Fuzzy sets and Boolean-valued universes

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

Set theory for fuzzy sets of higher level

Rough set theory for the interval-valued fuzzy information systems

A new fuzzy set theory, C-fuzzy set theory, is introduced in this paper. It is a particular case of the classical set theory and satisfies all formulas of the classical set theory. To add a limitation to C-fuzzy set system, in which all fuzzy sets must be “non-uniform inclusive” to each other, then it forms a family of sub-systems, the Z-fuzzy set family. It can be proved that the Z0-fuzzy set system, one of Z-fuzzy set systems, is equivalent to Zadeh’s fuzzy set system. Analysis shows that 1) Zadeh’s fuzzy set system defines the relations A = B and A ⊆ B between two fuzzy sets A and B as “∀u ∈ U, (μA ∈ (u) = μB(u))” and “∀u ∈ U, (μA(u) ≤ μB(u))” respectively is inappropriate, because it makes all fuzzy sets be “non-uniformly inclusive”; 2) it is also inappropriate to define two fuzzy sets’ union and intersection operations as the max and min of their grades of membership, because this prevents fuzzy set’s ability to correctly reflect different kinds of fuzzy phenomenon in the natural world. Then it has to work around the problem by invent unnatural functions that are hard to understand, such as augmenting max and min for union and intersection to min{a + b, 1} and max{a + b – 1, 0}, but these functions are incorrect on inclusive case. If both pairs of definitions are used together, not only are they unnatural, but also they are still unable to cover all possible set relationships in the natural world; and 3) it is incorrect to define the set complement as 1 – μA(u), because it can be proved that set complement cannot exist in Zadeh’s fuzzy set, and it causes confusion in logic and thinking. And it is seriously mistaken to believe that logics of fuzzy sets necessarily go against classical and normal thinking, logic, and conception. The C-fuzzy set theory proposed in this paper overcomes all of the above errors and shortcomings, and more reasonably reflects fuzzy phenomenon in the natural world. It satisfies all relations, formulas, and operations of the classical set theory. It is consistent with normal, natural, and classical thinking, logic, and concepts.

Researchers in science and engineering face various obstacles due to a lack of specific and full data. Many different approaches have been devised to deal with these restrictive requirements, but two notable schools of thought are the fuzzy set (FS) theory and the rough set (RS) theory, both of which have spawned many extensions and hybridizations. Although RS theory originated from an indiscernibility relation (also known as an equivalence relation in mathematics), emphasis rapidly shifted to similarity or coverings (and their fuzzy analogues). Many other hybrid schemes were suggested with this goal in mind. The gap between those concepts shrank because to this thorough analysis. Fuzzy set theory is a legitimate way to convey the ambiguity of assessment data, yet it is still inadequate for dealing with certain intricate problems in the actual world. In reality, decision makers will undoubtedly provide different kinds of ambiguous and nuanced assessments. Atanassov’s intuitionistic fuzzy set theory broadened the application of fuzzy set theory by imbuing it with an element of uncertainty. Sometimes in real life, you have to deal with a neutral element on top of the indeterminate one. Picture fuzzy sets were developed specifically for this purpose. Membership roles may be positive, neutral, or negative/refusal. In contrast, hesitant fuzzy sets and its hybrid models are useful when decision makers are on the fence about which option to choose. As a binary relation on a set, a graph is symmetric. It is a staple in mathematical modelling and is used in almost every scientific and technological discipline. Graph theory has been essential in the mathematical modelling and subsequent resolution of several real-world situations. Information about connections between things is often best represented using graph theory, which uses vertices to stand in for the items and edges for the relationships between them. The suggested dynamic algorithm is better to the static approach in dealing with the multidimensional dynamic changes of the hybrid incomplete decision system, according to a series of experiments carried out on nine UCI datasets.

Tolerance-based intuitionistic fuzzy-rough set approach for attribute reduction

Soft rough fuzzy sets and soft fuzzy rough sets

Hesitant fuzzy set is a generalization of the classical fuzzy set by returning a family of the membership degrees for each object in the universe. Since how to use the rough set model to solve fuzzy problems plays a crucial role in the development of the rough set theory, the fusion of hesitant fuzzy set and rough set is then firstly explored in this paper. Both constructive and axiomatic approaches are considered for this study. In constructive approach, the model of the hesitant fuzzy rough set is presented to approximate a hesitant fuzzy target through a hesitant fuzzy relation. In axiomatic approach, an operators-oriented characterization of the hesitant fuzzy rough set is presented, that is, hesitant fuzzy rough approximation operators are defined by axioms and then, different axiom sets of lower and upper hesitant fuzzy set-theoretic operators guarantee the existence of different types of hesitant fuzzy relations producing the same operators.

In this study, we establish a connection between rough soft set (Shabir et al., Knowl Base Syst 40:72–80, 2013) and fuzzy set. Based on the novel granulation structure called modified soft rough approximation space, fuzzy rough soft set is introduced. The important basic properties of fuzzy rough soft set are studied and supported by illustrative examples. Moreover lattice theory is studied on fuzzy rough soft set. The definitions and propositions presented in this paper enrich the soft set theory, rough set theory and fuzzy set theory, and also extend their application scopes. The paper ends with conclusions having future investigations of the study.

Soft set theory was first introduced by D. A. Molodtsov [D. Molodtsov, Soft set theory first results, Computers & Mathematics with Applications, 37(4–5) (1999) 19–31] in 1999 as an effective approach to deal with uncertainty. Over time, fuzzy soft set theory has evolved significantly, integrating concepts from both fuzzy set theory and soft set theory to enhance decision-making processes. Acharjee and Medhi [S. Acharjee and S. Medhi, Logical connectives of fuzzy soft set theory, New Mathematics and Natural Computation, 21(1) (2025) 339–352] introduced several advanced concepts related to logical connectives in fuzzy soft set theory. Building on Molodtsov’s foundational work, this paper explores new developments in fuzzy soft logic, extending the findings of Acharjee and Medhi. Specifically, we introduce the concepts of [Formula: see text]-fuzzy soft implication and [Formula: see text]-fuzzy soft bi-implication, along with their related results within the framework of fuzzy soft logic.

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

<p>The article presents the results of modeling the solution to the problem of determining the reliability of the components of a multiservice communication network (MCN) based on the theory of fuzzy sets. The main characteristics of the equipment that affect the reliability parameters of the MCN are given. To solve the problem of determining the reliability of MCN components based on the theory of fuzzy sets, a multiservice network is presented in the form of a hierarchical diagram, which shows the main components of each network level. A multiservice network is presented as a parameter of the U function. The reliable state of the MCN depends on the state of the equipment at the corresponding levels. The results of modeling the solution to the problem of determining the reliability of MCN components based on the theory of fuzzy sets are presented using the mathematical apparatus of the theory of fuzzy sets and fuzzy logic in MATLAB fuzzy logic toolbox, fuzzyTECH.</p>

Emerging trends in soft set theory and related topics.

A fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a fuzzy/crist set in a fuzzy approximation space. A fuzzy rough set algebra is a fuzzy set algebra with added dual pair of fuzzy rough approximation operators. In this paper, we study the mathematical structures of fuzzy rough set algebras in infinite universes of discourse. We first define the concept of fuzzy rough set algebras by the axiomatic approach. We then examine the properties of fuzzy rough approximation operators in different types of fuzzy rough set algebras. We also prove that if a system \(({\cal F}(U), \cap, \cup, \sim, L, H)\) is a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra then the derived system \(({\cal F}(U), \cap, \cup, \sim, LL, HH)\) is also a (respectively, a serial, a reflexive, a symmetric, a transitive, a topological, a similarity) fuzzy rough set algebra.KeywordsApproximation operatorsFuzzy rough setsFuzzy setsFuzzy rough set algebrasRough sets