Transforming Family Resemblance Concepts into Fuzzy Sets

2021 ◽  
pp. 004912412098619
Author(s):  
Francesco Veri
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Concept Formation ◽  
Family Resemblance ◽  
Prototype Theory ◽  
Power Mean

This article aims to clarify fundamental aspects of the process of assigning fuzzy scores to conditions based on family resemblance (FR) structures by considering prototype and set theories. Prototype theory and set theory consider FR structures from two different angles. Specifically, set theory links the conceptualization of FR to the idea of sufficient and INUS (Insufficient but Necessary part of a condition, which is itself Unnecessary but Sufficient for the result) sets. In contrast, concept membership in prototype theory is strictly linked to the notion of similarity (or resemblance) in relation to the prototype, which is the anchor of the ideational content of the concept. After an introductive section where I elucidate set-theoretic and prototypical aspects of concept formation, I individuate the axiomatic properties that identify the principles of transforming FR structures into fuzzy sets. Finally, I propose an algorithm based on the power mean that is able to operationalize FR structures considering both set-theoretic and prototype theory perspectives.

Intuitionistic Fuzzy Soft Ideals

2020 ◽  
pp. 91-104
Author(s):  
Shuker Khalil
Keyword(s):  
Social Science ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Sets ◽  
Real Life ◽  
Soft Sets ◽  
Vague Sets ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Sets Theory

The basic notions of soft sets theory are introduced by Molodtsov to deal with uncertainties when solving problems in practice as in engineering, social science, environment, and economics. This notion is convenient and easy to apply as it is free from the difficulties that appear when using other mathematical tools as theory of theory of fuzzy sets, rough sets, and theory of vague sets. The soft set theory has recently gaining significance for finding rational and logical solutions to various real-life problems, which involve uncertainty, impreciseness, and vagueness. The concepts of intuitionistic fuzzy soft left almost semigroups and the intuitionistic fuzzy soft ideal are introduced in this chapter, and some of their basic properties are studied.

scholarly journals A method for classification of red, blue, and green galaxies using fuzzy set theory

2020 ◽  
Vol 499 (1) ◽  
pp. L31-L35
Author(s):  
Biswajit Pandey
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Sloan Digital Sky Survey ◽  
Sky Survey ◽  
Blue Galaxies ◽  
Boundary Sets ◽  
Red Galaxies

ABSTRACT Red and blue galaxies are traditionally classified using some specific cuts in colour or other galaxy properties, which are supported by empirical arguments. The vagueness associated with such cuts are likely to introduce a significant contamination in these samples. Fuzzy sets are vague boundary sets that can efficiently capture the classification uncertainty in the absence of any precise boundary. We propose a method for classification of galaxies according to their colours using fuzzy set theory. We use data from the Sloan Digital Sky Survey (SDSS) to construct a fuzzy set for red galaxies with its members having different degrees of ‘redness’. We show that the fuzzy sets for the blue and green galaxies can be obtained from it using different fuzzy operations. We also explore the possibility of using fuzzy relation to study the relationship between different galaxy properties and discuss its strengths and limitations.

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 Set Theory and Topos Theory

1986 ◽  
Vol 29 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Michael Barr
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Topos Theory

AbstractThe relation between the categories of Fuzzy Sets and that of Sheaves is explored and the precise connection between them is explicated. In particular, it is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.

FUZZY SETS AND FUZZY RELATIONAL STRUCTURES AS CHU SPACES

2000 ◽  
Vol 08 (04) ◽  
pp. 471-479 ◽  
Author(s):  
BASIL K. PAPADOPOULOS ◽  
APOSTOLOS SYROPOULOS
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Linear Logic ◽  
Relational Structure ◽  
Mathematical Structures ◽  
Relational Structures ◽  
Chu Spaces

Chu spaces, which derive from the Chu construct of *-autonomous categories, can be used to represent most mathematical structures. Moreover, the logic of Chu spaces is linear logic. Most efforts to incorporate fuzzy set theory into the realm of linear logic are based on the assumption that fuzzy and linear negation are identical operations. We propose an incorporation based on the opposite assumption and we provide an interpretation of some linear connectives. Furthermore, we show that it is possible to represent any fuzzy relational structure as a Chu space by means of the functor G.

Set theory for fuzzy sets of higher level

1979 ◽  
Vol 2 (2) ◽  
pp. 125-151 ◽  
Author(s):  
Siegfried Gottwald
Keyword(s):  
Fuzzy Sets ◽  
Set Theory

scholarly journals Intuitionistic Fuzzy Soft Rough Set and Its Application in Decision Making

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Haidong Zhang ◽  
Lan Shu ◽  
Shilong Liao
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Intuitionistic Fuzzy Sets ◽  
Mathematical Tool ◽  
Intuitionistic Fuzzy ◽  
Rough Approximation ◽  
Soft Rough Set

The soft set theory, originally proposed by Molodtsov, can be used as a general mathematical tool for dealing with uncertainty. In this paper, we present concepts of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets, and investigate some properties of soft rough intuitionistic fuzzy sets and intuitionistic fuzzy soft rough sets in detail. Furthermore, classical representations of intuitionistic fuzzy soft rough approximation operators are presented. Finally, we develop an approach to intuitionistic fuzzy soft rough sets based on decision making and a numerical example is provided to illustrate the developed approach.

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