Intuitionistic fuzzy sets. Theory and applications
type: Item
instance of: scholarly article
title: Intuitionistic fuzzy sets. Theory and applications
zbMATH Open document ID: 0939.03057
published in: Studies in Fuzziness and Soft Computing
publication date: 1999-12-06T00:00:00Z
review text: The book is devoted to the intuitionistic fuzzy set theory and its applications. The exposition is organized in five chapters: 1. Intuitionistic fuzzy sets; 2. Interval-valued intuitionistic fuzzy sets; 3. Other extensions of intuitionistic fuzzy sets; 4. Elements of intuitionistic fuzzy logic; 5. Applications of intuitionistic fuzzy sets. An intuitionistic fuzzy set (IFS) \(A\) in \(E\) is defined as an object of the form \[ A= \{\langle x,\mu_A(x), \nu_A(x)\rangle/x\in E\}, \] where the functions \(\mu_A:E\to [0,1]\) and \(\nu_A: E\to[0, 1]\) define the degree of membership and the degree of non-membership for the elements \(x\in E\) satisfying the condition \(0\leq \mu_A(x)+ \nu_A(x)\leq 1\). The relationship between IFSs and ordinary fuzzy sets, as well as between IFSs and crisp sets is mentioned. The IFSs are natural extensions with higher describing abilities than the ordinary fuzzy sets. Operations (intersection, union, sum, etc.) and various relations (equality, inclusion, etc.) for IFSs are introduced and their properties are studied. The exposition is enriched with suitable geometric interpretations of operations between IFSs. The algebraic properties of the algebra \(\langle E^*,\cup, \cap\rangle\) are investigated, where \(E^*\) stays for all IFSs over the universe \(E\). Five kinds of Cartesian products are defined, their geometric interpretation is supplied, De Morgan's laws are given. The basic notion of Cartesian product leads to a family of related notions: intuitionistic fuzzy relation, index matrix, intuitionistic fuzzy graph. Various operators are defined on the IFSs: modal-like operators, topological operators, various \(\alpha\)- and \(\alpha, \beta\)-operators (equipped with geometric interpretations), identifying and unary operators, level operators. Their properties are investigated. The most important property of IFS, which have no counterpart in the ordinary fuzzy sets, is that modal-like operators can be defined over IFSs. At the end of C…
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 68T35
zbMATH DE Number: 1374047
zbMATH Keywords: intuitionistic fuzzy expert systems
zbMATH Keywords: intuitionistic fuzzy set theory
zbMATH Keywords: intuitionistic fuzzy logic
zbMATH Keywords: degree of membership
zbMATH Keywords: degree of non-membership
zbMATH Keywords: Cartesian product
zbMATH Keywords: intuitionistic fuzzy relation
zbMATH Keywords: intuitionistic fuzzy graph
zbMATH Keywords: modal-like operators
zbMATH Keywords: intuitionistic fuzzy models
zbMATH Keywords: neural networks
zbMATH Keywords: intuitionistic fuzzy systems
author: Krassimir Atanassov
reviewed by: Ketty Peeva
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1374047