AN OVERVIEW ON THE CONSTRUCTION OF FUZZY SET THEORIES

2005 ◽  
Vol 01 (03) ◽  
pp. 329-358 ◽  
Author(s):  
ANA PRADERA ◽  
ENRIC TRILLAS ◽  
ELOY RENEDO
Keyword(s):  
Fuzzy Set ◽  
Triangular Norms ◽  
Definition Of ◽  
Fuzzy Connectives ◽  
Standard Solutions

This paper analyzes the main issues involved in the construction of Fuzzy Set Theories. It reviews both standard solutions (based on the well-known triangular norms and conorms) as well as less the conventional proposals that provide alternative views on, for example, the definition of fuzzy connectives or the study of their properties.

REVISITING Xor-IMPLICATIONS: CLASSES OF FUZZY (CO)IMPLICATIONS BASED ON f-Xor (f-XNor) CONNECTIVES

2013 ◽  
Vol 21 (06) ◽  
pp. 899-925 ◽  
Author(s):  
BENJAMÍN BEDREGAL ◽  
RENATA HAX SANDER REISER ◽  
GRAÇALIZ PEREIRA DIMURO
Keyword(s):  
Boundary Conditions ◽  
Unit Interval ◽  
End Points ◽  
Triangular Norms ◽  
Fuzzy Implications ◽  
Classical Definition ◽  
Definition Of ◽  
Exclusive Or ◽  
Intrinsic Definition ◽  
Fuzzy Connectives

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.

scholarly journals Convexity of hesitant fuzzy sets based on aggregation functions

2020 ◽  
pp. 45-45
Author(s):  
Pedro Huidobro ◽  
Pedro Alonso ◽  
Vladimír Janis ◽  
Susana Montes
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Fuzzy Optimization ◽  
Hesitant Fuzzy Set ◽  
Hesitant Fuzzy Sets ◽  
Geometric Properties ◽  
Aggregation Functions ◽  
Definition Of

Convexity is one of the most important geometric properties of sets and a useful concept in many fields of mathematics, like optimization. As there are also important applications making use of fuzzy optimization, it is obvious that the studies of convexity are also frequent. In this paper we have extended the notion of convexity for hesitant fuzzy sets in order to fulfill some necessary properties. Namely, we have found an appropriate definition of convexity for hesitant fuzzy sets on any ordered universe based on aggregation functions such that it is compatible with the intersection, that is, the intersection of two convex hesitant fuzzy sets is a convex hesitant fuzzy set and it fulfills the cut worthy property.

scholarly journals Fuzzy Soft Multiset Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Shawkat Alkhazaleh ◽  
Abdul Razak Salleh
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Soft Set ◽  
Mathematical Tool ◽  
Basic Properties ◽  
Definition Of

In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsov's soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.

Evaluation of Quantified Statements Using Gradual Numbers

2011 ◽  
pp. 246-269 ◽  
Author(s):  
Ludovic Liétard ◽  
Daniel Rocacher
Keyword(s):  
Decision Making ◽  
Expert Systems ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Relational Databases ◽  
Fuzzy Numbers ◽  
Theoretical Background ◽  
Definition Of ◽  
Flexible Querying

This chapter is devoted to the evaluation of quantified statements which can be found in many applications as decision making, expert systems, or flexible querying of relational databases using fuzzy set theory. Its contribution is to introduce the main techniques to evaluate such statements and to propose a new theoretical background for the evaluation of quantified statements of type “Q X are A” and “Q B X are A.” In this context, quantified statements are interpreted using an arithmetic on gradual numbers from Nf, Zf, and Qf. It is shown that the context of fuzzy numbers provides a framework to unify previous approaches and can be the base for the definition of new approaches.

The Unique Representations of Fuzzy Entropy for a Finite Fuzzy Set

2013 ◽  
Vol 433-435 ◽  
pp. 766-769
Author(s):  
Ming Qing
Keyword(s):  
Fuzzy Set ◽  
Order Relation ◽  
Fuzzy Entropy ◽  
Simple Order ◽  
Fuzzy Degree ◽  
Definition Of

Many methods were presented to define fuzzy entropy to measure fuzzy degree of a fuzzy set and a variety of fuzzy entropy formulae were derived and constructed from the definitions of fuzzy entropy. In this paper, a new definition of fuzzy entropy is presented based on a simple order relation and computation formulae of fuzzy entropy is given. Then, the unique representations of fuzzy entropy are given by applying several set of reasonable conditions to fuzzy entropy.

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.

Fuzzy sets in modeling of patient’s disease states

2019 ◽  
Vol 0 (9/2019) ◽  
pp. 5-11
Author(s):  
Andrzej Ameljańczyk
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Medical Diagnosis ◽  
Medical Diagnostics ◽  
Medical Data ◽  
Medical Decisions ◽  
Disease States ◽  
Definition Of ◽  
Disease Unit

The paper concerns the mathematical modeling of patient’s disease states and disease unit patterns for the needs of algorithms supporting medical decisions. Due to the specificity of medical data and assessments in the modeling of patient’s disease states as well as diseases, the fuzzy set methodology was used. The paper presents a number of new characteristics of fuzzy sets allowing to assess the quality of medical diagnosis. In addition, a definition of a multi-aspect fuzzy set is presented, which may be useful in supporting medical diagnostics based on multi-criteria similarity models. The presented results can be used in the construction of algorithms for assessing the patient's state of health and mainly in the construction of algorithms for supporting diagnostic processes.

Territorial socio-ecological-economics systems in the field of tourism and service of Central Federal District: theoretical aspect

2015 ◽  
Vol 9 (3) ◽  
pp. 43-48 ◽  
Author(s):  
Мария Веслогузова ◽  
Mariya Vesloguzova ◽  
Лада Розанова ◽  
Lada Rozanova ◽  
Людмила Петрик ◽  
...  
Keyword(s):  
Fuzzy Set ◽  
Service Sector ◽  
Theoretical Aspect ◽  
Federal District ◽  
Estimation Algorithm ◽  
Initial Position ◽  
Economic Systems ◽  
Ecological Economic ◽  
Trade Offs ◽  
Definition Of

This article characterized the territorial socio-ecological-economic system, reveals the essence and sources offuzzi-ness of socio-ecological-economic processes and objects in the service sector, as well as approaches to their evaluation. The aim of the study is to reveal the nature of socio-ecological-economic systems, the disclosure of the nature, dynamics, level of mobility, interaction algorithms, the state of uncertainty and fuzziness of socio-ecological-economic processes and facilities. To achieve this goal, authors used the method of uncertain sets and algorithms L Zadeh, based on the concept of the appurtenance function μ(χ), which characterizes the degree of dependence of element «x» with specific vague set. In this initial position is accepted that the task of decision making in situations of uncertainty (including assessment tasks) in principle cannot be reduced to a strictly mathematical tasks. For this it is necessary to eliminate or reduce the uncertainty by introducing certain hypotheses, for example, in the form of the appurtenance function of fuzzy set or uncertain relationship. In evaluation of the areas it means that this procedure may not have the character of fully formal logical algorithm, and should rely heavily on logic-meaningful approaches and techniques of analysis. In the analysis authors appeal to informal and semi-formal research apparatus, the base of which constituted the systematic and synergetic communicative techniques and methods of the theory of expert evaluations of fuzzy set theory and the theory of trade-offs that allowed to formulate the definition of social, ecological and economic systems, characterize approaches to evaluation of these systems within the theory of fuzzy sets and making trade-offs, determine the estimation algorithm noted systems.

Medical Image Retrieval by Fuzzy Set Based Geometric Relationships

2010 ◽  
Vol 44-47 ◽  
pp. 3154-3158
Author(s):  
Hui Liu ◽  
Xiao Hui Xing
Keyword(s):  
Fuzzy Set ◽  
Medical Image ◽  
Field Model ◽  
Reference Object ◽  
New Approach ◽  
Markov Random ◽  
Fuzzy Pattern ◽  
Relational Graphs ◽  
Definition Of ◽  
Geometric Relationship

Modeling spatial context (e.g., autocorrelation) is a key challenge in classification and retrieval problems that arise in image processing regions. This work proposes a new approach for medical images retrieval enlightened by traditional Markov Random Field model and improve on it. Contrasting with previous work, this method relies on coping with the ambiguity of spatial relative position concepts: a new definition of the geometric relationship between two objects in a fuzzy set framework is proposed. This definition is based on a fuzzy pattern-matching approach, which comparing an object by the fuzzy set representation of the degree of position satisfaction to a reference object. Furthermore, Fuzzy Attributed Relational Graphs (FARGs) are used in this framework for the purpose of medical image similarity measurement.

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