or ∉ in the blank spaces :
a) 5……A
b) 8…....A
c) 0…....A
8. d) 4……A
e) 2……A
3. Write the following sets in roster form:
a) A = { x : x is an integer and -3<x<7 }
b) B = { x : x is a natural number less than 6}
c) C = { x : x is a prime number which is divisor of 60}
d) D = The set of all letters in the word TRIGNOMETRY
e) E = The set of all letters in the word BETTER
4
4. Write the following sets in the set builder form :
i. {3,6,9,12 }
ii. {2,4,8,16,32}
iii. {5,2,5,125,625}
iv. {2,4,6,….}
v. {1,4,9,…..100}
5. List all the elements of the following sets :
9. i. A = { x : x is an odd natural number }
ii. B = { x : x is an integer, x2
≤4 }
iii. C = { x : x is a letter in the word ‘LOYAL’ }
iv. D = { x : x is a month of a year not having 31 days}
6. Match the following :
i. {1,2,3,6} a. {x : x is a prime number and a divisor of 6}
ii. {2,3} b. {x : x is an odd natural number less than 10}
iii. {M,A,T,H,E,I,C,S} c. { x : x is a natural number and a divisor of 6}
iv. {1,3,5,7,9} d. { x : x is a letter of the word MATHEMATICS}
5
CHAPTER 2
TYPES OF SETS
10. Definition 1
A set having no elements is called empty set or void set or null set. It is usually denoted by { } or
ᶲ.
Example:
The set of all boys in a girls school is a null set.
Definition 2
A set having only one element is called a singleton set.
Example:
The set of all principals in a college is a singleton set.
NOTE :
{ }ᶲ is not an empty set but it is a singleton set.
Definition 3
A set which is empty or having finite number of elements is a finite set. Otherwise set is
infinite.
Examples:
i. S = { 2,4,6,8}is finite.
ii. Set of all students in a country is finite .
iii. Set of all points in a line is infinite.
iv. Set N of natural numbers is an infinite set
6
Order of a finite set : The number of elements of a finite set is called the order of that set.
Order of a set A is denoted by n(A).
NOTE :
11. order of a set is also known as the cardinal number of the set
Example :
1. Order of empty set is 0 and order of singleton set is 1.
2. If A = { x : x is a divisor of 20}
Then, A = {1,2,4,5,10,20}
n(A) = 6
Definition 4
Two sets A and B are said to be equal, if they contain same elements.
i.e ; A = B , if all elements of A are in B and all elements of B are in A.
Example :
I. The sets {-1,1} and {x : x2
– 1 = 0 } are equal.
II. A={x : xєR and x2
-3x + 2 = 0} are not equal sets since A = {1,2,-1,-2} and
B = {1,2}
III. The set of all letters in the word LAST and the set of all letters in the word SALT are
equal sets.
Definition 5
Two sets A and B are said to be equivalent if n(A) = n(B). i.e; they contain same number of
elements.
Example :
I. A = {1} and B = {2} are equivalent sets since n(A) = 1 and n(B) =1.
II. A = {1,2,3,4}, B = {a,b,c} are not equivalent.
NOTE :
All the equal sets are equivalent. But the converse is not true. i.e; all the equivalent sets need
not be equal sets.
7
Exercise
12. 1. Which of the following are examples of the null set.
a) Set of even prime number
b) {x : x is a natural number , x<5 and x>7}
c) Set of odd natural number divisible by 2}
d) {y : y is a point common to any two parallel line}
2. Which of the following sets are finite or infinite .
a) The set of months of a year
b) The set of prime number less than 99
c) {1,2,3,…99,100}
3. In the following statement whether A= B or not.
i. A= {a,b,c,d} , B= {d,c,b,a}
ii. A= {4,8,12,16} ,B= {8,4,16,18}
iii. A= {2,4,6,8,10} , B= {x : x is a positive even integer and x≤10}
8
CHAPTER 3
13. SUBSETS
Definition 1
The set A is called a subset of set B if every element of A is also an element of B.
We write it as A⊂B. If A is not a subset of B we write A⊄B .
If A ⊂ B, then B is called the superset of A. Symbolically we can write , A ⊂B if xєA ⇒
xєB
Example
1. A= {-1,2,5}; B= {3,-1,2,7,5}
Clearly A⊂B. But B⊄A.
2. The set of all odd natural numbers is a subset of the set of all natural numbers
3. Let A= { Vowels in the English alphabet}
B= {Letters in the English alphabet}
Then A is a subset of B and therefore B is the superset of A.
NOTE :
I. A⊂B and B⊂A if and only if A = B.
II. Every set is a subset of it self. i.e, A⊂A.
III. Empty set is a subset of every set. i.e, ᶲ⊂A
Definition 2
Let A and B be two sets such that A⊂B and A ≠ B.
9
Then A is called proper subset of B.
14. That is, there exists at least one element of B which is not in A.
If A is not a proper subset, it is called an improper subset.
NOTE :
I. Every set is an improper subset of itself
II. Every set except the null set will have minimum two subsets.
III. Empty set has only one subset.
IV. An element of a set cannot be a subset. That is, a⊂{a,b}is an incorrect statement.
*Subsets of set of Real numbers:
There are many important subsets of R.
The set of natural numbers,
N = {1,2,3,….}
The set of integers,
Z = {…,-3,-2,-1,0,1,2,3….}
The set of rational numbers,
Q = { x : x = p/q , p,qєZ and q ≠ 0}
The set of irrational numbers,
T = { x : xєR and x ∉Q}
Then , N⊂Z⊂Q, Q⊂R, T⊂R, N⊄T.
10
*Intervals as subsets of :
15. Let a,bєR and a< b.
We can define four intervals as follows.
1. Open interval (a,b) = { x : x єR, a <x <b}
2. Closed interval [a,b] = { x : x єR, a≤ x≤ b}
3. Closed but open interval [a,b) = { x : x єR, a≤ x<b}
4. Open but closed interval (a,b] = { x : x єR, a< x≤ b}
All these intervals are subset of R.
NOTE :
The set of real numbers can be written as an interval as (- ∞ ,∞ ).
The number (b - a) is called the length of any of the intervals (a,b) , [a,b] , [a,b) or (a,b].
Definition 3.
Power set of a set A is the set of all subsets of A. It is usually denoted by P(A).
Example :
• Let A = {1,2}
Then P(A) = { {1,2}, {1} ,{2} , ᶲ}, n(P(A)) = 4.
• If A = ᶲ ,P(A) = {ᶲ }. So n(P(A)) = 1.
NOTE:
• The elements of the power set are subsets.
• If n(A) = m, then the number of subsets of A =2m
.
In other words, if n(A) = m, then n(P(A)) = 2m
.
• If n(A) = m, then the number of proper subsets of A =2m
-1
11
Example
16. If A = {1,2,3,4}, then number of subsets of A = 24
= 16.
Number of proper subsets of A = 24
-1 = 15
n(P(A)) = 24
= 16
i.e, n(A) = m then n(P(A)) =2m
.
Definition 4
Universal set of given sets, is the superset of all sets under consideration. It is denoted
by U.
Example
• A = {x : x is an even natural number }
• B = {x : x is an odd natural number }
• C = {x : x is a prime number }
Then the universal set are considered as ,
• U = {x : x is an even natural number }
Definition
Relationships between the sets can be represented by means of diagram known as Venn
diagrams.
The universal set is represented usually by a rectangle. The elements of the sets are
written in their respective circles.
Example
If U = set of natural numbers less than 10 is the universal set of which A = The set of all
prime numbers less than 10 is a subset. The corresponding Venn diagram is given
below,
12
17. Exercise
1. Examine whether the following statements are true or false.
• {a,b} ⊄{b,c,a}
• {1,2,3}⊂{1,3,5}
• {a} є{a,b,c}
• {a} ⊂{a,b,c}
2. Let A = {1,2 {3,4}, 5}. Which of the following statements are incorrect and why?
• {3,4} ⊂ A
• {3,4}є A
• {2,4,5}є A
• {1,2,3⊂A
3. Write down all the subsets of the following sets.
• {a}
• {a,b}
13
18. • {1,2,3}
• {5,6,7,8}
4. How many elements has P(A) if A = ?ᶲ
5. Write the following intervals in the set builder form;
• (-3,0)
• [6,12]
• (6,12]
• [-23,5]
19. 14
CHAPTER 4
OPERATIONS ON SETS
Definition 1
The union of two sets A and B is the set C which consists of all those elements which are either
in A or in B( including those which are in both).
In symbols, we write
A∪B = { x: x ∈A or x∈ B}
Example
I. A = {1,2,3,4,5}, B = {3,4,5,6}
∴ A∪ B = {1,2,3,4,5,6 }
II. A = {x : x is an even natural number}
B = {x : x is an odd natural number }
∴ A ∪B = {x : x is a natural number}
III. A = { 1,2,3,4}, B = {1,3}
∴ A ∪B = {1,2,3,4}
Note that here B⊂A and A∪ B =A
The Venn diagram of A∪ B is as follows:
20. 15
*Some properties of the operation of union:
a. A ∪ B = B ∪A ( Commutative law)
b. (A ∪B) ∪ C = A∪ (B∪ C) (Associative law )
c. A∪ ᶲ = A (Law of identity element ,ᶲ is the identity of U )
d. A∪ A = A (Idempotent law )
e. U ∪A = U (Law of U )
.
Definition 2
The intersection of sets A and B is the set of all elements which are common to both A and B.
The symbol ∩ is used to denote the intersection.
Symbolically, we write A∩ B = { x : x ∈ A and x ∈B}.
Examples:
1) A = {1,2,3,4} , B = {3,4,5,6}
A ∩B = {3,4}
2) A = {a,b,c}, B = {a,b,c,d,e}
A ∩B = {a,b,c} = A
Note that here A⊂ B
*Some properties of operation of intersection:
i. A ∩B = B ∩A (Commutative law)
ii. (A∩ B)∩ C = A∩( B ∩C) (Associative law)
iii. ᶲ∩ A = ᶲ , U ∩A = A (Law of ᶲ and U )
iv. A∩ A = A (Idempotent law )
21. 16
v. A∩( B∪ C )= (A∩ B)∪ (A∩ C) (Distributive law)
The venn diagram of A∩B ,
Definition 3
Two sets A and B are called disjoint sets, if there is no element common to them.
That is, A∩ B = ᶲ.
The venn diagram of disjoint set ,
Example
A = {x : x is an even natural number }
B = {x : x is an odd natural number }
∴A∩ B = ᶲ.
Here , we say A and B are disjoint set.
22. 17
Definition 4
The difference of two sets A and B in this order is the set of elements that are in A and not in B.
i.e, A-B = { x : x ∈ A and x∉ B }
B-A = { x : x ∈ B and x∉ A }
Example
A = {1,2,3,4,5} , B = {4,5,6,7} then ,
A-B = {1,2,3} , B-A = {6,7}
The venn diagram of A-B , B-A, are as shown below,
1. A-B 2. B-A
a. A∪ B = (A- B)∪ (A∩B)∪( B-A)
b. A-B = A- (A ∩B)
c. A-B , B-A , and A∩ B are disjoint sets.
23. 18
Exercise
1. Find the union of each of the following pairs of sets;
i. X = {1,3,5} , Y = {1,2,3}
ii. A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number and less than 6}
2. If A = {1,2,3,4} , B = {3,4,5,6} , C = {5,6,7,8} and D = {7,8,9,10} ,find
i. A∪ B
ii. A∪C
iii. B∪ C
iv. B∪ D
v. A ∪B∪ C
vi. A∪ B∪D
vii. B∪ C∪ D
3. If A = {3,5,7,9,11} , B = {7,9,11,13} ,C = {11,13,15} and D = {15,17}. Find
i. A∩ B
ii. A∩ C
iii. A∩ D
iv. B∩D
v. A∩C∩D
4. If X = {a,b,c,d } , Y = {f,b,d,g}. find X-Y, Y-X, X∩Y.
24. 19
Definition 5
If A is any set and U is the universal set, then complement of A denoted by A is the set of allꞌ
elements that are in U and not in A.
i.e,A = { x: xꞌ ∈ U and x∉ A }
Obviously A = U – A.ꞌ
NOTE:
The complement of A is also denoted by AC
.
Example:
i. U = {1,2,3,4,5,6,7}
A = {1,3,5,7}, B = {2,3,4,6} then,
A = {2,4,6}, B = {1,5,7}.ꞌ ꞌ
ii. U = {x : x is a natural number}
A = {x : x is an even natural number}
Then A = {x : x is an odd natural number}ꞌ
The venn diagram of A is as given below,ꞌ
25. 20
*Some properties of complement sets:
a) A∪A = U and A∩A =ꞌ ꞌ ᶲ.
b) (A ) = Aꞌ ꞌ
c) ᶲꞌ. =U and U =ꞌ ᶲ.
d) De morgan’s laws
If A and B are any two subset of the universal set U, then
1. (A∪ B) = A ∩Bꞌ ꞌ ꞌ
2. (A∩B) = Aꞌ ꞌ∪Bꞌ
These two results are stated in words as,
The complement of the union of two sets is the intersection of their complements and the
complements of the intersection of two sets is the union of their complements. These are
called De morgan’s laws.
Exercise
1. If U = {a,b,c,d,e,f,g,h} . find the complements of the following sets.
• A = {a,b,c}
• B = {d,e,f,g}
• C = {a,c,e,g}
2. If U = {1,2,3,4,5,6,7,8,9} , A = {2,4,6,8}, B = { 2,3,5,7} . Verify that (A∪ B ) =ꞌ
A ∩B and (A∩B) = Aꞌ ꞌ ꞌ ꞌ∪Bꞌ
26. 21
CHAPTER 5
PRACTICAL PROBLEMS ON UNION AND INTERSECTION OF
TWO SETS
The following results are very useful in doing practical problems;
• If A and B are two disjoint sets,
then n(A∪ B) = n(A) + n(B)
• If A and B are any finite sets,
then n(A∪ B) = n(A) + n(B) –n(A∩B)
• If A,B,C are three disjoint sets ,
then n(A∪B∪C) = n(A) +n(B)+n(C)
• If A,B,C are any three finite sets,
then n(A∪B∪C) = n(A) + n(B) +n(C) – n(A∩B) – n(A∩C) – n(B∩C) + n(A∩B∩C)
We have already seen that
A∪ B = (A – B )∪ (A∩B )∪(B-A)
and A-B , A∩ B and B-A are disjoint set.
So, n(A∪B) = n(A - B) + n(B - A) + n(A∩ B)
Example 1: In a school there are 20 teachers who teach mathematics or physics. Of these 12
teach maths and 4 teach both physics and maths. How many teach physics ?
27. Solution : Let M denote the set of teachers who teach maths and P denote the set of teachers
who teach physics.
22
We have, n(M∪P) = 20, n(M) = 12, n(M∩P) = 4.
n(M∪P) = n(M) + n(P) --n(M∩P).
20 = 12 + n(P) – 4
∴n(P) = 12
Hence 12 teachers teach physics.
Example 2: In a class of 35 students , 24 like to play cricket and 16 like to play football. Also ,
each student likes to play at least one of the two games. How many students like to play both
cricket and football?
Solution : Let X be the set of student who like to play cricket and Y be the set of
students who like to play football. Then X∪Y is the set of students who like to play at
least one game and X∩Y is the set of students who like to play both games.
Given ,n(X) = 24 , n(Y) = 16 , n(X∪Y) = 35.
n(X∪Y) = n(X) + n(Y) –n(X∩Y)
35 = 24 + 16 - n(X∩Y)
∴n(X∩Y) = 5
Hence 5 students likes to play both games.
Exercise
1. If X and Y are two sets such that n(X) = 17, n(Y) = 23, and n(X∪ Y) = 38. Find
n(X ∩Y) ?
2. In a group of 400 people 250 can speak Hindi and 200 speak English. How many
people can speak both Hindi and English ?
28. 3. In a committee 50 people speak French 20 speak Spanish and 10 speak both
French and Spanish. How many speak at least one of these two language ?
23
Summary
This chapter deals with some basic definitions and operations involving sets. These are
summarized below ;
1. A set is a well defined collection of objects.
2. A set which does not contain any elements is called empty set
3. A set which consist of a definite number of elements is called finite set. Other wise the
set is called infinite set.
4. Two sets A and B are said to be equal, if they have exactly the same elements.
5. A set A is said to be subset of a set B , if every element of A is also an element of B.
6. A power set of a set A is collection of all subsets of A.
7. The union of two sets A and B is the set of all those elements which are either in A or
in B.
8. The intersection of two sets A and B is the set of all elements which are common.
9. The complement of a subset A of universal set U is the set of all elements of U which
are not in A.
10. If A and B are finite sets such that A∩ B =ᶲ. ,then n(A∪B) =n(A) + n(B).
11. If A∩B ≠ᶲ , then n(A∪B) = n(A) + n(B) –n(A∩ B).