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.
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 provides information about visually representing sets using different methods. It discusses listing sets using curly brackets, representing sets using set-builder notation with conditions, and displaying sets using Venn diagrams. The document also covers labeling sets, finding cardinality, types of sets such as empty, finite, infinite, and singleton sets, and relationships between sets like equal, equivalent, subset, and superset.
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 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.
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.
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.
This document discusses sets and set theory. It defines what a set is, the different types of sets, and how sets are represented. The key points are:
- A set is a collection of distinct objects, called elements or members. Sets can be finite or infinite.
- There are different types of sets including empty sets, which have no elements, singleton sets, which have one element, and finite sets, which have a fixed number of elements.
- Sets can be represented in roster form by listing elements between curly brackets or in set builder form using a rule to describe the elements.
- Equivalent sets have the same number of elements, while infinite sets have an indefinite number of
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 provides information about visually representing sets using different methods. It discusses listing sets using curly brackets, representing sets using set-builder notation with conditions, and displaying sets using Venn diagrams. The document also covers labeling sets, finding cardinality, types of sets such as empty, finite, infinite, and singleton sets, and relationships between sets like equal, equivalent, subset, and superset.
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 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.
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.
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.
This document discusses sets and set theory. It defines what a set is, the different types of sets, and how sets are represented. The key points are:
- A set is a collection of distinct objects, called elements or members. Sets can be finite or infinite.
- There are different types of sets including empty sets, which have no elements, singleton sets, which have one element, and finite sets, which have a fixed number of elements.
- Sets can be represented in roster form by listing elements between curly brackets or in set builder form using a rule to describe the elements.
- Equivalent sets have the same number of elements, while infinite sets have an indefinite number of
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.
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.
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.
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
- A set is a collection of distinct objects called elements or members. Sets can contain numbers, people, animals, letters, or other sets.
- There are different types of sets including finite, infinite, singleton, null/empty, equivalent, equal, overlapping, disjoint, and subset.
- Key properties of sets are discussed such as cardinality/cardinality, union, intersection, elements, and relationships between sets like subset and equality.
- Different types of numbers are defined including natural numbers, integers, rational numbers, irrational numbers, prime numbers, and real numbers. Empty sets, phi symbol, and whether zero is positive or negative are also covered.
This document discusses sets and set operations. It defines key concepts such as:
- Sets can be represented in descriptive form, set builder form, and roster form.
- Universal sets, subsets, proper subsets, power sets, unions, intersections, complements, disjoint sets, differences, and symmetric differences of sets.
- Examples of how to use formulas involving sets and set operations to solve problems, such as finding the size of an intersection given other set information.
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.
Chapter 2 Mathematical Language and Symbols.pdfRaRaRamirez
This document discusses mathematical language and symbols. It defines key concepts such as sets, relations, functions, and binary operations. Sets are collections of distinct objects that can be defined using a roster or rule. Relations pair elements between two sets. A function is a special type of relation where each input is paired with exactly one output. Binary operations take two inputs from a set and return an output in that same set. Common properties of binary operations include commutativity and associativity.
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 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.
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.
The document defines some basic concepts of sets including:
- A set is a well-defined collection of objects that can be represented in statement, roster, or set-builder form.
- Standard sets in math include the sets of natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- An element belongs to a set if it is contained within the set. Sets are usually represented by capital letters and elements by lowercase letters.
- Sets can be empty, equal, equivalent, finite or infinite, singleton, subsets of other sets, or universal sets containing all elements of other sets.
This document provides an overview of sets and related concepts in discrete mathematics. Some key points covered include:
- A set is an unordered collection of distinct objects. Sets can contain numbers, words, or other sets. Order and duplicates do not matter.
- Sets are specified using curly brackets and listing elements, set-builder notation, ellipses, or capital letters. Membership is denoted using the symbol ∈.
- Basic set relationships include subsets, proper subsets, equality, the empty set, unions, and intersections. Power sets contain all possible subsets.
- Tuples are ordered lists used to specify locations in n-dimensional spaces. Cartesian products combine elements from multiple sets into ordered pairs
This document defines and explains sets and related concepts such as subsets, the universal set, the null set, cardinality of sets, set notation, equivalent sets, subsets, and the number of possible subsets. It provides examples to illustrate these concepts such as defining sets using listing elements and rules, identifying subsets, drawing Venn diagrams, and calculating the number of subsets. Key terms defined include sets, subsets, universal set, null set, and cardinality. Notation and symbols used to represent sets and relationships between sets are also explained.
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
Set theory- Introduction, symbols with its meaningDipakMahurkar1
The document provides information about the concepts of set theory that will be covered in the Discrete Mathematics and Information Theory course. It defines basic set operations like union, intersection, complement and subset. It explains notation for set membership, empty set, universal set and Venn diagrams. Examples are given for each concept to illustrate the set operations and relationships between different sets.
The document provides an introduction to the concept of sets in mathematics. It defines what a set is as a collection of distinct objects, called elements or members, that fall under a certain condition. It discusses different types of sets such as well-defined sets, which have clear criteria for inclusion of elements, and not well-defined sets, which do not. It also covers set notation and symbols used to represent sets and define membership and non-membership of elements. Key topics include using curly brackets to enclose set elements, the element-of symbol to indicate membership, and the empty set symbol. The document demonstrates set concepts through examples and encourages readers to practice determining whether given sets are well-defined or not.
The document defines and provides examples of different types of functions including:
- Domain, codomain, and range of a function
- Injective, surjective, and bijective functions
- Into, one-to-one into, many-to-one, and many-to-one onto functions
It also discusses recurrence relations and recursively defined functions, providing examples of how functions can be defined recursively by building on previous terms.
This document discusses mathematical concepts related to relations including:
1. The inverse of a relation R-1, which relates elements in the opposite direction as R.
2. The composition of two relations R and S, denoted R◦S or RS, which relates elements related by both R and S.
3. Matrices can represent relations and be used to calculate their composition.
4. A partial order relation on a set A is a relation that is reflexive, antisymmetric, and transitive. Examples of partial order relations include set inclusion and the less than or equal to relation on real numbers.
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.
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.
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.
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
- A set is a collection of distinct objects called elements or members. Sets can contain numbers, people, animals, letters, or other sets.
- There are different types of sets including finite, infinite, singleton, null/empty, equivalent, equal, overlapping, disjoint, and subset.
- Key properties of sets are discussed such as cardinality/cardinality, union, intersection, elements, and relationships between sets like subset and equality.
- Different types of numbers are defined including natural numbers, integers, rational numbers, irrational numbers, prime numbers, and real numbers. Empty sets, phi symbol, and whether zero is positive or negative are also covered.
This document discusses sets and set operations. It defines key concepts such as:
- Sets can be represented in descriptive form, set builder form, and roster form.
- Universal sets, subsets, proper subsets, power sets, unions, intersections, complements, disjoint sets, differences, and symmetric differences of sets.
- Examples of how to use formulas involving sets and set operations to solve problems, such as finding the size of an intersection given other set information.
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.
Chapter 2 Mathematical Language and Symbols.pdfRaRaRamirez
This document discusses mathematical language and symbols. It defines key concepts such as sets, relations, functions, and binary operations. Sets are collections of distinct objects that can be defined using a roster or rule. Relations pair elements between two sets. A function is a special type of relation where each input is paired with exactly one output. Binary operations take two inputs from a set and return an output in that same set. Common properties of binary operations include commutativity and associativity.
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 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.
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.
The document defines some basic concepts of sets including:
- A set is a well-defined collection of objects that can be represented in statement, roster, or set-builder form.
- Standard sets in math include the sets of natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
- An element belongs to a set if it is contained within the set. Sets are usually represented by capital letters and elements by lowercase letters.
- Sets can be empty, equal, equivalent, finite or infinite, singleton, subsets of other sets, or universal sets containing all elements of other sets.
This document provides an overview of sets and related concepts in discrete mathematics. Some key points covered include:
- A set is an unordered collection of distinct objects. Sets can contain numbers, words, or other sets. Order and duplicates do not matter.
- Sets are specified using curly brackets and listing elements, set-builder notation, ellipses, or capital letters. Membership is denoted using the symbol ∈.
- Basic set relationships include subsets, proper subsets, equality, the empty set, unions, and intersections. Power sets contain all possible subsets.
- Tuples are ordered lists used to specify locations in n-dimensional spaces. Cartesian products combine elements from multiple sets into ordered pairs
This document defines and explains sets and related concepts such as subsets, the universal set, the null set, cardinality of sets, set notation, equivalent sets, subsets, and the number of possible subsets. It provides examples to illustrate these concepts such as defining sets using listing elements and rules, identifying subsets, drawing Venn diagrams, and calculating the number of subsets. Key terms defined include sets, subsets, universal set, null set, and cardinality. Notation and symbols used to represent sets and relationships between sets are also explained.
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
Set theory- Introduction, symbols with its meaningDipakMahurkar1
The document provides information about the concepts of set theory that will be covered in the Discrete Mathematics and Information Theory course. It defines basic set operations like union, intersection, complement and subset. It explains notation for set membership, empty set, universal set and Venn diagrams. Examples are given for each concept to illustrate the set operations and relationships between different sets.
The document provides an introduction to the concept of sets in mathematics. It defines what a set is as a collection of distinct objects, called elements or members, that fall under a certain condition. It discusses different types of sets such as well-defined sets, which have clear criteria for inclusion of elements, and not well-defined sets, which do not. It also covers set notation and symbols used to represent sets and define membership and non-membership of elements. Key topics include using curly brackets to enclose set elements, the element-of symbol to indicate membership, and the empty set symbol. The document demonstrates set concepts through examples and encourages readers to practice determining whether given sets are well-defined or not.
The document defines and provides examples of different types of functions including:
- Domain, codomain, and range of a function
- Injective, surjective, and bijective functions
- Into, one-to-one into, many-to-one, and many-to-one onto functions
It also discusses recurrence relations and recursively defined functions, providing examples of how functions can be defined recursively by building on previous terms.
This document discusses mathematical concepts related to relations including:
1. The inverse of a relation R-1, which relates elements in the opposite direction as R.
2. The composition of two relations R and S, denoted R◦S or RS, which relates elements related by both R and S.
3. Matrices can represent relations and be used to calculate their composition.
4. A partial order relation on a set A is a relation that is reflexive, antisymmetric, and transitive. Examples of partial order relations include set inclusion and the less than or equal to relation on real numbers.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
This document provides an overview of propositional logic:
1. It defines propositions as statements that can be either true or false, and propositional variables connected by logical connectives like AND and OR.
2. It explains the five main connectives - negation, conjunction, disjunction, implication, and biconditional - and provides their truth tables.
3. It discusses well-formed formulas, tautologies, contradictions, and contingencies. Equivalences between logical statements and duality principles are also covered.
4. Normal forms like CNF and DNF are defined, along with concepts like minterms and maxterms. Rules of inference for building logical arguments are outlined
A subgroup is a subset of a group that is also a group. A normal subgroup is a subgroup where left and right cosets are equal. The intersection of two normal subgroups is also a normal subgroup. A permutation is a one-to-one mapping from a set to itself. Permutations form a group. A cyclic permutation has a single generator element. The length of a cycle of an element in a permutation is the order of that element. A ring is a set with two binary operations, addition and multiplication, satisfying certain properties. Integral domains have no zero divisors. A field has nonzero multiplication and every nonzero element has a multiplicative inverse.
The document discusses different types of algebraic structures including semigroups, monoids, groups, and abelian groups. It defines each structure based on what axioms they satisfy such as closure, associativity, identity element, and inverses. Examples are given of sets that satisfy each structure under different binary operations like addition, multiplication, subtraction and division. The properties of algebraic structures like commutativity, associativity, identity, inverses and cancellation laws are also explained.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document discusses database management systems (DBMS). It defines a DBMS as system software that allows users to create, manage, and access databases. A DBMS provides a systematic way for end users to create, read, update, and delete data in a database. It also serves as an interface between databases and users or application programs, ensuring data is organized and accessible. The document outlines some key components of a DBMS, including users, data, DBMS software, and database applications. It also describes several advantages of using a DBMS, such as improved data mapping and access, reduced data redundancy, data independence and consistency, and enhanced security features.
The document is a presentation on multimedia given by Miss Nandini Sharma at Shri Ram College of Engineering and Management in Palwal. It defines multimedia and discusses multimedia input and output devices. It also covers applications of multimedia, frameworks, authoring tools, distribution networks and techniques like animation, morphing and video on demand.
Number System, Positional and non-positional number system, conversion number system from binary to another base and vice versa, decimal to another base and vice versa, convert another base than 10 to another base than 10, binary arithmetic operation such as binary addition, subtraction, multiplication, division
Computer Network Notes (Handwritten) UNIT 2NANDINI SHARMA
Data link layer: flow control, error control, line discipline, stop and wait, sliding window protocol, stop and wait arq, sliding window arq, BSC, HDLC, bit stuffing, elemenary data link protocol etc
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
An improved modulation technique suitable for a three level flying capacitor ...IJECEIAES
This research paper introduces an innovative modulation technique for controlling a 3-level flying capacitor multilevel inverter (FCMLI), aiming to streamline the modulation process in contrast to conventional methods. The proposed
simplified modulation technique paves the way for more straightforward and
efficient control of multilevel inverters, enabling their widespread adoption and
integration into modern power electronic systems. Through the amalgamation of
sinusoidal pulse width modulation (SPWM) with a high-frequency square wave
pulse, this controlling technique attains energy equilibrium across the coupling
capacitor. The modulation scheme incorporates a simplified switching pattern
and a decreased count of voltage references, thereby simplifying the control
algorithm.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
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What is set (in mathematics)?
The collection of well-defined distinct objects is known as a set. The word well-defined refers to a
specific property which makes it easy to identify whether the given object belongs to the set or not.
The word ‘distinct’ means that the objects of a set must be all different.
For example:
1. The collection of children in class VII whose weight exceeds 35 kg represents a set.
2. The collection of all the intelligent children in class VII does not represent a set because the word
intelligent is vague. What may appear intelligent to one person may not appear the same to another
person.
Elements of Set:
The different objects that form a set are called the elements of a set. The elements of the set are
written in any order and are not repeated. Elements are denoted by small letters.
Notation of a Set:
A set is usually denoted by capital letters and elements are denoted by small letters
If x is an element of set A, then we say x ϵ A. [x belongs to A]
If x is not an element of set A, then we say x ∉ A. [x does not belong to A]
For example:
The collection of vowels in the English alphabet.
Solution :
Let us denote the set by V, then the elements of the set are a, e, i, o, u or we can say, V = [a, e, i, o,
u].
We say a ∈ V, e ∈ V, i ∈ V, o ∈ V and u ∈ V.
Also, we can say b ∉ V, c ∉ v, d ∉ v, etc.
In representation of a set the following three methods are commonly used:
(i) Statement form method
(ii) Roster or tabular form method
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(iii) Rule or set builder form method
1. Statement form:
In this, well-defined description of the elements of the set is given and the same are enclosed in curly
brackets.
For example:
(i) The set of odd numbers less than 7 is written as: {odd numbers less than 7}.
(ii) A set of football players with ages between 22 years to 30 years.
(iii) A set of numbers greater than 30 and smaller than 55.
(iv) A set of students in class VII whose weights are more than your weight.
2. Roster form or tabular form:
In this, elements of the set are listed within the pair of brackets { } and are separated by commas.
For example:
(i) Let N denote the set of first five natural numbers.
Therefore, N = {1, 2, 3, 4, 5} → Roster Form
(ii) The set of all vowels of the English alphabet.
Therefore, V = {a, e, i, o, u} → Roster Form
(iii) The set of all odd numbers less than 9.
Therefore, X = {1, 3, 5, 7} → Roster Form
(iv) The set of all natural number which divide 12.
Therefore, Y = {1, 2, 3, 4, 6, 12} → Roster Form
(v) The set of all letters in the word MATHEMATICS.
Therefore, Z = {M, A, T, H, E, I, C, S} → Roster Form
(vi) W is the set of last four months of the year.
Therefore, W = {September, October, November, December} → Roster Form
Note:
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The order in which elements are listed is immaterial but elements must not be repeated.
3. Set builder form:
In this, a rule, or the formula or the statement is written within the pair of brackets so that the set is well
defined. In the set builder form, all the elements of the set, must possess a single property to become the
member of that set.
In this form of representation of a set, the element of the set is described by using a symbol ‘x’ or any other
variable followed by a colon The symbol ‘:‘ or ‘|‘ is used to denote such that and then we write the property
possessed by the elements of the set and enclose the whole description in braces. In this, the colon stands for
‘such that’ and braces stand for ‘set of all’.
For example:
(i) Let P is a set of counting numbers greater than 12;
the set P in set-builder form is written as :
P = {x : x is a counting number and greater than 12}
or
P = {x | x is a counting number and greater than 12}
This will be read as, 'P is the set of elements x such that x is a counting number and is greater than 12'.
Note:
The symbol ':' or '|' placed between 2 x's stands for such that.
(ii) Let A denote the set of even numbers between 6 and 14. It can be written in the set builder form
as;
A = {x|x is an even number, 6 < x < 14}
or A = {x : x ∈ P, 6 < x < 14 and P is an even number}
(iii) If X = {4, 5, 6, 7} . This is expressed in roster form.
Let us express in set builder form.
X = {x : x is a natural number and 3 < x < 8}
(iv) The set A of all odd natural numbers can be written as
A = {x : x is a natural number and x = 2n + 1 for n ∈ W}
Solved example using the three methods of representation of a set:
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The set of integers lying between -2 and 3.
Statement form: {I is a set of integers lying between -2 and 3}
Roster form: I = {-1, 0, 1, 2}
Set builder form: I = {x : x ∈ I, -2 < x < 3}
What are the different notations in sets?
To learn about sets we shall use some accepted notations for the familiar sets of numbers.
Some of the different notations used in sets are:
∈
∉
: or |
∅
n(A)
∪
∩
N
W
I or Z
Z+
Q
Q+
R
R+
C
Belongs to
Does not belongs to
Such that
Null set or empty set
Cardinal number of the set A
Union of two sets
Intersection of two sets
Set of natural numbers = {1, 2, 3, ……}
Set of whole numbers = {0, 1, 2, 3, ………}
Set of integers = {………, -2, -1, 0, 1, 2, ………}
Set of all positive integers
Set of all rational numbers
Set of all positive rational numbers
Set of all real numbers
Set of all positive real numbers
Set of all complex numbers
These are the different notations in sets generally required while solving various types of problems on sets.
Note:
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(i) The pair of curly braces { } denotes a set. The elements of set are written inside a pair of curly braces
separated by commas.
(ii) The set is always represented by a capital letter such as; A, B, C, …….. .
(iii) If the elements of the sets are alphabets then these elements are written in small letters.
(iv) The elements of a set may be written in any order.
(v) The elements of a set must not be repeated.
(vi) The Greek letter Epsilon ‘∈’ is used for the words ‘belongs to’, ‘is an element of’, etc.
Therefore, x ∈ A will be read as ‘x belongs to set A’ or ‘x is an element of the set A'.
(vii) The symbol ‘∉’ stands for ‘does not belongs to’ also for ‘is not an element of’.
Therefore, x ∉ A will read as ‘x does not belongs to set A’ or ‘x is not an element of the set A'.
The standard sets of numbers can be expressed in all the three forms of representation of a set i.e.,
statement form, roster form, set builder form.
1. N = Natural numbers
= Set of all numbers starting from 1 → Statement form
= Set of all numbers 1, 2, 3, ………..
= {1, 2, 3, …….} → Roster form
= {x :x is a counting number starting from 1} → Set builder form
Therefore, the set of natural numbers is denoted by N i.e., N = {1, 2, 3, …….}
2. W = Whole numbers
= Set containing zero and all natural numbers → Statement form
= {0, 1, 2, 3, …….} → Roster form
= {x :x is a zero and all natural numbers} → Set builder form
Therefore, the set of whole numbers is denoted by W i.e., W = {0, 1, 2, .......}
3. Z or I = Integers
= Set containing negative of natural numbers, zero and the natural numbers
→ Statement form
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= {………, -3, -2, -1, 0, 1, 2, 3, …….} → Roster form
= {x :x is a containing negative of natural numbers, zero and the natural numbers}
→ Set builder form
Therefore, the set of integers is denoted by I or Z i.e., I = {...., -2, -1, 0, 1, 2, ….}
4. E = Even natural numbers.
= Set of natural numbers, which are divisible by 2 → Statement form
= {2, 4, 6, 8, ……….} → Roster form
= {x :x is a natural number, which are divisible by 2} → Set builder form
Therefore, the set of even natural numbers is denoted by E i.e., E = {2, 4, 6, 8,.......}
5. O = Odd natural numbers.
= Set of natural numbers, which are not divisible by 2 → Statement form
= {1, 3, 5, 7, 9, ……….} → Roster form
= {x :x is a natural number, which are not divisible by 2} → Set builder form
Therefore, the set of odd natural numbers is denoted by O i.e., O = {1, 3, 5, 7, 9,.......}
Therefore, almost every standard sets of numbers can be expressed in all the three methods as discussed above.
What are the different types of sets?
The different types of sets are explained below with examples.
Empty Set or Null Set:
A set which does not contain any element is called an empty set, or the null set or the void set and it
is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set,
since the number of elements in an empty set is finite, i.e., 0.
For example: (a) The set of whole numbers less than 0.
(b) Clearly there is no whole number less than 0.
Therefore, it is an empty set.
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(c) N = {x : x ∈ N, 3 < x < 4}
• Let A = {x : 2 < x < 3, x is a natural number}
Here A is an empty set because there is no natural number between
2 and 3.
• Let B = {x : x is a composite number less than 4}.
Here B is an empty set because there is no composite number less than 4.
Note:
∅ ≠ {0} ∴ has no element.
{0} is a set which has one element 0.
The cardinal number of an empty set, i.e., n(∅) = 0
Singleton Set:
A set which contains only one element is called a singleton set.
For example:
• A = {x : x is neither prime nor composite}
It is a singleton set containing one element, i.e., 1.
• B = {x : x is a whole number, x < 1}
This set contains only one element 0 and is a singleton set.
• Let A = {x : x ∈ N and x² = 4}
Here A is a singleton set because there is only one element 2 whose square is 4.
• Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is even, i.e., 2.
Finite Set:
A set which contains a definite number of elements is called a finite set. Empty set is also called a
finite set.
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For example:
• The set of all colors in the rainbow.
• N = {x : x ∈ N, x < 7}
• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}
Infinite Set:
The set whose elements cannot be listed, i.e., set containing never-ending elements is called an
infinite set.
For example:
• Set of all points in a plane
• A = {x : x ∈ N, x > 1}
• Set of all prime numbers
• B = {x : x ∈ W, x = 2n}
Note:
All infinite sets cannot be expressed in roster form.
For example:
The set of real numbers since the elements of this set do not follow any particular pattern.
Cardinal Number of a Set:
The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by
n(A).
For example:
• A {x : x ∈ N, x < 5}
A = {1, 2, 3, 4}
Therefore, n(A) = 4
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• B = set of letters in the word ALGEBRA
B = {A, L, G, E, B, R}
Therefore, n(B) = 6
Equivalent Sets:
Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The
symbol for denoting an equivalent set is ‘↔’.
For example:
A = {1, 2, 3} Here n(A) = 3
B = {p, q, r} Here n(B) = 3
Therefore, A ↔ B
Equal sets:
Two sets A and B are said to be equal if they contain the same elements. Every element of A is an
element of B and every element of B is an element of A.
For example:
A = {p, q, r, s}
B = {p, s, r, q}
Therefore, A = B
The various types of sets and their definitions are explained above with the help of examples.
The relations are stated between the pairs of sets. Learn to state, giving reasons whether the
following sets are equivalent or equal, disjoint or overlapping.
Equal Set:
Two sets A and B are said to be equal if all the elements of set A are in set B and vice versa. The
symbol to denote an equal set is =.
A = B means set A is equal to set B and set B is equal to set A.
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For example;
A = {2, 3, 5}
B = {5, 2, 3}
Here, set A and set B are equal sets.
Equivalent Set:
Two sets A and B are said to be equivalent sets if they contain the same number of elements. The
symbol to denote equivalent set is ↔.
A ↔ means set A and set B contain the same number of elements.
For example;
A = {p, q, r}
B = {2, 3, 4}
Here, we observe that both the sets contain three elements.
Notes:
Equal sets are always equivalent.
Equivalent sets may not be equal.
Disjoint Sets:
Two sets A and B are said to be disjoint, if they do not have any element in common.
For example;
A = {x : x is a prime number}
B = {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.
Overlapping sets:
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Two sets A and B are said to be overlapping if they contain at least one element in common.
For example;
• A = {a, b, c, d}
B = {a, e, i, o, u}
• X = {x : x ∈ N, x < 4}
Y = {x : x ∈ I, -1 < x < 4}
Here, the two sets contain three elements in common, i.e., (1, 2, 3)
The above explanations will help us to find whether the pairs of sets are equal sets or equivalent sets,
disjoint sets or overlapping sets.
The relations are stated between the pairs of sets. Learn to state, giving reasons whether the
following sets are equivalent or equal, disjoint or overlapping.
Equal Set:
Two sets A and B are said to be equal if all the elements of set A are in set B and vice versa. The
symbol to denote an equal set is =.
A = B means set A is equal to set B and set B is equal to set A.
For example;
A = {2, 3, 5}
B = {5, 2, 3}
Here, set A and set B are equal sets.
Equivalent Set:
Two sets A and B are said to be equivalent sets if they contain the same number of elements. The
symbol to denote equivalent set is ↔.
A ↔ means set A and set B contain the same number of elements.
For example;
A = {p, q, r}
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B = {2, 3, 4}
Here, we observe that both the sets contain three elements.
Notes:
Equal sets are always equivalent.
Equivalent sets may not be equal.
Disjoint Sets:
Two sets A and B are said to be disjoint, if they do not have any element in common.
For example;
A = {x : x is a prime number}
B = {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.
Overlapping sets:
Two sets A and B are said to be overlapping if they contain at least one element in common.
For example;
• A = {a, b, c, d}
B = {a, e, i, o, u}
• X = {x : x ∈ N, x < 4}
Y = {x : x ∈ I, -1 < x < 4}
Here, the two sets contain three elements in common, i.e., (1, 2, 3)
The above explanations will help us to find whether the pairs of sets are equal sets or equivalent sets,
disjoint sets or overlapping sets.
Definition of Subset:
If A and B are two sets, and every element of set A is also an element of set B, then A is called a
subset of B and we write it as A ⊆ B or B ⊇ A
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The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’
• Every set is a subset of itself, i.e., A ⊂ A, B ⊂ B.
• Empty set is a subset of every set.
• Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’.
• A ⊆ B means A is a subset of B or A is contained in B.
• B ⊆ A means B contains A.
For example;
1. Let A = {2, 4, 6}
B = {6, 4, 8, 2}
Here A is a subset of B
Since, all the elements of set A are contained in set B.
But B is not the subset of A
Since, all the elements of set B are not contained in set A.
Notes:
If ACB and BCA, then A = B, i.e., they are equal sets.
Every set is a subset of itself.
Null set or ∅ is a subset of every set.
2. The set N of natural numbers is a subset of the set Z of integers and we write N ⊂ Z.
3. Let A = {2, 4, 6}
B = {x : x is an even natural number less than 8}
Here A ⊂ B and B ⊂ A.
Hence, we can say A = B
4. Let A = {1, 2, 3, 4}
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B = {4, 5, 6, 7}
Here A ⊄ B and also B ⊄ C
[⊄ denotes ‘not a subset of’]
Super Set:
Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote ‘is a super set of’
For example;
A = {a, e, i, o, u}
B = {a, b, c, ............., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A
Proper Subset:
If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The
symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B.
For example;
1. A = {1, 2, 3, 4}
Here n(A) = 4
B = {1, 2, 3, 4, 5}
Here n(B) = 5
We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A.
So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B
Notes:
No set is a proper subset of itself.
Null set or ∅ is a proper subset of every set.
2. A = {p, q, r}
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B = {p, q, r, s, t}
Here A is a proper subset of B as all the elements of set A are in set B and also A ≠ B.
Notes:
No set is a proper subset of itself.
Empty set is a proper subset of every set.
Power Set:
The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A),
every element is a set.
For example;
If A = {p, q} then all the subsets of A will be
P(A) = {∅, {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = 4 = 22
In general, n[P(A)] = 2m where m is the number of elements in set A.
Universal Set
A set which contains all the elements of other given sets is called a universal set. The symbol for
denoting a universal set is ∪ or ξ.
For example;
1. If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]
2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal set is a
set of all integers.
3. If A = {a, b, c} B = {d, e} C = {f, g, h, i}
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then U = {a, b, c, d, e, f, g, h, i} can be taken as universal set.
Number of Subsets of a given Set:
If a set contains ‘n’ elements, then the number of subsets of the set is 2nn.
Number of Proper Subsets of the Set:
If a set contains ‘n’ elements, then the number of proper subsets of the set is 2nn - 1.
If A = {p, q} the proper subsets of A are [{ }, {p}, {q}]
⇒ Number of proper subsets of A are 3 = 222 - 1 = 4 - 1
In general, number of proper subsets of a given set = 2mm - 1, where m is the number of elements.
For example:
1. If A {1, 3, 5}, then write all the possible subsets of A. Find their numbers.
Solution:
The subset of A containing no elements - { }
The subset of A containing one element each - {1} {3} {5}
The subset of A containing two elements each - {1, 3} {1, 5} {3, 5}
The subset of A containing three elements - {1, 3, 5)
Therefore, all possible subsets of A are { }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}
Therefore, number of all possible subsets of A is 8 which is equal 233.
Proper subsets are = { }, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}
Number of proper subsets are 7 = 8 - 1 = 233 - 1
2. If the number of elements in a set is 2, find the number of subsets and proper subsets.
Solution:
Number of elements in a set = 2
Then, number of subsets = 222 = 4
Also, the number of proper subsets = 222 - 1
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= 4 – 1 = 3
3. If A = {1, 2, 3, 4, 5}
then the number of proper subsets = 255 - 1
= 32 - 1 = 31 {Take [2nn - 1]}
and power set of A = 255 = 32 {Take [2nn]}
Definition of operations on sets:
When two or more sets combine together to form one set under the given conditions, then operations
on sets are carried out.
What are the four basic operations on sets?
Solution:
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Definition of Union of Sets:
Union of two given sets is the smallest set which contains all the elements of both the sets.
To find the union of two given sets A and B is a set which consists of all the elements of A and all the
elements of B such that no element is repeated.
The symbol for denoting union of sets is ‘∪’.
For example;
Let set A = {2, 4, 5, 6}
and set B = {4, 6, 7, 8}
Taking every element of both the sets A and B, without repeating any element, we get a new set = {2, 4,
5, 6, 7, 8}
This new set contains all the elements of set A and all the elements of set B with no repetition of elements
and is named as union of set A and B.
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The symbol used for the union of two sets is ‘∪’.
Therefore, symbolically, we write union of the two sets A and B is A ∪ B which means A union B.
Therefore, A ∪ B = {x : x ∈ A or x ∈ B}
Solved examples to find union of two given sets:
1. If A = {1, 3, 7, 5} and B = {3, 7, 8, 9}. Find union of two set A and B.
Solution:
A ∪ B = {1, 3, 5, 7, 8, 9}
No element is repeated in the union of two sets. The common elements 3, 7 are taken only once.
2. Let X = {a, e, i, o, u} and Y = {ф}. Find union of two given sets X and Y.
Solution:
X ∪ Y = {a, e, i, o, u}
Therefore, union of any set with an empty set is the set itself.
3. If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6, 8}.
(i) Find the union of sets P and Q
(ii) Find the union of two set P and R
(iii) Find the union of the given sets Q and R
Solution:
(i) Union of sets P and Q is P ∪ Q
The smallest set which contains all the elements of set P and all the elements of set Q is {0, 2, 3, 4, 5, 6, 7,
9, 12}.
(ii) Union of two set P and R is P ∪ R
The smallest set which contains all the elements of set P and all the elements of set R is {2, 3, 4, 5, 6, 7,
8}.
(iii) Union of the given sets Q and R is Q ∪ R
The smallest set which contains all the elements of set Q and all the elements of set R is {0, 2, 3, 4, 6, 8,
9, 12}.
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Notes:
A and B are the subsets of A ∪ B
The union of sets is commutative, i.e., A ∪ B = B ∪ A.
The operations are performed when the sets are expressed in roster form.
Some properties of the operation of union:
(i) A∪B = B∪A (Commutative law)
(ii) A∪(B∪C) = (A∪B)∪C (Associative law)
(iii) A ∪ ϕ = A (Law of identity element, is the identity of ∪)
(iv) A∪A = A (Idempotent law)
(v) U∪A = U (Law of ∪) ∪ is the universal set.
Notes:
A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is always the set itself.
Definition of Intersection of Sets:
Intersection of two given sets is the largest set which contains all the elements that are common to
both the sets.
To find the intersection of two given sets A and B is a set which consists of all the elements which
are common to both A and B.
The symbol for denoting intersection of sets is ‘∩‘.
For example:
Let set A = {2, 3, 4, 5, 6}
and set B = {3, 5, 7, 9}
In this two sets, the elements 3 and 5 are common. The set containing these common elements i.e.,
{3, 5} is the intersection of set A and B.
The symbol used for the intersection of two sets is ‘∩‘.
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Therefore, symbolically, we write intersection of the two sets A and B is A ∩ B which means A
intersection B.
The intersection of two sets A and B is represented as A ∩ B = {x : x ∈ A and x ∈ B}
Solved examples to find intersection of two given sets:
1. If A = {2, 4, 6, 8, 10} and B = {1, 3, 8, 4, 6}. Find intersection of two set A and B.
Solution:
A ∩ B = {4, 6, 8}
Therefore, 4, 6 and 8 are the common elements in both the sets.
2. If X = {a, b, c} and Y = {ф}. Find intersection of two given sets X and Y.
Solution:
X ∩ Y = { }
3. If set A = {4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15, 18} and set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
(i) Find the intersection of sets A and B.
(ii) Find the intersection of two set B and C.
(iii) Find the intersection of the given sets A and C.
Solution:
(i) Intersection of sets A and B is A ∩ B
Set of all the elements which are common to both set A and set B is {6, 12}.
(ii) Intersection of two set B and C is B ∩ C
Set of all the elements which are common to both set B and set C is {3, 6, 9}.
(iii) Intersection of the given sets A and C is A ∩ C
Set of all the elements which are common to both set A and set C is {4, 6, 8, 10}.
Notes:
A ∩ B is a subset of A and B.
Intersection of a set is commutative, i.e., A ∩ B = B ∩ A.
Operations are performed when the set is expressed in the roster form.
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Some properties of the operation of intersection
(i) A∩B = B∩A (Commutative law)
(ii) (A∩B)∩C = A∩ (B∩C) (Associative law)
(iii) ϕ ∩ A = ϕ (Law of ϕ)
(iv) U∩A = A (Law of ∪)
(v) A∩A = A (Idempotent law)
(vi) A∩(B∪C) = (A∩B) ∪ (A∩C) (Distributive law) Here ∩ distributes over ∪
Also, A∪(B∩C) = (AUB) ∩ (AUC) (Distributive law) Here ∪ distributes over ∩
Notes:
A ∩ ϕ = ϕ ∩ A = ϕ i.e. intersection of any set with the empty set is always the empty set.
How to find the difference of two sets?
If A and B are two sets, then their difference is given by A - B or B - A.
• If A = {2, 3, 4} and B = {4, 5, 6}
A - B means elements of A which are not the elements of B.
i.e., in the above example A - B = {2, 3}
In general, B - A = {x : x ∈ B, and x ∉ A}
• If A and B are disjoint sets, then A – B = A and B – A = B
Solved examples to find the difference of two sets:
1. A = {1, 2, 3} and B = {4, 5, 6}.
Find the difference between the two sets:
(i) A and B
(ii) B and A
Solution:
The two sets are disjoint as they do not have any elements in common.
(i) A - B = {1, 2, 3} = A
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(ii) B - A = {4, 5, 6} = B
2. Let A = {a, b, c, d, e, f} and B = {b, d, f, g}.
Find the difference between the two sets:
(i) A and B
(ii) B and A
Solution:
(i) A - B = {a, c, e}
Therefore, the elements a, c, e belong to A but not to B
(ii) B - A = {g)
Therefore, the element g belongs to B but not A.
3. Given three sets P, Q and R such that:
P = {x : x is a natural number between 10 and 16},
Q = {y : y is a even number between 8 and 20} and
R = {7, 9, 11, 14, 18, 20}
(i) Find the difference of two sets P and Q
(ii) Find Q - R
(iii) Find R - P
(iv) Find Q – P
Solution:
According to the given statements:
P = {11, 12, 13, 14, 15}
Q = {10, 12, 14, 16, 18}
R = {7, 9, 11, 14, 18, 20}
(i) P – Q = {Those elements of set P which are not in set Q}
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= {11, 13, 15}
(ii) Q – R = {Those elements of set Q not belonging to set R}
= {10, 12, 16}
(iii) R – P = {Those elements of set R which are not in set P}
= {7, 9, 18, 20}
(iv) Q – P = {Those elements of set Q not belonging to set P}
= {10, 16, 18}
In complement of a set if ξ be the universal set and A a subset of ξ, then the complement of A is the
set of all elements of ξ which are not the elements of A.
Symbolically, we denote the complement of A with respect to ξ as A’.
For Example; If ξ = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 7} find A'.
Solution:
We observe that 2, 4, 5, 6 are the only elements of ξ which do not belong to A.
Therefore, A' = {2, 4, 5, 6}
Note:
The complement of a universal set is an empty set.
The complement of an empty set is a universal set.
The set and its complement are disjoint sets.
For Example;
1. Let the set of natural numbers be the universal set and A is a set of even natural numbers,
then A' {x: x is a set of odd natural numbers}
2. Let ξ = The set of letters in the English alphabet.
A = The set of consonants in the English alphabet
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then A' = The set of vowels in the English alphabet.
3. Show that;
(a) The complement of a universal set is an empty set.
Let ξ denote the universal set, then
ξ' = The set of those elements which are not in ξ.
= empty set = ϕ
Therefore, ξ = ϕ so the complement of a universal set is an empty set.
(b) A set and its complement are disjoint sets.
Let A be any set then A' = set of those elements of ξ which are not in A'.
Let x ∉ A, then x is an element of ξ not contained in A'
So x ∉ A'
Therefore, A and A' are disjoint sets.
Therefore, Set and its complement are disjoint sets
Similarly, in complement of a set when U be the universal set and A is a subset of U. Then the
complement of A is the set all elements of U which are not the elements of A.
Symbolically, we write A' to denote the complement of A with respect to U.
Thus, A' = {x : x ∈ U and x ∉ A}
Obviously A' = {U - A}
For Example; Let U = {2, 4, 6, 8, 10, 12, 14, 16}
A = {6, 10, 4, 16}
A' = {2, 8, 12, 14}
We observe that 2, 8, 12, 14 are the only elements of U which do not belong to A.
Some properties of complement sets
(i) A ∪ A' = A' ∪ A = ∪ (Complement law)
(ii) (A ∩ B') = ϕ (Complement law)
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(iii) (A ∪ B) = A' ∩ B' (De Morgan’s law)
(iv) (A ∩ B)' = A' ∪ B' (De Morgan’s law)
(v) (A')' = A (Law of complementation)
(vi) ϕ' = ∪ (Law of empty set
(vii) ∪' = ϕ and universal set)
What is the cardinal number of a set?
The number of distinct elements in a finite set is called its cardinal number. It is denoted as n(A) and
read as ‘the number of elements of the set’.
For example:
(i) Set A = {2, 4, 5, 9, 15} has 5 elements.
Therefore, the cardinal number of set A = 5. So, it is denoted as n(A) = 5.
(ii) Set B = {w, x, y, z} has 4 elements.
Therefore, the cardinal number of set B = 4. So, it is denoted as n(B) = 4.
(iii) Set C = {Florida, New York, California} has 3 elements.
Therefore, the cardinal number of set C = 3. So, it is denoted as n(C) = 3.
(iv) Set D = {3, 3, 5, 6, 7, 7, 9} has 5 element.
Therefore, the cardinal number of set D = 5. So, it is denoted as n(D) = 5.
(v) Set E = { } has no element.
Therefore, the cardinal number of set D = 0. So, it is denoted as n(D) = 0.
Note:
(i) Cardinal number of an infinite set is not defined.
(ii) Cardinal number of empty set is 0 because it has no element.
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Solved examples on Cardinal number of a set:
1. Write the cardinal number of each of the following sets:
(i) X = {letters in the word MALAYALAM}
(ii) Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}
(iii) Z = {natural numbers between 20 and 50, which are divisible by 7}
Solution:
(i) Given, X = {letters in the word MALAYALAM}
Then, X = {M, A, L, Y}
Therefore, cardinal number of set X = 4, i.e., n(X) = 4
(ii) Given, Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}
Then, Y = {5, 6, 7, 11, 13, 8}
Therefore, cardinal number of set Y = 6, i.e., n(Y) = 6
(iii) Given, Z = {natural numbers between 20 and 50, which are divisible by 7}
Then, Z = {21, 28, 35, 42, 49}
Therefore, cardinal number of set Z = 5, i.e., n(Z) = 5
2. Find the cardinal number of a set from each of the following:
(i) P = {x | x ∈ N and x22 < 30}
(ii) Q = {x | x is a factor of 20}
Solution:
(i) Given, P = {x | x ∈ N and x22 < 30}
Then, P = {1, 2, 3, 4, 5}
Therefore, cardinal number of set P = 5, i.e., n(P) = 5
(ii) Given, Q = {x | x is a factor of 20}
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Then, Q = {1, 2, 4, 5, 10, 20}
Therefore, cardinal number of set Q = 6, i.e., n(Q) = 6
Cardinal Properties of Sets:
We have already learnt about the union, intersection and difference of sets. Now, we will go through
some practical problems on sets related to everyday life.
If A and B are finite sets, then
• n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
If A ∩ B = ф , then n(A ∪ B) = n(A) + n(B)
It is also clear from the Venn diagram that
• n(A - B) = n(A) - n(A ∩ B)
• n(B - A) = n(B) - n(A ∩ B)
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Problems on Cardinal Properties of Sets
1. If P and Q are two sets such that P ∪ Q has 40 elements, P has 22 elements and Q has 28 elements,
how many elements does P ∩ Q have?
Solution:
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Given n(P ∪ Q) = 40, n(P) = 18, n(Q) = 22
We know that n(P U Q) = n(P) + n(Q) - n(P ∩ Q)
So, 40 = 22 + 28 - n(P ∩ Q)
40 = 50 - n(P ∩ Q)
Therefore, n(P ∩ Q) = 50 – 40
= 10
2. In a class of 40 students, 15 like to play cricket and football and 20 like to play cricket. How many
like to play football only but not cricket?
Solution:
Let C = Students who like cricket
F = Students who like football
C ∩ F = Students who like cricket and football both
C - F = Students who like cricket only
F - C = Students who like football only.
n(C) = 20 n(C ∩ F) = 15 n (C U F) = 40 n (F) = ?
n(C ∪ F) = n(C) + n(F) - n(C ∩ F)
40 = 20 + n(F) - 15
40 = 5 + n(F)
40 – 5 = n(F)
Therefore, n(F)= 35
Therefore, n(F - C) = n(F) - n (C ∩ F)
= 35 – 15
= 20
Therefore, Number of students who like football only but not cricket = 20
More problems on cardinal properties of sets
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3. There is a group of 80 persons who can drive scooter or car or both. Out of these, 35 can drive
scooter and 60 can drive car. Find how many can drive both scooter and car? How many can drive
scooter only? How many can drive car only?
Solution:
Let S = {Persons who drive scooter}
C = {Persons who drive car}
Given, n(S ∪ C) = 80 n(S) = 35 n(C) = 60
Therefore, n(S ∪ C) = n(S) + n(C) - n(S ∩ C)
80 = 35 + 60 - n(S ∩ C)
80 = 95 - n(S ∩ C)
Therefore, n(S∩C) = 95 – 80 = 15
Therefore, 15 persons drive both scooter and car.
Therefore, the number of persons who drive a scooter only = n(S) - n(S ∩ C)
= 35 – 15
= 20
Also, the number of persons who drive car only = n(C) - n(S ∩ C)
= 60 - 15
= 45
4. It was found that out of 45 girls, 10 joined singing but not dancing and 24 joined singing. How
many joined dancing but not singing? How many joined both?
Solution:
Let S = {Girls who joined singing}
D = {Girls who joined dancing}
Number of girls who joined dancing but not singing = Total number of girls - Number of girls who
joined singing
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45 – 24
= 21
Now, n(S - D) = 10 n(S) =24
Therefore, n(S - D) = n(S) - n(S ∩ D)
⇒ n(S ∩ D) = n(S) - n(S - D)
= 24 - 10
= 14
Therefore, number of girls who joined both singing and dancing is 14.
What are Venn Diagrams?
Pictorial representations of sets represented by closed figures are called set diagrams or Venn
diagrams.
Venn diagrams are used to illustrate various operations like union, intersection and difference.
We can express the relationship among sets through this in a more significant way.
In this,
• A rectangle is used to represent a universal set.
• Circles or ovals are used to represent other subsets of the universal set.
Venn diagrams in different situations
• If a set A is a subset of set B, then the circle representing set A is drawn inside the circle
representing set B.
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• If set A and set B have some elements in common, then to represent them, we draw two circles
which are overlapping.
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• If set A and set B are disjoint, then they are represented by two non-intersecting circles.
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In this diagrams, the universal set is represented by a rectangular region and its subsets by circles inside the
rectangle. We represented disjoint set by disjoint circles and intersecting sets by intersecting circles.
Multisets
A multiset is an unordered collection of elements, in which the multiplicity of an element may be
one or more than one or zero. The multiplicity of an element is the number of times the element
repeated in the multiset. In other words, we can say that an element can appear any number of
times in a set.
Example:
1. A = {l, l, m, m, n, n, n, n}
2. B = {a, a, a, a, a, c}
Operations on Multisets
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1. Union of Multisets: The Union of two multisets A and B is a multiset such that the
multiplicity of an element is equal to the maximum of the multiplicity of an element in A and B
and is denoted by A ∪ B.
Example:
1. Let A = {l, l, m, m, n, n, n, n}
2. B = {l, m, m, m, n},
3. A ∪ B = {l, l, m, m, m, n, n, n, n}
2. Intersections of Multisets: The intersection of two multisets A and B, is a multiset such that
the multiplicity of an element is equal to the minimum of the multiplicity of an element in A and
B and is denoted by A ∩ B.
Example:
1. Let A = {l, l, m, n, p, q, q, r}
2. B = {l, m, m, p, q, r, r, r, r}
3. A ∩ B = {l, m, p, q, r}.
3. Difference of Multisets: The difference of two multisets A and B, is a multiset such that the
multiplicity of an element is equal to the multiplicity of the element in A minus the multiplicity
of the element in B if the difference is +ve, and is equal to 0 if the difference is 0 or negative
Example:
1. Let A = {l, m, m, m, n, n, n, p, p, p}
2. B = {l, m, m, m, n, r, r, r}
3. A - B = {n, n, p, p, p}
4. Sum of Multisets: The sum of two multisets A and B, is a multiset such that the multiplicity
of an element is equal to the sum of the multiplicity of an element in A and B
Example
1. Let A = {l, m, n, p, r}
2. B = {l, l, m, n, n, n, p, r, r}
3. A + B = {l, l, l, m, m, n, n, n, n, p, p, r, r, r}
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5. Cardinality of Sets: The cardinality of a multiset is the number of distinct elements in a
multiset without considering the multiplicity of an element
Example:
1. A = {l, l, m, m, n, n, n, p, p, p, p, q, q, q}
The cardinality of the multiset A is 5.
Ordered Set
It is defined as the ordered collection of distinct objects.
Example:
1. Roll no {3, 6, 7, 8, 9}
2. Week Days {S, M, T, W, W, TH, F, S, S}
Ordered Pairs
An Ordered Pair consists of two elements such that one of them is designated as the first member
and other as the second member.
(a, b) and (b, a) are two different ordered pair. An ordered triple can also be written regarding an
ordered pair as {(a, b) c}
An ordered Quadrable is an ordered pair {(((a, b), c) d)} with the first element as ordered triple.
An ordered n-tuple is an ordered pair where the first component is an ordered (n - 1) tuples, and
the nth
element is the second component.
1. {(n -1), n}
Example: