scholarly journals On Characterization of Rough Type-2 Fuzzy Sets

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Tao Zhao ◽  
Zhenbo Wei
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Rough Sets ◽  
Approximation Space ◽  
Fuzzy Approximation ◽  
Approximation Operators ◽  
Sets Theory

Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the generalized Pawlak approximation space are also introduced. Some basic properties of the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators are discussed. The connections between special crisp binary relations and generalized rough type-2 fuzzy approximation operators are further examined. The axiomatic characterization of generalized rough type-2 fuzzy approximation operators is also presented. Finally, the attribute reduction of type-2 fuzzy information systems is investigated.

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.

Granular Computing

2011 ◽  
pp. 774-780
Author(s):  
Georg Peters
Keyword(s):  
Probability Theory ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Granular Computing ◽  
Information Granularity ◽  
Relationship Of ◽  
The Relationship

It is well accepted that in many real life situations information is not certain and precise but rather uncertain or imprecise. To describe uncertainty probability theory emerged in the 17th and 18th century. Bernoulli, Laplace and Pascal are considered to be the fathers of probability theory. Today probability can still be considered as the prevalent theory to describe uncertainty. However, in the year 1965 Zadeh seemed to have challenged probability theory by introducing fuzzy sets as a theory dealing with uncertainty (Zadeh, 1965). Since then it has been discussed whether probability and fuzzy set theory are complementary or rather competitive (Zadeh, 1995). Sometimes fuzzy sets theory is even considered as a subset of probability theory and therefore dispensable. Although the discussion on the relationship of probability and fuzziness seems to have lost the intensity of its early years it is still continuing today. However, fuzzy set theory has established itself as a central approach to tackle uncertainty. For a discussion on the relationship of probability and fuzziness the reader is referred to e.g. Dubois, Prade (1993), Ross et al. (2002) or Zadeh (1995). In the meantime further ideas how to deal with uncertainty have been suggested. For example, Pawlak introduced rough sets in the beginning of the eighties of the last century (Pawlak, 1982), a theory that has risen increasing attentions in the last years. For a comparison of probability, fuzzy sets and rough sets the reader is referred to Lin (2002). Presently research is conducted to develop a Generalized Theory of Uncertainty (GTU) as a framework for any kind of uncertainty whether it is based on probability, fuzziness besides others (Zadeh, 2005). Cornerstones in this theory are the concepts of information granularity (Zadeh, 1979) and generalized constraints (Zadeh, 1986). In this context the term Granular Computing was first suggested by Lin (1998a, 1998b), however it still lacks of a unique and well accepted definition. So, for example, Zadeh (2006a) colorfully calls granular computing “ballpark computing” or more precisely “a mode of computation in which the objects of computation are generalized constraints”.

scholarly journals A Hesitant Fuzzy Set Approach to Ideal Theory in Γ-Semigroups

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Mohammad Y. Abbasi ◽  
Aakif F. Talee ◽  
Sabahat A. Khan ◽  
Kostaq Hila
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Ideal Theory ◽  
Hesitant Fuzzy Set ◽  
Fuzzy Sets Theory ◽  
Hesitant Fuzzy Sets ◽  
Sets Theory

We apply the hesitant fuzzy sets theory to Γ-semigroups and provide some characterizations of hesitant fuzzy left (right and bi-) ideals. We introduce the hesitant fuzzy left (resp., right and two-sided) ideal, hesitant fuzzy bi-ideal, and hesitant fuzzy interior ideal in Γ-semigroup and study some properties of them. Finally, a characterization of a simple Γ-semigroup by means of a hesitant fuzzy simple Γ-semigroup is obtained.

Uncertainty and Vagueness Concepts in Decision Making

2011 ◽  
pp. 901-909
Author(s):  
Georg Peters
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
17Th Century ◽  
Basic Principles ◽  
In The Beginning

One of the main challenges in decision making is how to deal with uncertainty and vagueness. The classic uncertainty concept is probability, which goes back to the 17th century. Possible candidates for the title of father of probability are Bernoulli, Laplace, and Pascal. Some 40 years ago, Zadeh (1965) introduced the concept of fuzziness, which is sometimes interpreted as one form of probability. However, we will show that the terms fuzzy and probability are complementary. Recently, in the beginning of the ’80s, Pawlak (1982) suggested rough sets to manage uncertainties. The objective of this article is to give a basic introduction into probability, fuzzy set, and rough set theory and show their potential in dealing with uncertainty and vagueness. The article is structured as follows. In the next three sections we will discuss the basic principles of probability, fuzzy sets, and rough sets, and their relationship with each other. The article concludes with a short summary.

scholarly journals Further Study of MultigranulationT-Fuzzy Rough Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Wentao Li ◽  
Xiaoyan Zhang ◽  
Wenxin Sun
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Approximation Space ◽  
Fuzzy Rough Sets ◽  
Fuzzy Rough Set ◽  
Upper Approximation ◽  
Fuzzy Approximation ◽  
Approximation Operators ◽  
Special Instance

The optimistic multigranulationT-fuzzy rough set model was established based on multiple granulations underT-fuzzy approximation space by Xu et al., 2012. From the reference, a natural idea is to consider pessimistic multigranulation model inT-fuzzy approximation space. So, in this paper, the main objective is to make further studies according to Xu et al., 2012. The optimistic multigranulationT-fuzzy rough set model is improved deeply by investigating some further properties. And a complete multigranulationT-fuzzy rough set model is constituted by addressing the pessimistic multigranulationT-fuzzy rough set. The full important properties of multigranulationT-fuzzy lower and upper approximation operators are also presented. Moreover, relationships between multigranulation and classicalT-fuzzy rough sets have been studied carefully. From the relationships, we can find that theT-fuzzy rough set model is a special instance of the two new types of models. In order to interpret and illustrate optimistic and pessimistic multigranulationT-fuzzy rough set models, a case is considered, which is helpful for applying these theories to practical issues.

scholarly journals On Fuzzy Rough Sets and Their Topological Structures

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Weidong Tang ◽  
Jinzhao Wu ◽  
Dingwei Zheng
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Fuzzy Relations ◽  
Fuzzy Rough Sets ◽  
Approximation Spaces ◽  
Topological Structures ◽  
Fuzzy Approximation ◽  
Approximation Operators

The core concepts of rough set theory are information systems and approximation operators of approximation spaces. Approximation operators draw close links between rough set theory and topology. This paper is devoted to the discussion of fuzzy rough sets and their topological structures. Fuzzy rough approximations are further investigated. Fuzzy relations are researched by means of topology or lower and upper sets. Topological structures of fuzzy approximation spaces are given by means of pseudoconstant fuzzy relations. Fuzzy topology satisfying (CC) axiom is investigated. The fact that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies satisfying (CC) axiom is proved, the concept of fuzzy approximating spaces is introduced, and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained, which illustrates that we can research fuzzy relations or fuzzy approximation spaces by means of topology and vice versa. Moreover, fuzzy pseudoclosure operators are examined.

scholarly journals Relational Formal Characterization of Rough Sets

2013 ◽  
Vol 21 (1) ◽  
pp. 55-64
Author(s):  
Adam Grabowski
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Relational Structures ◽  
Approximation Operators ◽  
Mizar Mathematical Library ◽  
Direct Correspondence

Summary The notion of a rough set, developed by Pawlak [10], is an important tool to describe situation of incomplete or partially unknown information. In this article, which is essentially the continuation of [6], we try to give the characterization of approximation operators in terms of ordinary properties of underlying relations (some of them, as serial and mediate relations, were not available in the Mizar Mathematical Library). Here we drop the classical equivalence- and tolerance-based models of rough sets [12] trying to formalize some parts of [19] following also [18] in some sense (Propositions 1-8, Corr. 1 and 2; the complete description is available in the Mizar script). Our main problem was that informally, there is a direct correspondence between relations and underlying properties, in our approach however [7], which uses relational structures rather than relations, we had to switch between classical (based on pure set theory) and abstract (using the notion of a structure) parts of the Mizar Mathematical Library. Our next step will be translation of these properties into the pure language of Mizar attributes.

Measures of uncertainty for a fuzzy probabilistic approximation space

2021 ◽  
pp. 1-24
Author(s):  
Lijun Chen ◽  
Damei Luo ◽  
Pei Wang ◽  
Zhaowen Li ◽  
Ningxin Xie
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fuzzy Relation ◽  
Point Of View ◽  
Approximation Space ◽  
Fuzzy Approximation ◽  
Entropy Measurement ◽  
Fuzzy Relation Matrix ◽  
Relation Matrix

 An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

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