This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
This document provides an introduction to power system reliability concepts including definitions, categories of reliability studies, and basic concepts from statistics, probability, and set theory that are important for reliability analysis. It discusses two categories of reliability studies: system adequacy related to sufficient generation, transmission, and distribution facilities to meet demand, and system security related to the ability to respond to disturbances. The document then provides an overview of key concepts from statistics like frequency, mean, median, mode, and standard deviation. It also covers basic concepts from set theory like subsets, unions, intersections, and complements. Finally, it discusses fundamental probability concepts like sample space, events, mutually exclusive and independent events, and how to calculate probabilities using addition and multiplication rules.
This document discusses soft computing and fuzzy sets. It begins by defining soft computing as being tolerant of imprecision and focusing on approximation rather than precise outputs. Fuzzy sets are introduced as a tool of soft computing that allow for graded membership in sets rather than binary membership. Key concepts regarding fuzzy sets are explained, including fuzzy logic operations, fuzzy numbers, and fuzzy variables. Linear programming problems are discussed and how they can be modeled as fuzzy linear programming problems to account for imprecision in the coefficients and constraints.
The document discusses fuzzy logic and fuzzy sets. It defines fuzzy sets as sets with non-crisp boundaries where elements have degrees of membership between 0 and 1 rather than simply belonging or not belonging. It outlines some key concepts of fuzzy sets including membership functions, basic types of fuzzy sets over discrete and continuous universes, and set-theoretic operations like union, intersection, and complement for fuzzy sets.
Notes on (T, S)-Intuitionistic Fuzzy Subhemirings of a HemiringIRJET Journal
This document discusses (T,S)-intuitionistic fuzzy subhemirings of a hemiring. It begins with introducing some key definitions including T-fuzzy subhemiring, anti S-fuzzy subhemiring, (T,S)-intuitionistic fuzzy subhemiring, and the product of intuitionistic fuzzy subsets. It then presents some properties of (T,S)-intuitionistic fuzzy subhemirings such as the intersection of two (T,S)-intuitionistic fuzzy subhemirings forming another (T,S)-intuitionistic fuzzy subhemiring, and the product of (T,S)-intuitionistic fuzzy subhemirings of two hemir
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...ijfls
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point
theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for
the approximate controllability of the system.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...Wireilla
This summary provides the key details about the document in 3 sentences:
The document discusses approximate controllability results for impulsive linear fuzzy stochastic differential equations under nonlocal conditions. It presents sufficient conditions for the approximate controllability of such systems using Banach fixed point theorems, stochastic analysis, and fuzzy processes. The main result establishes approximate controllability of impulsive linear fuzzy stochastic differential equations by verifying assumptions on the system using fixed point theorems.
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).pptDediTriLaksono1
The document defines various concepts related to fuzzy sets and fuzzy logic. It defines the support, core, normality, crossover points, and other properties of fuzzy sets. It also defines operations on fuzzy sets like union, intersection, complement, and algebraic operations. It discusses the extension principle for mapping fuzzy sets through functions. It provides examples of applying the extension principle and compositions of fuzzy relations. Finally, it discusses linguistic variables and modifiers like hedges, negation, and connectives that are used to modify terms in a linguistic variable.
This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the
equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct
a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of
fuzzy congruence of group G (respectively ring R) is isomorphic to the lattice of fuzzy normal subgroup of G
(respectively fuzzy ideals of R).In Yuan Boond Wu wangrning investigated the relationship between the fuzzy
ideals and the fuzzy congruences on a distributive lattice and obtained that the lattice of fuzzy ideals is
isomorphic to the lattice of fuzzy congruences on a generalized Boolean algebra. Fuzzy group theory can be
used to describe, symmetries and permutation in nature and mathematics. The fuzzy group is one of the oldest
branches of abstract algebra. For example group can be used is classify to all of the forms chemical crystal can
take. Group can be used to count the number of non-equivalent objects and permutation or symmetries. For
example, the number of different is switching functions of n, variable when permutation of the input are
allowed. Beside crystallography and combinatory group have application of quantum mechanics.
This document provides an introduction to power system reliability concepts including definitions, categories of reliability studies, and basic concepts from statistics, probability, and set theory that are important for reliability analysis. It discusses two categories of reliability studies: system adequacy related to sufficient generation, transmission, and distribution facilities to meet demand, and system security related to the ability to respond to disturbances. The document then provides an overview of key concepts from statistics like frequency, mean, median, mode, and standard deviation. It also covers basic concepts from set theory like subsets, unions, intersections, and complements. Finally, it discusses fundamental probability concepts like sample space, events, mutually exclusive and independent events, and how to calculate probabilities using addition and multiplication rules.
This document discusses soft computing and fuzzy sets. It begins by defining soft computing as being tolerant of imprecision and focusing on approximation rather than precise outputs. Fuzzy sets are introduced as a tool of soft computing that allow for graded membership in sets rather than binary membership. Key concepts regarding fuzzy sets are explained, including fuzzy logic operations, fuzzy numbers, and fuzzy variables. Linear programming problems are discussed and how they can be modeled as fuzzy linear programming problems to account for imprecision in the coefficients and constraints.
The document discusses fuzzy logic and fuzzy sets. It defines fuzzy sets as sets with non-crisp boundaries where elements have degrees of membership between 0 and 1 rather than simply belonging or not belonging. It outlines some key concepts of fuzzy sets including membership functions, basic types of fuzzy sets over discrete and continuous universes, and set-theoretic operations like union, intersection, and complement for fuzzy sets.
Notes on (T, S)-Intuitionistic Fuzzy Subhemirings of a HemiringIRJET Journal
This document discusses (T,S)-intuitionistic fuzzy subhemirings of a hemiring. It begins with introducing some key definitions including T-fuzzy subhemiring, anti S-fuzzy subhemiring, (T,S)-intuitionistic fuzzy subhemiring, and the product of intuitionistic fuzzy subsets. It then presents some properties of (T,S)-intuitionistic fuzzy subhemirings such as the intersection of two (T,S)-intuitionistic fuzzy subhemirings forming another (T,S)-intuitionistic fuzzy subhemiring, and the product of (T,S)-intuitionistic fuzzy subhemirings of two hemir
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...ijfls
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point
theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for
the approximate controllability of the system.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...Wireilla
This summary provides the key details about the document in 3 sentences:
The document discusses approximate controllability results for impulsive linear fuzzy stochastic differential equations under nonlocal conditions. It presents sufficient conditions for the approximate controllability of such systems using Banach fixed point theorems, stochastic analysis, and fuzzy processes. The main result establishes approximate controllability of impulsive linear fuzzy stochastic differential equations by verifying assumptions on the system using fixed point theorems.
00_1 - Slide Pelengkap (dari Buku Neuro Fuzzy and Soft Computing).pptDediTriLaksono1
The document defines various concepts related to fuzzy sets and fuzzy logic. It defines the support, core, normality, crossover points, and other properties of fuzzy sets. It also defines operations on fuzzy sets like union, intersection, complement, and algebraic operations. It discusses the extension principle for mapping fuzzy sets through functions. It provides examples of applying the extension principle and compositions of fuzzy relations. Finally, it discusses linguistic variables and modifiers like hedges, negation, and connectives that are used to modify terms in a linguistic variable.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex fuzzy relation are introduced.
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
This document summarizes a lecture on fuzzy logic and neural networks. It introduces fuzzy sets and compares them to classical or crisp sets. Key concepts covered include fuzzy set representation using membership functions, common membership function types like triangular and trapezoidal, fuzzy set operations, and properties of fuzzy and crisp sets. Examples are provided to demonstrate calculating membership values and performing operations on fuzzy sets.
The document discusses fuzzy sets and fuzzy logic. It defines fuzzy as meaning not clear or precise, with blurred outlines. Fuzzy sets allow partial membership in a set, whereas classical sets have binary membership. Fuzzy sets are represented by membership functions that can take on values between 0 and 1. Common fuzzy set operations like union, intersection, and complement are defined. Fuzzy logic is then introduced as a way to represent imprecise concepts and approximate reasoning, extending conventional binary logic to allow intermediate truth values.
This document provides an overview of fuzzy logic and fuzzy set theory. It discusses how fuzzy sets allow for partial membership rather than crisp binary membership in sets. Various membership functions are described, including triangular, trapezoidal, Gaussian, and sigmoidal functions. Properties of fuzzy sets like support, convexity, and symmetry are defined. Finally, fuzzy set operations analogous to classical set operations are mentioned.
Mathematical Foundations for Machine Learning and Data MiningMadhavRao65
This document provides an overview of a presentation on mathematical foundations and various topics in mathematics including linear algebra, probability and statistics, calculus, and optimization. It discusses Google's PageRank algorithm and computed tomography as examples of mathematical foundations. It also provides examples of finding the best apartment based on criteria and recognizing patterns for biometric identification using machine learning models. Various concepts in linear algebra are defined such as vectors, vector spaces, subspaces, spanning sets, linear independence, basis and dimension. Examples of spanning sets, linear independence, and whether certain sets are subspaces are given. References for magnetohydrodynamic modeling of blood flow are also provided.
The document discusses fuzzy logic and fuzzy sets. It begins by explaining fuzzy logic is used to model imprecise concepts and dependencies using natural language terms. It then defines fuzzy variables, universes of discourse, and fuzzy sets which have membership functions assigning a degree of membership between 0 and 1. Operations on fuzzy sets like intersection, union, and complement are also covered. The document also discusses fuzzy rules, relations, and approximate reasoning using max-min inference.
The document discusses alpha cuts and their properties in fuzzy sets. It defines alpha cuts as crisp sets containing elements of the universal set whose membership degree in the fuzzy set is greater than or equal to alpha. The higher the alpha value, the smaller the alpha cut set. It also discusses support, core, and height of fuzzy sets. Support is the crisp set of all elements with non-zero membership, core those with membership 1, and height the highest alpha value of a non-empty alpha cut. Examples are given to illustrate key fuzzy set operations and concepts.
This document discusses L-fuzzy sub λ-groups. Key points:
- It defines L-fuzzy sub λ-groups and anti L-fuzzy sub λ-groups, which combine fuzzy set theory with lattice ordered group theory.
- Properties of L-fuzzy sub λ-groups are investigated, such as conditions under which a subset is a sub λ-group. The intersection of two L-fuzzy sub λ-groups is also an L-fuzzy sub λ-group.
- A relationship is established between an L-fuzzy sub λ-group and its complement, which must be an anti L-fuzzy sub λ-group.
- Level subsets of
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
This document provides definitions and notation for set theory concepts. It defines what a set is, ways to describe sets (explicitly by listing elements or implicitly using set builder notation), and basic set relationships like subset, proper subset, union, intersection, complement, power set, and Cartesian product. It also discusses Russell's paradox and defines important sets like the natural numbers. Key identities for set operations like idempotent, commutative, associative, distributive, De Morgan's laws, and complement laws are presented. Proofs of identities using logical equivalences and membership tables are demonstrated.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses signal-space analysis and representation of bandpass signals. It can be summarized as follows:
1) A bandpass real signal x(t) can be represented using its complex envelope x(t) and carrier frequency fc. This results in an in-phase (I) and quadrature-phase (Q) representation of the signal.
2) Signals can be viewed as vectors in a vector space. Basic algebra concepts like groups, fields, and vector spaces are introduced.
3) Key concepts discussed include orthonormal bases, projection theorems, Gram-Schmidt orthonormalization, and representing signals in inner product spaces which allows defining notions of length and angle between signals.
The document discusses the origins and evolution of fuzzy logic, beginning with fuzzy set theory proposed by Zadeh in 1965 which aimed to represent vagueness in natural language using fuzzy sets with non-crisp boundaries. It explains key concepts in fuzzy logic like membership functions, fuzzy set operations, fuzzy relations and compositions. The document also compares classical sets with crisp boundaries to fuzzy sets and contrasts crisp logic with fuzzy logic which allows for degrees of truth between 0 and 1.
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
This document discusses combining rough set theory and formal concept analysis by introducing rough set approximation operators on concept lattices. It begins with an overview of classical rough set theory and formal concept analysis. It then defines rough set approximations on a concept lattice using the notions of a formal concept and concept lattice from formal concept analysis. The key points are:
1) Rough set theory approximates an undefinable set through lower and upper definable sets, while formal concept analysis models relationships between objects and properties through formal concepts and concept lattices.
2) A formal concept is a pair consisting of a set of objects (extension) and a set of properties (intension) that are functionally dependent.
3) A concept lattice
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex fuzzy relation are introduced.
The document discusses eigenvalues and eigenvectors. It defines an eigenvalue problem as finding scale constants (λ) and nonzero vectors (X) such that when a square matrix (A) multiplies a vector (X), it produces a vector in the same direction but scaled by λ. The characteristic polynomial is used to find the eigenvalues by setting its determinant equal to 0. Once the eigenvalues are obtained, the corresponding eigenvectors can be found by solving the homogeneous system (A - λI)X = 0. Examples are provided to demonstrate finding the eigenvalues and eigenvectors of different matrices.
This document summarizes a lecture on fuzzy logic and neural networks. It introduces fuzzy sets and compares them to classical or crisp sets. Key concepts covered include fuzzy set representation using membership functions, common membership function types like triangular and trapezoidal, fuzzy set operations, and properties of fuzzy and crisp sets. Examples are provided to demonstrate calculating membership values and performing operations on fuzzy sets.
The document discusses fuzzy sets and fuzzy logic. It defines fuzzy as meaning not clear or precise, with blurred outlines. Fuzzy sets allow partial membership in a set, whereas classical sets have binary membership. Fuzzy sets are represented by membership functions that can take on values between 0 and 1. Common fuzzy set operations like union, intersection, and complement are defined. Fuzzy logic is then introduced as a way to represent imprecise concepts and approximate reasoning, extending conventional binary logic to allow intermediate truth values.
This document provides an overview of fuzzy logic and fuzzy set theory. It discusses how fuzzy sets allow for partial membership rather than crisp binary membership in sets. Various membership functions are described, including triangular, trapezoidal, Gaussian, and sigmoidal functions. Properties of fuzzy sets like support, convexity, and symmetry are defined. Finally, fuzzy set operations analogous to classical set operations are mentioned.
Mathematical Foundations for Machine Learning and Data MiningMadhavRao65
This document provides an overview of a presentation on mathematical foundations and various topics in mathematics including linear algebra, probability and statistics, calculus, and optimization. It discusses Google's PageRank algorithm and computed tomography as examples of mathematical foundations. It also provides examples of finding the best apartment based on criteria and recognizing patterns for biometric identification using machine learning models. Various concepts in linear algebra are defined such as vectors, vector spaces, subspaces, spanning sets, linear independence, basis and dimension. Examples of spanning sets, linear independence, and whether certain sets are subspaces are given. References for magnetohydrodynamic modeling of blood flow are also provided.
The document discusses fuzzy logic and fuzzy sets. It begins by explaining fuzzy logic is used to model imprecise concepts and dependencies using natural language terms. It then defines fuzzy variables, universes of discourse, and fuzzy sets which have membership functions assigning a degree of membership between 0 and 1. Operations on fuzzy sets like intersection, union, and complement are also covered. The document also discusses fuzzy rules, relations, and approximate reasoning using max-min inference.
The document discusses alpha cuts and their properties in fuzzy sets. It defines alpha cuts as crisp sets containing elements of the universal set whose membership degree in the fuzzy set is greater than or equal to alpha. The higher the alpha value, the smaller the alpha cut set. It also discusses support, core, and height of fuzzy sets. Support is the crisp set of all elements with non-zero membership, core those with membership 1, and height the highest alpha value of a non-empty alpha cut. Examples are given to illustrate key fuzzy set operations and concepts.
This document discusses L-fuzzy sub λ-groups. Key points:
- It defines L-fuzzy sub λ-groups and anti L-fuzzy sub λ-groups, which combine fuzzy set theory with lattice ordered group theory.
- Properties of L-fuzzy sub λ-groups are investigated, such as conditions under which a subset is a sub λ-group. The intersection of two L-fuzzy sub λ-groups is also an L-fuzzy sub λ-group.
- A relationship is established between an L-fuzzy sub λ-group and its complement, which must be an anti L-fuzzy sub λ-group.
- Level subsets of
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
This document provides definitions and notation for set theory concepts. It defines what a set is, ways to describe sets (explicitly by listing elements or implicitly using set builder notation), and basic set relationships like subset, proper subset, union, intersection, complement, power set, and Cartesian product. It also discusses Russell's paradox and defines important sets like the natural numbers. Key identities for set operations like idempotent, commutative, associative, distributive, De Morgan's laws, and complement laws are presented. Proofs of identities using logical equivalences and membership tables are demonstrated.
Errors in the Discretized Solution of a Differential Equationijtsrd
We study the error in the derivatives of an unknown function. We construct the discretized problem. The local truncation and global errors are discussed. The solution of discretized problem is constructed. The analytical and discretized solutions are compared. The two solution graphs are described by using MATLAB software. Wai Mar Lwin | Khaing Khaing Wai "Errors in the Discretized Solution of a Differential Equation" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd27937.pdfPaper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/27937/errors-in-the-discretized-solution-of-a-differential-equation/wai-mar-lwin
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses signal-space analysis and representation of bandpass signals. It can be summarized as follows:
1) A bandpass real signal x(t) can be represented using its complex envelope x(t) and carrier frequency fc. This results in an in-phase (I) and quadrature-phase (Q) representation of the signal.
2) Signals can be viewed as vectors in a vector space. Basic algebra concepts like groups, fields, and vector spaces are introduced.
3) Key concepts discussed include orthonormal bases, projection theorems, Gram-Schmidt orthonormalization, and representing signals in inner product spaces which allows defining notions of length and angle between signals.
The document discusses the origins and evolution of fuzzy logic, beginning with fuzzy set theory proposed by Zadeh in 1965 which aimed to represent vagueness in natural language using fuzzy sets with non-crisp boundaries. It explains key concepts in fuzzy logic like membership functions, fuzzy set operations, fuzzy relations and compositions. The document also compares classical sets with crisp boundaries to fuzzy sets and contrasts crisp logic with fuzzy logic which allows for degrees of truth between 0 and 1.
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
This document discusses combining rough set theory and formal concept analysis by introducing rough set approximation operators on concept lattices. It begins with an overview of classical rough set theory and formal concept analysis. It then defines rough set approximations on a concept lattice using the notions of a formal concept and concept lattice from formal concept analysis. The key points are:
1) Rough set theory approximates an undefinable set through lower and upper definable sets, while formal concept analysis models relationships between objects and properties through formal concepts and concept lattices.
2) A formal concept is a pair consisting of a set of objects (extension) and a set of properties (intension) that are functionally dependent.
3) A concept lattice
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
Similar to Fuzzy Sets decision making under information of uncertainty (20)
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
Predictably Improve Your B2B Tech Company's Performance by Leveraging DataKiwi Creative
Harness the power of AI-backed reports, benchmarking and data analysis to predict trends and detect anomalies in your marketing efforts.
Peter Caputa, CEO at Databox, reveals how you can discover the strategies and tools to increase your growth rate (and margins!).
From metrics to track to data habits to pick up, enhance your reporting for powerful insights to improve your B2B tech company's marketing.
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This is the webinar recording from the June 2024 HubSpot User Group (HUG) for B2B Technology USA.
Watch the video recording at https://youtu.be/5vjwGfPN9lw
Sign up for future HUG events at https://events.hubspot.com/b2b-technology-usa/
Learn SQL from basic queries to Advance queriesmanishkhaire30
Dive into the world of data analysis with our comprehensive guide on mastering SQL! This presentation offers a practical approach to learning SQL, focusing on real-world applications and hands-on practice. Whether you're a beginner or looking to sharpen your skills, this guide provides the tools you need to extract, analyze, and interpret data effectively.
Key Highlights:
Foundations of SQL: Understand the basics of SQL, including data retrieval, filtering, and aggregation.
Advanced Queries: Learn to craft complex queries to uncover deep insights from your data.
Data Trends and Patterns: Discover how to identify and interpret trends and patterns in your datasets.
Practical Examples: Follow step-by-step examples to apply SQL techniques in real-world scenarios.
Actionable Insights: Gain the skills to derive actionable insights that drive informed decision-making.
Join us on this journey to enhance your data analysis capabilities and unlock the full potential of SQL. Perfect for data enthusiasts, analysts, and anyone eager to harness the power of data!
#DataAnalysis #SQL #LearningSQL #DataInsights #DataScience #Analytics
Global Situational Awareness of A.I. and where its headedvikram sood
You can see the future first in San Francisco.
Over the past year, the talk of the town has shifted from $10 billion compute clusters to $100 billion clusters to trillion-dollar clusters. Every six months another zero is added to the boardroom plans. Behind the scenes, there’s a fierce scramble to secure every power contract still available for the rest of the decade, every voltage transformer that can possibly be procured. American big business is gearing up to pour trillions of dollars into a long-unseen mobilization of American industrial might. By the end of the decade, American electricity production will have grown tens of percent; from the shale fields of Pennsylvania to the solar farms of Nevada, hundreds of millions of GPUs will hum.
The AGI race has begun. We are building machines that can think and reason. By 2025/26, these machines will outpace college graduates. By the end of the decade, they will be smarter than you or I; we will have superintelligence, in the true sense of the word. Along the way, national security forces not seen in half a century will be un-leashed, and before long, The Project will be on. If we’re lucky, we’ll be in an all-out race with the CCP; if we’re unlucky, an all-out war.
Everyone is now talking about AI, but few have the faintest glimmer of what is about to hit them. Nvidia analysts still think 2024 might be close to the peak. Mainstream pundits are stuck on the wilful blindness of “it’s just predicting the next word”. They see only hype and business-as-usual; at most they entertain another internet-scale technological change.
Before long, the world will wake up. But right now, there are perhaps a few hundred people, most of them in San Francisco and the AI labs, that have situational awareness. Through whatever peculiar forces of fate, I have found myself amongst them. A few years ago, these people were derided as crazy—but they trusted the trendlines, which allowed them to correctly predict the AI advances of the past few years. Whether these people are also right about the next few years remains to be seen. But these are very smart people—the smartest people I have ever met—and they are the ones building this technology. Perhaps they will be an odd footnote in history, or perhaps they will go down in history like Szilard and Oppenheimer and Teller. If they are seeing the future even close to correctly, we are in for a wild ride.
Let me tell you what we see.
Open Source Contributions to Postgres: The Basics POSETTE 2024ElizabethGarrettChri
Postgres is the most advanced open-source database in the world and it's supported by a community, not a single company. So how does this work? How does code actually get into Postgres? I recently had a patch submitted and committed and I want to share what I learned in that process. I’ll give you an overview of Postgres versions and how the underlying project codebase functions. I’ll also show you the process for submitting a patch and getting that tested and committed.
2. 1. Introduction
Uncertainty
When A is a fuzzy set and x is a relevant object, the proposition
“x is a member of A” is not necessarily either true or false. It may
be true only to some degree, the degree to which x is actually a
member of A.
For example: the weather today
Sunny: If we define any cloud cover of 25% or less is sunny.
This means that a cloud cover of 26% is not sunny?
“Vagueness” should be introduced.
2
3. The crisp set v.s. the fuzzy set
The crisp set is defined in such a way as to partition the individuals in some
given universe of discourse into two groups: members and nonmembers.
However, many classification concepts do not exhibit this characteristic.
For example, the set of tall people, expensive cars, or sunny days.
A fuzzy set can be defined mathematically by assigning to each possible
individual in the universe of discourse a value representing its grade of
membership in the fuzzy set.
For example: a fuzzy set representing our concept of sunny might assign a
degree of membership of 1 to a cloud cover of 0%, 0.8 to a cloud cover of
20%, 0.4 to a cloud cover of 30%, and 0 to a cloud cover of 75%.
3
4. 2. Fuzzy sets: basic types
A membership function:
A characteristic function: the values assigned to the elements
of the universal set fall within a specified range and indicate
the membership grade of these elements in the set.
Larger values denote higher degrees of set membership.
A set defined by membership functions is a fuzzy set.
The most commonly used range of values of membership
functions is the unit interval [0,1].
The universal set X is always a crisp set.
Notation:
The membership function of a fuzzy set A is denoted by :
Alternatively, the function can be denoted by A and has the form
We use the second notation.
A
]
1
,
0
[
:
X
A
4
]
1
,
0
[
:
X
A
7. 2. Fuzzy sets: basic types
An example:
Define the seven levels of education:
7
Highly
educated (0.8)
Very highly
educated (0.5)
8. 2. Fuzzy sets: basic types
Several fuzzy sets representing linguistic concepts such as low, medium,
high, and so one are often employed to define states of a variable. Such a
variable is usually called a fuzzy variable.
For example:
8
9. 2. Fuzzy sets: basic types
Given a universal set X, a fuzzy set is defined by a function of
the form
This kind of fuzzy sets are called ordinary fuzzy sets.
Interval-valued fuzzy sets:
The membership functions of ordinary fuzzy sets are often overly
precise.
We may be able to identify appropriate membership functions
only approximately.
Interval-valued fuzzy sets: a fuzzy set whose membership
functions does not assign to each element of the universal set
one real number, but a closed interval of real numbers between
the identified lower and upper bounds.
]
1
,
0
[
:
X
A
9
]),
1
,
0
([
:
X
A
Power set
11. 2. Fuzzy sets: basic types
Fuzzy sets of type 2:
: the set of all ordinary fuzzy sets that can be defined with
the universal set [0,1].
is also called a fuzzy power set of [0,1].
11
12. 2. Fuzzy sets: basic types
Discussions:
The primary disadvantage of interval-value fuzzy sets,
compared with ordinary fuzzy sets, is computationally more
demanding.
The computational demands for dealing with fuzzy sets of type
2 are even greater then those for dealing with interval-valued
fuzzy sets.
This is the primary reason why the fuzzy sets of type 2 have
almost never been utilized in any applications.
12
13. 3. Fuzzy sets: basic concepts
Consider three fuzzy sets that represent the concepts of a young,
middle-aged, and old person. The membership functions are
defined on the interval [0,80] as follows:
13
Find line passing through
(x,y) and (20,1):
1/[35-20] = y/[35-x]
15. 3. Fuzzy sets: basic concepts
-cut and strong -cut
Given a fuzzy set A defined on X and any number
the -cut and strong -cut are the crisp sets:
The -cut of a fuzzy set A is the crisp set that contains all
the elements of the universal set X whose membership
grades in A are greater than or equal to the specified value
of .
The strong -cut of a fuzzy set A is the crisp set that
contains all the elements of the universal set X whose
membership grades in A are only greater than the specified
value of .
15
],
1
,
0
[
17. 3. Fuzzy sets: basic concepts
A level set of A:
The set of all levels that represent distinct -cuts of a
given fuzzy set A.
For example:
17
]
1
,
0
[
18. 3. Fuzzy sets: basic concepts
For example: consider the discrete approximation D2 of fuzzy set
A2
18
19. 3 Fuzzy sets: basic concepts
The standard complement of fuzzy set A with respect to the
universal set X is defined for all by the equation
Elements of X for which are called equilibrium points of A.
For example, the equilibrium points of A2 in Fig. 1.7 are 27.5 and 52.5.
X
x
)
(
)
( x
A
x
A
19
)
(
1
)
( x
A
x
A
20. 3. Fuzzy sets: basic concepts
Given two fuzzy sets, A and B, their standard intersection and union
are defined for all by the equations
where min and max denote the minimum operator and the
maximum operator, respectively.
X
x
20
)],
(
),
(
max[
)
)(
(
)],
(
),
(
min[
)
)(
(
x
B
x
A
x
B
A
x
B
x
A
x
B
A
21. 3. Fuzzy sets: basic concepts
Another example:
A1, A2, A3 are normal.
B and C are subnormal.
B and C are convex.
are not
convex.
21
2
1 A
A
B
3
2 A
A
C
C
B
C
B
and
Normality and convexity
may be lost when we
operate on fuzzy sets by
the standard operations
of intersection and
complement.
22. 3. Fuzzy sets: basic concepts
Discussions:
Normality and convexity
may be lost when we
operate on fuzzy sets by
the standard operations of
intersection and
complement.
The fuzzy intersection and
fuzzy union will satisfies all
the properties of the
Boolean lattice listed in
Table 1.1 except the low of
contradiction and the low of
excluded middle.
22
23. 3. Fuzzy sets: basic concepts
The law of contradiction
To verify that the law of contradiction is violated for fuzzy sets, we
need only to show that
is violated for at least one .
This is easy since the equation is obviously violated for any value
, and is satisfied only for
0
)]
(
1
),
(
min[
x
A
x
A
X
x
23
)
1
,
0
(
)
(
x
A }.
1
,
0
{
)
(
x
A
A
A
24. 3. Fuzzy sets: basic concepts
To verify the law of absorption,
This requires showing that
is satisfied for all .
Consider two cases:
(1)
(2)
)
(
)
( x
B
x
A
)
(
)
( x
B
x
A
24
A
B
A
A
)
(
)
(
)]]
(
),
(
min[
),
(
max[ x
A
x
B
x
A
x
A
X
x
)
(
)]
(
),
(
max[
)]]
(
),
(
min[
),
(
max[ x
A
x
A
x
A
x
B
x
A
x
A
)
(
)]
(
),
(
max[
)]]
(
),
(
min[
),
(
max[ x
A
x
B
x
A
x
B
x
A
x
A
)
(
)]]
(
),
(
min[
),
(
max[ x
A
x
B
x
A
x
A
25. 3. Fuzzy sets: basic concepts
Given two fuzzy set
we say that A is a subset of B and write iff
for all .
)
(
)
( x
B
x
A
X
x
25
B
A
any
for
and
iff B
B
A
A
B
A
B
A
27. 27
Let s = [i(1),i(2),..,i(k)] be a subsequence of [1,2,…,n] and let
s* = [i(k+1), i(k+2),…, i(n)] be the sequence complementary to
[i(1),i(2),..,i(k)].
The projection of n-ary fuzzy relation R on U(s) = U(i1) U(i2) .. U(ik)
denoted Proj[U(s)](R) is k-ary fuzzy relation
{((u(i(1)),u(i(2)),…u(i(k))), sup [R](u(1),u(2),…u(n))}
u(i(k+1), u(i(k+2)), … u(i(n))
Example: Let’s take relation R – less than (previous page).
Proj[U1](R) = {(0,1),(10, 0.9), (20, 0.7), (30, 0.5),…..}
The converse of the projection of n-ary relation is called a cylindrical
extension.
Let R be k-ary fuzzy relation on U(s) = U(i1) U(i2) .. U(ik).
A cylindrical extension of R in U = U(1) U(2) … U(n) is
C(R)= {(u(1),u(2),..u(n)): [R](u(i1),u(i2),…u(i(n)))}.
31. 31
Let R be fuzzy relation on U(1) U(2) … U(r), and S be fuzzy
relation on U(s) U(s+1) … U(n).
Let {i1, i2,.., ik}= ({1,2…,r}- {s, s+1,…,n}) ({s, s+1,…,n}- {1,2,…,r})
Symmetric difference
The composition of R and S denoted by RS is defined as:
Proj[U(i1), U(i2), …, U(ik)](c(R)c(S)).
Example: R = Fast Less_Than
33. Conception of Fuzzy Logic
Many decision-making and problem-solving
tasks are too complex to be defined precisely
however, people succeed by using imprecise
knowledge
Fuzzy logic resembles human reasoning in its
use of approximate information and
uncertainty to generate decisions.
34. 34
Natural Language
Consider:
Joe is tall -- what is tall?
Joe is very tall -- what does this differ from tall?
Natural language (like most other activities in
life and indeed the universe) is not easily
translated into the absolute terms of 0 and 1.
“false” “true”
35. 35
Fuzzy Logic
An approach to uncertainty that combines
real values [0…1] and logic operations
Fuzzy logic is based on the ideas of fuzzy set
theory and fuzzy set membership often found
in natural (e.g., spoken) language.
36. 36
Example: “Young”
Example:
Ann is 28, 0.8 in set “Young”
Bob is 35, 0.1 in set “Young”
Charlie is 23, 1.0 in set “Young”
Unlike statistics and probabilities, the degree
is not describing probabilities that the item is
in the set, but instead describes to what
extent the item is the set.
37. 37
Membership function of fuzzy logic
Age
25 40 55
Young Old
1
Middle
0.5
DOM
Degree of
Membership
Fuzzy values
Fuzzy values have associated degrees of membership in the set.
0
40. Benefits of fuzzy logic
You want the value to switch gradually as
Young becomes Middle and Middle becomes
Old. This is the idea of fuzzy logic.
41. 41
Fuzzy Set Operations
Fuzzy union (): the union of two fuzzy sets
is the maximum (MAX) of each element from
two sets.
E.g.
A = {1.0, 0.20, 0.75}
B = {0.2, 0.45, 0.50}
A B = {MAX(1.0, 0.2), MAX(0.20, 0.45), MAX(0.75, 0.50)}
= {1.0, 0.45, 0.75}
42. 42
Fuzzy intersection (): the intersection of two
fuzzy sets is just the MIN of each element
from the two sets.
E.g.
A B = {MIN(1.0, 0.2), MIN(0.20, 0.45), MIN(0.75,
0.50)} = {0.2, 0.20, 0.50}
43. 43
Fuzzy Set Operations
The complement of a fuzzy variable with
DOM x is (1-x).
Complement ( _c): The complement of a
fuzzy set is composed of all elements’
complement.
Example.
Ac = {1 – 1.0, 1 – 0.2, 1 – 0.75} = {0.0, 0.8, 0.25}
44. 44
Crisp Relations
Ordered pairs showing connection between two
sets:
(a,b): a is related to b
(2,3) are related with the relation “<“
Relations are set themselves
< = {(1,2), (2, 3), (2, 4), ….}
Relations can be expressed as matrices
…
45. 45
Fuzzy Relations
Triples showing connection between two sets:
(a,b,#): a is related to b with degree #
Fuzzy relations are set themselves
Fuzzy relations can be expressed as matrices
…
47. 47
Where is Fuzzy Logic used?
Fuzzy logic is used directly in very few
applications.
Most applications of fuzzy logic use it as the
underlying logic system for decision support
systems.
48. 48
Fuzzy Expert System
Fuzzy expert system is a collection of
membership functions and rules that are
used to reason about data.
Usually, the rules in a fuzzy expert system
are have the following form:
“if x is low and y is high then z is medium”
51. 51
Fuzzification
Establishes the fact base of the fuzzy system. It identifies the
input and output of the system, defines appropriate IF THEN
rules, and uses raw data to derive a membership function.
Consider an air conditioning system that determine the best
circulation level by sampling temperature and moisture levels.
The inputs are the current temperature and moisture level.
The fuzzy system outputs the best air circulation level: “none”,
“low”, or “high”. The following fuzzy rules are used:
1. If the room is hot, circulate the air a lot.
2. If the room is cool, do not circulate the air.
3. If the room is cool and moist, circulate the air slightly.
A knowledge engineer determines membership functions that map
temperatures to fuzzy values and map moisture measurements to fuzzy
values.
52. 52
Inference
Evaluates all rules and determines their truth values.
If an input does not precisely correspond to an IF
THEN rule, partial matching of the input data is used
to interpolate an answer.
Continuing the example, suppose that the system has
measured temperature and moisture levels and mapped them
to the fuzzy values of .7 and .1 respectively. The system now
infers the truth of each fuzzy rule. To do this a simple method
called MAX-MIN is used. This method sets the fuzzy value of
the THEN clause to the fuzzy value of the IF clause. Thus, the
method infers fuzzy values of 0.7, 0.1, and 0.1 for rules 1, 2,
and 3 respectively.
53. 53
Composition
Combines all fuzzy conclusions obtained by inference
into a single conclusion. Since different fuzzy rules
might have different conclusions, consider all rules.
Continuing the example, each inference suggests a different
action
rule 1 suggests a "high" circulation level
rule 2 suggests turning off air circulation
rule 3 suggests a "low" circulation level.
A simple MAX-MIN method of selection is used where the
maximum fuzzy value of the inferences is used as the final
conclusion. So, composition selects a fuzzy value of 0.7 since
this was the highest fuzzy value associated with the inference
conclusions.
54. 54
Defuzzification
Convert the fuzzy value obtained from composition
into a “crisp” value. This process is often complex
since the fuzzy set might not translate directly into a
crisp value.Defuzzification is necessary, since
controllers of physical systems require discrete
signals.
Continuing the example, composition outputs a fuzzy value of
0.7. This imprecise value is not directly useful since the air
circulation levels are “none”, “low”, and “high”. The
defuzzification process converts the fuzzy output of 0.7 into
one of the air circulation levels. In this case it is clear that a
fuzzy output of 0.7 indicates that the circulation should be set
to “high”.
55. 55
Defuzzification
There are many defuzzification methods. Two of the
more common techniques are the centroid and
maximum methods.
In the centroid method, the crisp value of the output
variable is computed by finding the variable value of
the center of gravity of the membership function for
the fuzzy value.
In the maximum method, one of the variable values
at which the fuzzy subset has its maximum truth
value is chosen as the crisp value for the output
variable.
57. 57
Fuzzification
Two Inputs (x, y) and one output (z)
Membership functions:
low(t) = 1 - ( t / 10 )
high(t) = t / 10
Low High
1
0
t
X=0.32 Y=0.61
0.32
0.68
Low(x) = 0.68, High(x) = 0.32, Low(y) = 0.39, High(y) = 0.61
Crisp Inputs
58. 58
Create rule base
Rule 1: If x is low AND y is low Then z is high
Rule 2: If x is low AND y is high Then z is low
Rule 3: If x is high AND y is low Then z is low
Rule 4: If x is high AND y is high Then z is high
61. 61
Defuzzification
Center of Gravity
Low High
1
0
0.61
0.39
t
Crisp output
Max
Min
Max
Min
dt
t
f
dt
t
tf
C
)
(
)
(
Center of Gravity
62. 62
A Real Fuzzy Logic System
The subway in Sendai, Japan uses a fuzzy
logic control system developed by Serji
Yasunobu of Hitachi.
It took 8 years to complete and was finally put
into use in 1987.
63. 63
Control System
Based on rules of logic obtained from train
drivers so as to model real human decisions
as closely as possible
Task: Controls the speed at which the train
takes curves as well as the acceleration and
braking systems of the train
64. 64
The results of the fuzzy logic controller for the
Sendai subway are excellent!!
The train movement is smoother than most
other trains
Even the skilled human operators who
sometimes run the train cannot beat the
automated system in terms of smoothness or
accuracy of stopping
69. 69
Example II
if temperature is cold and oil is cheap
then heating is high
Linguistic
Variable
Linguistic
Variable
Linguistic
Variable
Linguistic
Value
Linguistic
Value
Linguistic
Value
cold cheap
high
70. 70
Definition [Zadeh 1973]
A linguistic variable is characterized by a quintuple
, ( ), , ,
x T x U G M
Name
Term Set
Universe
Syntactic Rule
Semantic Rule
71. 71
Example
A linguistic variable is characterized by a quintuple
, ( ), , ,
x T x U G M
age
old, very old, not so old,
(age) more or less young,
quite young, very young
G
[0, 100]
old
(old) , ( ) [0,100]
M u u u
1
2
old
0 [0,50]
( ) 50
1 [50,100]
5
u
u u
u
Example semantic rule:
74. 74
A B
A B
A B A B
1
1
0
0
1
0
1
0
1
0
1
1
A B A B
1
1
0
0
1
0
1
0
1
0
1
1
1 ( ) ( )
( , )
( ) otherwise
A B
A B
B
x y
x y
y
( , ) max 1 ( ), ( )
A B A B
x y x x
75. 75
A B If A then B
A A is true
B is true
B
A B
A
B
Modus Ponens
A B A B
1
1
0
0
1
0
1
0
1
0
1
1
76. 76
If x is A then y is B.
antecedent
or
premise
consequence
or
conclusion
A B
77. 77
Examples
If x is A then y is B.
A B
If pressure is high, then volume is small.
If the road is slippery, then driving is dangerous.
If a tomato is red, then it is ripe.
If the speed is high, then apply the brake a little.
78. 78
Fuzzy Rules as Relations
If x is A then y is B.
, ,
R A B
x y x y
R
A fuzzy rule can be defined
as a binary relation with MF
Depends on how
to interpret A B
A B
79. 79
Interpretations of A B
A
B
A entails B
x
x
y
A coupled with B
A
B
x
x
y
, , ?
R A B
x y x y
80. 80
Interpretations of A B
B
A entails B
x
x
y
A coupled with B
A
B
x
x
y
, , ?
R A B
x y x y
81. 81
Interpretations of A B
A
B
A entails B
x
x
y
A coupled with B
A
B
x
x
y
, , ?
R A B
x y x y
A entails B (not A or B)
• Material implication
• Propositional calculus
• Extended propositional calculus
• Generalization of modus ponens
R A B A B
( )
R A B A A B
( )
R A B A B B
1 ( ) ( )
( , )
( ) otherwise
A B
R
B
x y
x y
y
82. 82
Interpretations of A B
, , ?
R A B
x y x y
A entails B (not A or B)
• Material implication
• Propositional calculus
• Extended propositional calculus
• Generalization of modus ponens
R A B A B
( )
R A B A A B
( )
R A B A B B
1 ( ) ( )
( , )
( ) otherwise
A B
R
B
x y
x y
y
( , ) max 1 ( ), ( )
R A B
x y x x
( , ) max 1 ( ),min ( ), ( )
R A A B
x y x x x
( , ) max 1 max ( ), ( ) , ( )
R A B B
x y x x x
84. 84
Fuzzy Reasoning
Single Rule with Single Antecedent
Rule:
Fact:
Conclusion:
if x is A then y is B
x is A’
y is B’
( )
x
x
A A’
y
( )
y
B
85. 85
Fuzzy Reasoning
Single Rule with Single Antecedent
Rule:
Fact:
Conclusion:
if x is A then y is B
x is A’
y is B’
( )
x
x
A A’
y
( )
y
B
( ) max min ( ), ( , )
B x A R
y x x y
( ) ( , )
x A R
x x y
( , ) ( ) ( )
R A B
x y x y
( ) ( ) ( )
x A A B
x x y
( ) ( ) ( )
x A A B
x x y
B
Firing
Strength Firing Strength
Max-Min Composition
86. 86
Fuzzy Reasoning
Single Rule with Single Antecedent
Rule:
Fact:
Conclusion:
if x is A then y is B
x is A’
y is B’
( )
x
x
A A’
y
( )
y
B
( ) max min ( ), ( , )
B x A R
y x x y
( ) ( , )
x A R
x x y
( )
B A A B
( , ) ( ) ( )
R A B
x y x y
( ) ( ) ( )
x A A B
x x y
( ) ( ) ( )
x A A B
x x y
B
Max-Min Composition
87. 87
Fuzzy Reasoning
Single Rule with Multiple Antecedents
Rule:
Fact:
Conclusion:
if x is A and y is B then z is C
x is A and y is B
z is C
88. 88
Fuzzy Reasoning
Single Rule with Multiple Antecedents
Rule:
Fact:
Conclusion:
if x is A and y is B then z is C
x is A’ and y is B’
z is C’
( )
x
x
A A’
y
( )
y
B
B’
z
( )
z
C
89. 89
Fuzzy Reasoning
Single Rule with Multiple Antecedents
( )
x
x
A A’
y
( )
y
B
B’
z
( )
z
C
Rule:
Fact:
Conclusion:
if x is A and y is B then z is C
x is A’ and y is B’
z is C’
, ,
( ) max min ( , ), ( , , )
C x y A B R
y x y x y z
R A B C
( , , ) ( , , )
R A B C
x y z x y z
( ) ( ) ( )
A B C
x y z
, , ( , ) ( , , )
x y A B R
x y x y z
, ( ) ( ) ( ) ( ) ( )
x y A B A B C
x y x y z
( ) ( ) ( ) ( ) ( )
x A A y B B C
x x y y z
Firing Strength
C
Max-Min Composition
90. 90
90
Fuzzy Reasoning
Single Rule with Multiple Antecedents
( )
x
x
A A’
y
( )
y
B
B’
z
( )
z
C
Rule:
Fact:
Conclusion:
if x is A and y is B then z is C
x is A’ and y is B’
z is C’
, ,
( ) max min ( , ), ( , , )
C x y A B R
y x y x y z
R A B C
( , , ) ( , , )
R A B C
x y z x y z
( ) ( ) ( )
A B C
x y z
, , ( , ) ( , , )
x y A B R
x y x y z
, ( ) ( ) ( ) ( ) ( )
x y A B A B C
x y x y z
( ) ( ) ( ) ( ) ( )
x A A y B B C
x x y y z
Firing Strength
C
Max-Min Composition
C A B A B C
91. 91
Fuzzy Reasoning
Multiple Rules with Multiple Antecedents
Rule1:
Fact:
Conclusion:
if x is A1 and y is B1 then z is C1
x is A’ and y is B’
z is C’
Rule2: if x is A2 and y is B2 then z is C2
92. 92
Fuzzy Reasoning
Multiple Rules with Multiple Antecedents
Rule1:
Fact:
Conclusion:
if x is A1 and y is B1 then z is C1
x is A’ and y is B’
z is C’
Rule2: if x is A2 and y is B2 then z is C2
( )
x
x
A1
A’
( )
z
z
C1
( )
y
y
B1
( )
x
x
A2
( )
y
y
B2
( )
z
z
C2
A’
B’
B’
93. 93
Fuzzy Reasoning
Multiple Rules with Multiple Antecedents
Rule1:
Fact:
Conclusion:
if x is A1 and y is B1 then z is C1
x is A’ and y is B’
z is C’
Rule2: if x is A2 and y is B2 then z is C2
( )
x
x
A1
A’
( )
z
z
C1
( )
y
y
B1
( )
x
x
A2
( )
y
y
B2
( )
z
z
C2
A’
B’
B’
( )
z
z
Max
1
C
2
C
1 2
C C C
1 2
C A B R R
1 2
A B R A B R
1 2
C C
Max-Min Composition