Chapter 3
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This document discusses knowledge reduction in rough set theory. It defines core and reduct concepts. The core is the most important part of knowledge that cannot be eliminated. A reduct is a minimal set of attributes that provides the same classification as the full set. It also discusses relative reducts and cores with respect to a given subset. Generalized definitions are provided for reducing categories of knowledge rather than individual attributes.
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This document discusses rough set theory, which is used to model vagueness and uncertainty when precise definitions of sets are not possible given available information. Key points:
1. Rough sets use upper and lower approximations to represent sets that cannot be precisely defined based on available knowledge. The lower approximation includes all elements that surely belong to the set, while the upper approximation includes elements that possibly belong.
2. Rough sets have properties like boundary regions that characterize their imprecision. Measures like roughness quantify the degree of uncertainty in approximating sets.
3. Membership in a rough set is characterized by upper and lower membership relations. Properties and examples are provided to illustrate rough set concepts.
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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.
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1. A set is a well-defined collection of distinct objects called elements or members. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties of elements.
2. The empty set, denoted by Ø, is a set with no elements. Finite sets have a definite number of elements while infinite sets have an unlimited number of elements.
3. Subsets contain elements that are also elements of another set. The power set of a set contains all possible subsets of that set including the empty set and the original set itself.
CBSE Class X - Mathematics Set Theory Topic
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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
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Visit Edvie.com for more topics
Mathematics set theory presentation.
This document provides an overview of set theory concepts including:
1. It defines what a set is and introduces some key terms like members, elements, and operations between sets.
2. It outlines several ways to represent sets including using rosters/lists and describing characteristic properties.
3. It discusses set notation where sets are denoted by capital letters and explains membership.
4. It describes different types of sets such as finite, infinite, null, singleton, and disjoint sets.
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This document provides an introduction to sequences and series. It begins with definitions of sequences, series, arithmetic progressions, geometric progressions and their key terms. It then presents several examples of finding terms in arithmetic and geometric progressions. The document also defines arithmetic mean and geometric mean, and discusses how to find the sum of terms in an arithmetic progression using formulas. Overall, the document serves as an introductory guide to common concepts involving sequences and progressions.
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This document discusses rough set theory, which is used to model vagueness and uncertainty when precise definitions of sets are not possible given available information. Key points:
1. Rough sets use upper and lower approximations to represent sets that cannot be precisely defined based on available knowledge. The lower approximation includes all elements that surely belong to the set, while the upper approximation includes elements that possibly belong.
2. Rough sets have properties like boundary regions that characterize their imprecision. Measures like roughness quantify the degree of uncertainty in approximating sets.
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This slide include the introduction to sets or set theory.
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--History
--Definition of sets with examples
-- point to remember about sets
--belongs to functions
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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.
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1. A set is a well-defined collection of distinct objects called elements or members. Sets can be represented in roster form by listing elements within curly brackets or in set-builder form using properties of elements.
2. The empty set, denoted by Ø, is a set with no elements. Finite sets have a definite number of elements while infinite sets have an unlimited number of elements.
3. Subsets contain elements that are also elements of another set. The power set of a set contains all possible subsets of that set including the empty set and the original set itself.
CBSE Class X - Mathematics Set Theory Topic
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
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This document provides an overview of set theory concepts including:
1. It defines what a set is and introduces some key terms like members, elements, and operations between sets.
2. It outlines several ways to represent sets including using rosters/lists and describing characteristic properties.
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sum of arithmetic sequence s
Here are the steps to solve the problems:
1. Given: a1 = 15 seats, d = 3 additional seats per row, n = 40 rows
Solve for an:
an = a1 + (n-1)d
= 15 + (40-1)3
= 15 + 117
= 132
Solve for S40:
S40 = n/2(a1 + an)
= 40/2(15 + 132)
= 40/2(147)
= 2940
2. Given: a1 = Php 1000, d = Php 300 additional sales per week, n = 15 weeks
Solve for an:
an = a1 +
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arithmetic means
Here are the answers to the arithmetic sequences:
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2. 4, 8, 12, 10
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The document discusses different types of dependencies between knowledge:
1. Total dependency, where one knowledge (Q) is fully derivable from another knowledge (P).
2. Independent knowledge, where knowledge P and Q are unrelated and neither depends on the other.
3. Partial dependency, where only some parts of knowledge Q are derivable from knowledge P. The document defines partial dependency as occurring at a degree k between 0 and 1, with 1 being total dependency.
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The document discusses the language of sets and how to define them. There are two main ways to define a set: using the roster method, which lists all the elements, and using set-builder notation, which describes the elements. It provides examples of defining sets in both ways and exercises for the students to practice defining sets. It also discusses elements of a set and what makes a set well-defined by being unambiguous about whether an object belongs to the set.
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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.
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sum of arithmetic sequence s
Here are the steps to solve the problems:
1. Given: a1 = 15 seats, d = 3 additional seats per row, n = 40 rows
Solve for an:
an = a1 + (n-1)d
= 15 + (40-1)3
= 15 + 117
= 132
Solve for S40:
S40 = n/2(a1 + an)
= 40/2(15 + 132)
= 40/2(147)
= 2940
2. Given: a1 = Php 1000, d = Php 300 additional sales per week, n = 15 weeks
Solve for an:
an = a1 +
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1. A set is a collection of distinct objects or elements that have some common property. Sets are represented using curly braces and capital letters. Elements within a set should not be repeated.
2. There are two main methods to define a set - the roster method which lists all elements, and the set builder method which describes the property defining membership.
3. There are several types of sets including finite sets with a set number of elements, infinite sets, singleton sets with one element, empty sets with no elements, equal sets with the same elements, and subsets which are sets within other sets.
arithmetic means
Here are the answers to the arithmetic sequences:
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5. 11, 17, 13, 9, 11
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Grade 10 Math Module 1 searching for patterns, sequence and series
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This document discusses knowledge representation and classification in rough set theory. It defines key concepts such as knowledge base, classification, elementary and basic categories. A knowledge base consists of a universe of objects and a set of classifications over those objects. Classifications are equivalence relations that partition the universe into categories. Elementary categories come from individual classifications, while basic categories are intersections of elementary categories. The document also discusses equivalence, generalization and specialization of knowledge bases based on comparisons of their category structures.
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2. Independent knowledge, where knowledge P and Q are unrelated and neither depends on the other.
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Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
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This document introduces the concept of fuzzy sets, which are classes of objects that have a continuum of grades of membership rather than crisp criteria for inclusion. Fuzzy sets are characterized by membership functions that assign each object a value between 0 and 1 indicating its grade of membership. Operations like union, intersection, and complementation are extended from ordinary sets to fuzzy sets. Basic properties of these operations are established, including identities analogous to those for ordinary sets. Fuzzy sets provide a framework for dealing with imprecisely defined classes that play an important role in human thinking and domains like pattern recognition.
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Math lesson for Grade 7 students
Topic: Basic Concept of Sets
05 tat math class xi_r&f-1_for website
The document defines key concepts related to relations and functions:
1. Ordered pairs, Cartesian products, and domains and ranges of relations are introduced. An ordered pair lists two objects in a particular order, while a Cartesian product is the set of all ordered pairs with the first element from one set and the second from another.
2. The domain of a relation is the set of first elements in its ordered pairs, while the range is the set of second elements. Inverse relations are also defined.
3. Several examples illustrate calculating Cartesian products of sets, determining the domains and ranges of given relations, and finding inverse relations.
Lesson 1.1 basic ideas of sets part 2
This document discusses key concepts related to sets including:
- Cardinality refers to the number of elements in a set and is denoted by n(A).
- Two sets are equivalent if they have the same number of elements.
- A set A is a subset of set B if every element of A is also an element of B, denoted by A ⊆ B.
- A set A is a proper subset of set B if A is a subset of B but the sets are not equal, denoted by A ⊂ B.
- The universal set U refers to all elements being considered in a problem.
- The complement of a set A, denoted by A', refers to
Explore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematics with this comprehensive PowerPoint presentation. Whether you are a student delving into the world of discrete structures or an enthusiast eager to understand the fundamentals, this presentation serves as an insightful guide.
piya_new__1_ (1).pdf
- The document introduces the concept of soft* IF lower and upper rough approximation spaces and their application in data mining and medical decision making. It provides definitions for soft sets, intuitionistic fuzzy sets, rough sets, and soft rough approximation spaces.
- Soft* IF lower and upper rough approximation operators are defined on intuitionistic fuzzy sets over a soft intuitionistic fuzzy approximation space. These operators allow for the approximation of intuitionistic fuzzy sets and determine whether a set is soft definable or a soft rough intuitionistic fuzzy set.
- The paper aims to apply these concepts of soft* IF lower and upper rough approximation to data mining and medical decision making by using MATLAB programming to show an example.
Set
This document provides an overview of topics covered in a course on logic and reasoning in mathematics. It begins with an introduction to set theory and explains that sets can be defined by listing elements or with set-builder notation. Subsets, proper subsets, complements, and the number of subsets in a finite set are also discussed. The role of logic and reasoning is emphasized throughout mathematical facts and procedures.
Partial ordering in soft set context
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
Sets-newest.pptx
The document defines basic concepts about sets including:
- A set is a collection of distinct objects called elements or members.
- The roster method defines a set by listing its elements within curly brackets.
- It is conventional to use capital letters to name sets.
- The cardinality of a set refers to the number of elements in the set.
- Two sets are equal if they contain the exact same elements regardless of order.
- A subset is a set whose elements are also elements of another set.
Subsets Definition Types, Properties and Example Questions.pdf
What are the types and properties of subsets? Read this blog to learn the definition, types, properties of subsets with practice example questions and solutions.
Sets functions-sequences-exercises
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
Chapter1 Formal Language and Automata Theory
This document provides an introduction and outline for a course on Formal Language Theory. The course will cover topics like set theory, relations, mathematical induction, graphs and trees, strings and languages. It will then introduce formal grammars including regular grammars, context-free grammars and pushdown automata. The course is divided into 5 chapters: Basics, Introduction to Grammars, Regular Languages, Context-Free Languages, and Pushdown Automata. The Basics chapter provides an overview of formal vs natural languages and reviews concepts like sets, relations, functions, and mathematical induction.
Crisp set
The document discusses classical or crisp set theory. Some key points:
1) Classical set theory deals with sets that have definite membership - an element either fully belongs to a set or not. This is represented by true/false or yes/no.
2) A set is a well-defined collection of objects. The universal set is the overall context within which sets are defined.
3) Set operations like union, intersection, complement and difference are used to combine or relate sets according to specific rules.
4) Properties like commutativity, associativity and distributivity define the logical behavior of sets under different operations.
Similar to Chapter 3 (20)
Rough sets and fuzzy rough sets in Decision Making
Rough sets and fuzzy rough sets in Decision Making
Set theory- Introduction, symbols with its meaning
Set theory- Introduction, symbols with its meaning
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Q1 Week 1 Lesson -Concepts of Sets and Operation on Sets.pptx
Explore the foundational concepts of sets in discrete mathematics
Explore the foundational concepts of sets in discrete mathematics
Subsets Definition Types, Properties and Example Questions.pdf
Subsets Definition Types, Properties and Example Questions.pdf
Chapter 3
- 1. Reduction of Knowledge Chapter 3
- 10. Take a closer look !