Basic concepts and history of fuzzy set theory and fuzzy logic

2004 ◽  
Author(s):  
George Klir ◽  
Bo Yuan
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
History Of ◽  
Basic Concepts

scholarly journals Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 432 ◽  
Author(s):  
Vilém Novák
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Type Theory ◽  
Rough Set Theory ◽  
Basic Concepts ◽  
Alternative Set

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.

Prehistory, Emergence, and Evolution of Fuzzy Logic

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Twentieth Century ◽  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Academic Community ◽  
Early Research ◽  
Critical Stage ◽  
Stage Of Development ◽  
History Of

This chapter first identifies the rare instances throughout the long history of classical logic when the principle of bivalence was challenged and shows that they all have been rather inconsequential. It then briefly examines the early research on many-valued logics during the first half the twentieth century, and describes in some detail circumstances that led to the emergence of fuzzy set theory and fuzzy logic in the mid-1960s. This is followed by characterizing the evolving attitudes toward fuzzy logic, especially within the academic community, and by a summary of major and well-documented debates between members of the emerging fuzzy logic community and opponents of fuzzy logic. Finally, the chapter describes how the supporting infrastructure for fuzzy logic evolved during the early and rather critical stage of development of fuzzy logic.

Fuzzy Set Theory Applied to Accounting Sciences

2020 ◽  
Vol 16 (01) ◽  
pp. 1-16
Author(s):  
Carmen Lozano ◽  
Enriqueta Mancilla-Rendón
Keyword(s):  
Social Sciences ◽  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Hamming Distance ◽  
Great Acceptance ◽  
Classical Mathematics ◽  
Triangular Numbers ◽  
And Mathematics

Fuzzy set theory and fuzzy logic have been successfully developed in engineering and mathematics. However, these concepts have found great acceptance in social sciences in recent years since they provide an answer to those problems in the real world that cannot be modeled using classical mathematics. In this paper, we propose a new methodology for accounting science based on fuzzy triangular numbers. The methodology uses Hamming distance between fuzzy triangular numbers and arithmetic operations to evaluate corporate governance of multinational public stock corporations (PSCs) in the telecommunications sector.

Fuzzy Logic in the Broad Sense

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Human Reasoning ◽  
Human Beings ◽  
Real Numbers ◽  
New Directions ◽  
Gradual Development

The chapter begins by introducing the important and useful distinction between the research agendas of fuzzy logic in the narrow and the broad senses. The chapter deals with the latter agenda, whose ultimate goal is to employ intuitive fuzzy set theory for emulating commonsense human reasoning in natural language and other unique capabilities of human beings. Restricting to standard fuzzy sets, whose membership degrees are real numbers in the unit interval [0,1], the chapter describes how this broad agenda has become increasingly specific via the gradual development of standard fuzzy set theory and the associated fuzzy logic. An overview of currently recognized nonstandard fuzzy sets, which open various new directions in fuzzy logic, is presented in the last section of this chapter.

scholarly journals Fuzzy Insurance

1990 ◽  
Vol 20 (1) ◽  
pp. 33-55 ◽  
Author(s):  
Jean Lemaire
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Life Insurance ◽  
Fuzzy Set Theory ◽  
Fuzzy Numbers ◽  
Fuzzy Decision Making ◽  
Fuzzy Decision ◽  
Basic Concepts ◽  
Definition Of

AbstractFuzzy set theory is a recently developed field of mathematics, that introduces sets of objects whose boundaries are not sharply defined. Whereas in ordinary Boolean algebra an element is either contained or not contained in a given set, in fuzzy set theory the transition between membership and non-membership is gradual. The theory aims at modelizing situations described in vague or imprecise terms, or situations that are too complex or ill-defined to be analysed by conventional methods. This paper aims at presenting the basic concepts of the theory in an insurance framework. First the basic definitions of fuzzy logic are presented, and applied to provide a flexible definition of a “preferred policyholder” in life insurance. Next, fuzzy decision-making procedures are illustrated by a reinsurance application, and the theory of fuzzy numbers is extended to define fuzzy insurance premiums.

scholarly journals THE PROBLEM OF OBJECTIVE ASSESSMENT OF THE STUDENT’S KNOWLEDGE

2021 ◽  
Vol 7 (73) (1) ◽  
pp. 77-85
Author(s):  
N. V. Apatova ◽  
A. I. Gaponov
Keyword(s):  
Fuzzy Logic ◽  
Knowledge Acquisition ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Educational Material ◽  
Specific Situation ◽  
Mathematical Apparatus ◽  
Subjective Factor ◽  
Hierarchy Analysis

In the modern pedagogical literature a large number of works are devoted to the issue of evaluating students mastered knowledge. Ensuring the quality of diagnostics of the level of knowledge acquisition by students of educational organizations is based on an adequate, reasonable, acceptable and accessible mathematical apparatus. The adequacy and validity of the mathematical apparatus allows you to significantly reduce the influence of the subjective factor on the assessment of knowledge acquisition and minimize objections to its application. On the other hand, acceptability makes it possible to use a specific situation under consideration as a diagnosis, taking into account only its inherent nuances. Accessibility also means the ability to use the proposed mathematical algorithm by teachers who have the necessary minimum of elements of the proposed mathematical apparatus. The article considers the possibility of determining the level of development of the studied topic by students on the basis of five factors of knowledge acquisition: 1. Understanding; 2. Recognition; 3.Reproduction; 4. Application; 5. Creativity. In this case, the calculations are performed by using a simplified method of hierarchy analysis and fuzzy set theory in order to reduce the subjectivity inherent in the rating system of evaluation, the use of elements of the hierarchy analysis method and fuzzy logic is proposed. The fuzzy set theory algorithm is implemented using the fuzzy Logic Toolbox application package of the MATLAB software environment. Using the method of hierarchy analysis is to aggregate scores based on the application of a matrix of paired comparisons. The synthesis of these methods allows us to obtain a fairly effective comparative assessment of students development of educational material. On the basis of the considered methodology for evaluating students assimilation of educational material, it is suggested that it is possible to reduce the influence of a subjective factor to a minimum.

scholarly journals Fuzzy Logic in Aircraft Onboard Systems Reliability Evaluation—A New Approach

Sensors ◽  
2021 ◽  
Vol 21 (23) ◽  
pp. 7913
Author(s):  
Andrzej Żyluk ◽  
Konrad Kuźma ◽  
Norbert Grzesik ◽  
Mariusz Zieja ◽  
Justyna Tomaszewska
Keyword(s):  
Fuzzy Logic ◽  
Reliability Analysis ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probabilistic Model ◽  
Expert Knowledge ◽  
Operation Time ◽  
Theory Approach ◽  
Operational Data

This paper is a continuation of research into the possibility of using fuzzy logic to assess the reliability of a selected airborne system. The research objectives include an analysis of statistical data, a reliability analysis in the classical approach, a reliability analysis in the fuzzy set theory approach, and a comparison of the obtained results. The system selected for the investigation was the aircraft gun system. In the first step, after analysing the statistical (operational) data, reliability was assessed using a classical probabilistic model in which, on the basis of the Weibull distribution fitted to the operational data, the basic reliability characteristics were determined, including the reliability function for the selected aircraft system. The second reliability analysis, in a fuzzy set theory approach, was conducted using a Mamdani Type Fuzzy Logic Controller developed in the Matlab software with the Fuzzy Logic Toolbox package. The controller was designed on the basis of expert knowledge obtained by a survey. Based on the input signals in the form of equipment operation time (number of flying hours), number of shots performed (shots), and the state of equipment corrosion (corrosion), the controller determines the reliability of air armament. The final step was to compare the results obtained from two methods: classical probabilistic model and fuzzy logic. The authors have proved that the reliability model using fuzzy logic can be used to assess the reliability of aircraft airborne systems.

Konsep Himpunan Fuzzy pada Paradoks Russel

2020 ◽  
Vol 2 (2) ◽  
pp. 189
Author(s):  
Muhammad Abdy ◽  
Awi Dassa ◽  
Sri Julia Nensi
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Binary Logic ◽  
True Value ◽  
Degree Of Membership ◽  
The Law

Himpunan fuzzy menggunakan dasar logika fuzzy untuk menyatakan suatu objek menjadi anggota dengan derajat keanggotaan ( ), tetapi Logika fuzzy melanggar hukum logika biner sehingga muncul anggapan bahwa logika fuzzy memiliki masalah yang sama dengan paradoks. Tetapi nilai kebenarana logika fuzzy tergantung dari derajat keanggotaan yang dimilikinya sehingga dapat ditarik sebuah kesimpulan dari besar darajat keanggotaan tersebut, sedangkan paradoks nilai kebenarannya tidak dapat ditarik kesimpulan apapun.  Paradoks merupakan bentuk kritik landasan yang bertujuan untuk mengungkapkan dan menentukan inkonsistensi atau kontradiksi yang dihasilkan dari beberapa eksperimen mental dalam matematika, salah satu paradoks yang terkenal dalam kritik landasan teori himpunan adalah paradok Russel  Pemecahan paradoks Russel dengan menggunakan konsep teori himpunan fuzzy diperoleh derajat keanggotaan  adalah 0.5 merupakan pernyataan setengah benar (half true) dan  adalah 0.5 jugan merupakan pernyataan setengah benar (half true). Kata kunci: Logika fuzzy, himpunan fuzzy, paradoks, paradoks Russel.Fuzzy sets use the basis of fuzzy logic to declare an object to be a member with the degree of membership ( ), but fuzzy logic violates the law of binary logic so that the assumption arises that fuzzy logic has the same problem with paradox. But the true value of fuzzy logic depends on the degree of membership it has so that a conclusion can be drawn from the large membership ranks, while the paradox of its value cannot be drawn any conclusions. The paradox is a form of ground criticism that aims to express and determine the inconsistencies or contradictions that result from several mental experiments in mathematics, one of the paradoxes that is well-known in critics of set theory is Russel's paradox . The paradoxical solution of Russell by using fuzzy set theory concepts is that the degree of  membership is 0.5 and  is 0.5.Keywords: Fuzzy Logic, fuzzy set, paradox, Russel paradox.

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