Zadeh's Fuzzy Set Research Articles

Rough set model based on axiomatic fuzzy set

Both fuzzy set and rough set are important mathematical tools to describe incomplete and uncertain information, and they are highly complementary to each other. What is more, most fuzzy rough sets are obtained by combining Zadeh fuzzy sets and Pawlak rough sets. There are few reports about the combination of axiomatic fuzzy sets and Pawlak rough sets. For this reason, we propose the axiomatic fuzzy rough sets (namely rough set model with respect to the axiomatic fuzzy set) establishing on fuzzy membership space. In this paper, we first present a similarity description method based on vague partitions. Then the concept of similarity operator is proposed to describe uncertainty in the fuzzy approximation space. Finally, some characterizations concerning upper and lower approximation operators are shown, including basic properties. Furthermore, we give a algorithm to verify the effectiveness and efficiency of the model.

Beyond Comparative Poetics: An Evaluation of The Nucleo-genre Paradigm

The dominance of Western theories in an age of globalization and swift technological advances, which brought to collective consciousness manifestations of literary and artistic genres in distant geographical places that were once viewed as primitive or exotic, necessitates raising the question of comparative poetics anew. During the second half of the twentieth century, scholars such as James J.Y. Liu, Earl Miner, Stephen Owen, and Wai-lim Yip engaged in comparing Western to Eastern poetics. However, their work fell short of reaching a unifying approach to the study of world poetics. This is the gap that Alaa Abd al-Hadi, a thinker, poet and postcolonial critic, came to fill out. Abd al-Hadi’s Nucleo-genre Paradigm is universally oriented. Based on qualitative logic and Lutfi Zadeh's fuzzy sets, it posits that there are essential elements common to various cultural manifestations of a single genre across time and space. It is these common elements, which help decide whether a certain manifestation belongs to a certain genre, that allow for cross cultural communication and understanding while respecting cultural specificities, manifested by an infinite number of aesthetic elements. Unlike the other approaches to the field of comparative poetics, Nucleo-genre Paradigm distinguishes between two levels of genre, the poetic level of production and the aesthetic level of reception, and gives primacy to reception, which is often disregarded in genre theories. In this paper, an attempt is made to evaluate the contribution of the Nucleo-genre Paradigm beyond comparative poetics.

Mathematical Models of Diagnostic Information Granules Generated by Scaling Intuitionistic Fuzzy Sets

The paper presents a certain class of the mathematical models of diagnostic information granules describing the fuzzy symptoms-faults relationship. A certain fuzzy diagnostic information retrieval system is described as an application of an expert diagnostic system. Symptoms and faults are fuzzy, and with some scaling of the symptom-fault concept pair values. These value pairs can be considered as intuitionistic fuzzy sets for the space of diagnosed objects. In this article, for scaling intuitionistic fuzzy sets (n-ScIFS), the deductive theory is formulated. There the intuitionistic fuzzy sets (IFSs) and the Pythagorean fuzzy sets (PFSs) are generalized to the n-ScIFS objects. The membership and non-membership values, as standard, can be described by the 1:1 scale or the quadratic function scale. However, any power scale n>2 can be used. In this paper, any n-Sc scaling functions retaining the IFSs properties are considered. The n-ScIFS theory introduces a conceptual apparatus analogous to the classical theory of Zadeh fuzzy sets and Yager PFSs, consistently striving, for the first time, to formulate the relational structure of n-ScIFSs as a model of a certain information granule system called here the diagnostic granule system. In addition, power- and linear-repeatable diagnostic granules are defined in the n-ScIFSs structure for serial or parallel diagnosis processes. The information granule base is determined and a diagnostic granule system model produced by this information granule base is shown. Certain algorithms have been given to establish the semantic language of diagnosis describing the system of diagnostic information granules.

Representation of a Crisp Set as a Pair of Dual Fuzzy Sets

Expert knowledge representations often fail to determine compatibility levels on all objects, and these levels are represented for a certain sampling of universe. The samplings for the fuzzy terms of the linguistic variable, whose compatibility functions are aggregated according to a certain problem, may also be different. In such a case, neither L.A. Zadeh’s analysis of fuzzy sets and even the dual forms of developing today R.R. Yager’s q-rung orthopair fuzzy sets cannot provide the necessary aggregations. This fact, as a given, can be considered as a source of new types of information, in order to obtain different levels of compatibility according to Zadeh, presented throughout the universe. This source of information can be represented as a pair ⟨A, fA⟩, where there is some crisp subset of the universe A that determines the sampling of objects from the universe, and a function fA determines the compatibility levels of the elements of that sampling. It is a notion of split fuzzy set, constructed in this article, that allows for the semantic representation and aggregation of such information. This notion is again and again based on the notion of Zadeh fuzzy set. In particular, the operation of splitting a crisp subset into dual fuzzy sets is introduced. Definitions of set operations on split dual fuzzy-sets are presented in the paper. The proofs are also presented that follow naturally from definitions and previous results. An example of MADM is presented for illustration of the application of splitting operation.

The n-Pythagorean Fuzzy Sets

The following paper presents deductive theories of n-Pythagorean fuzzy sets (n-PFS). N-PFS objects are a generalization of the intuitionistic fuzzy sets (IFSs) and the Yager Pythagorean fuzzy sets (PFSs). Until now, the values of membership and non-membership functions have been described on a one-to-one scale and a quadratic function scale. There is a symmetry between the values of this membership and non-membership functions. The scales of any power functions are used here in order to increase the scope of the decision-making problems. The theory of n-PFS introduces a conceptual apparatus analogous to the classic theory of Zadeh fuzzy sets, consistently striving to correctly define the n-PFS algebra.

Integrations of q-Rung Orthopair Fuzzy Continuous Information

Yager's q-rung orthopair fuzzy sets (q-ROFSs), which extend Zadeh's fuzzy sets, use the membership and nonmembership functions to describe things’ vague characteristics, and the sum of the q th-power for the membership and nonmembership functions is less than or equal to 1. More recently, some scholars have proposed a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. However, so far, there is no research on aggregating q-rung orthopair fuzzy continuous information. Thus, we proposed q-rung orthopair fuzzy definite integrals (q-ROFDIs) to fill this vacancy. First, we further study the operations of q-rung orthopair fuzzy numbers (q-ROFNs) that are the core of q-ROFSs. We also introduce the limit of a q-ROFN sequence. Subsequently, we construct the q-ROFDIs step-by-step, give their concrete values, and discuss their integrability criteria from two perspectives. From the perspectives of modern analysis and the operational laws of q-ROFNs, we investigate the q-ROFDIs in detail, which are concise and considerably different from the investigative techniques of the previous research on aggregating continuous information. Finally, a practical example is provided to show the effectiveness, elasticity, and superiority of the q-ROFDIs via comparing them with the existing methods.

Diagram Illustrations of Aggregation Operations for the Intuitionistic Fuzzy Values

Atanassov (Intuitionistic fuzzy set, Fuzzy Sets and Systems 20 (1986) 87–96) proposed the intuitionistic fuzzy set (A-IFS) that is a generalization of Zadeh's fuzzy set. The essential elements of an A-IFS are intuitionistic fuzzy values (IFVs), each of which is depicted by a membership degree and a non-membership degree. The existing literature only gives the explanation of the meaning and the aggregation operators of IFVs, which is very hard and non-visualized for us to comprehend. To overcome the above shortcomings, this paper puts forward the diagram method to analyze the structures of the basic aggregation operations of IFVs. In addition, this paper utilizes the diagram method which is similar to joint probability distribution to describe the components of aggregation operations of IFVs, and each IFV depicts a 3-dimensional vector which is regarded as the marginal distributions of the joint probability distribution.

An activation detection based similarity measure for intuitionistic fuzzy sets

Intuitionistic fuzzy sets (IF-sets), with mechanisms to represent both the degree of membership and hesitancy of a given entity with respect to a concept under consideration, have been proven to be a useful extension to Zadeh's fuzzy set theory. Noteworthy efforts by various researchers have been devoted to defining a robust similarity measure for a given pair of IF-sets, as we often need to quantify the similarity between given entities in application domains ranging from medical diagnosis to multiple criteria decision making. These efforts have shown that it is highly non-trivial to construct a truly robust IF-set similarity measure with easy-to-understand interpretations. In this article, grounded on native concepts from activation detection in medical image analysis, a model for determining the degree of similarity between IF-sets is proposed. An IF-set similarity measure (termed the activation detection based similarity measure) is then systematically built from this model. We show that the proposed measure produces results that are intuitively appealing, easy to understand, and can be robustly interpreted. Moreover, we demonstrate that the proposed measure obeys standard conventions regarding set definition in the classical setting, and is equivalent to the Jaccard's similarity measure as we transition from the intuitionistic fuzzy setting to the classical setting. The source code of the numerical implementation of the proposed measure is available from the author upon request.

BI-IDEALS OF ORDERED SEMIGROUPS BASED ON THE INTERVAL-VALUED FUZZY POINT

Interval-valued fuzzy set theory (advanced generalization of Zadeh's fuzzy sets) is a more generalized theory that can deal with real world problems more precisely than ordinary fuzzy set theory. In this paper, we introduce the notion of generalized quasi-coincident with () relation of an interval-valued fuzzy point with an interval-valued fuzzy set. In fact, this new concept is a more generalized form of quasi-coincident with relation of an interval-valued fuzzy point with an interval-valued fuzzy set. Applying this newly defined idea, the notion of an interval-valued -fuzzy bi-ideal is introduced. Moreover, some characterizations of interval-valued -fuzzy bi-ideals are described. It is shown that an interval-valued -fuzzy bi-ideal is an interval-valued fuzzy bi-ideal by imposing a condition on interval-valued fuzzy subset. Finally, the concept of implication-based interval-valued fuzzy bi-ideals, characterizations of an interval-valued fuzzy bi-ideal and an interval-valued -fuzzy bi-ideal are considered.

Definite Integrals of Atanassov's Intuitionistic Fuzzy Information

Atanassov's intuitionistic fuzzy set (A-IFS) is a generalized form of Zadeh's fuzzy set, and the basic elements of an A-IFS are intuitionistic fuzzy numbers (IFNs). Recently, lots of aggregation techniques have been proposed for fusing IFNs. However, they only deal with a limited number of IFNs that take the form of discrete information. In this paper, we will first apply the definite integral to give the notion of definite integration for IFNs and investigate a lot of novel integral operators and, then, utilize these integral operators to get some new aggregation operators that can aggregate the IFNs spreading all over an area, which means that each point in a 2-D plane is an IFN that we want to aggregate. The new techniques can help us to deal with more complicated intuitionistic fuzzy information.

On the entropy for Atanassov's intuitionistic fuzzy sets: An interpretation from the perspective of amount of knowledge

Given the fact that totally unknown information cannot be expressed by Zadeh's fuzzy sets but can well be represented by Atanassov's intuitionistic fuzzy sets (A-IFSs), we provide in this paper a new idea, from the perspective of amount of knowledge, that the entropy for such information should be unique so that it may differ significantly from the entropy of a particular fuzzy element of which the membership and non-membership values are both equal to 0.5. Motivated by this idea, we present a more strict axiomatic definition of entropy on A-IFSs, and then develop a new entropy measure of A-IFSs that satisfies the refined axioms, with the aim of overcoming some drawbacks and ambiguities the existing entropy brings about and building a new entropy measurement model in the A-IFSs context to show some unique features of A-IFSs from the viewpoint of a specific purpose, notably those related to decision making. This allows us to capture the intrinsic amount of knowledge and the additional features that may be relevant when making decisions, and ultimately may be able to help us distinguish such particular cases in which the membership values are equal to the corresponding non-membership values in the A-IFSs context. Finally, a real-life example is provided for an in-depth discussion on the application of the developed models in decision making under uncertainty.

Neutrosophic Crisp Open Set and Neutrosophic Crisp Continuity via Neutrosophic Crisp Ideals

Abstract —The focus of this paper is to propose a new notion of neutrosophic crisp sets via neutrosophic crisp ideals and to study some basic operations and results in neutrosophic crisp topological spaces. Also, neutrosophic crisp L-openness and neutrosophic crisp L- continuity are considered as a generalizations for a crisp and fuzzy concepts. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Finally, we define and study two different types of neutrosophic crisp functions. Index Terms —Neutrosophic Crisp Set; Neutrosophic Crisp Ideals; Neutrosophic Crisp L-open Sets; Neutrosophic Crisp L- Continuity; Neutrosophic Sets. I. I NTRODUCTION The fuzzy set was introduced by Zadeh [20] in 1965, where each element had a degree of membership. In 1983 the intuitionstic fuzzy set was introduced by K. Atanassov [1, 2, 3] as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. Salama et al [11] defined intuitionistic fuzzy ideal and neutrosophic ideal for a set and generalized the concept of fuzzy ideal concepts, first initiated by Sarker [19]. Smarandache [16, 17, 18] defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. Neutrosophic sets have been investigated by Salama et al. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In this paper is to introduce and study some new neutrosophic crisp notions via neutrosophic crisp ideals. Also, neutrosophic crisp L-openness and neutrosophic crisp L- continuity are considered. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Recently, we define and study two different types of neutrosophic crisp functions. The paper unfolds as follows. The next section briefly introduces some definitions related to , implies neutrosophic set theory and some terminologies of neutrosophic crisp set and neutrosophic crisp ideal. Section 3 presents neutrosophic crisp L- open and neutrosophic crisp L- closed sets. Section 4 presents neutrosophic crisp L–continuous functions. Conclusions appear in the last section. II. P

Neutrosophic Crisp Open Set and Neutrosophic Crisp Continuity via Neutrosophic Crisp Ideals

The focus of this paper is to propose a new notion of neutrosophic crisp sets via neutrosophic crisp ideals and to study some basic operations and results in neutrosophic crisp topological spaces. Also, neutrosophic crisp L-openness and neutrosophic crisp Lcontinuity are considered as a generalizations for a crisp and fuzzy concepts. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Finally, we define and study two different types of neutrosophic crisp functions. Index Terms—Neutrosophic Crisp Set; Neutrosophic Crisp Ideals; Neutrosophic Crisp L-open Sets; Neutrosophic Crisp L- Continuity; Neutrosophic Sets. I. INTRODUCTION The fuzzy set was introduced by Zadeh [20] in 1965, where each element had a degree of membership. In 1983 the intuitionstic fuzzy set was introduced by K. Atanassov [1, 2, 3] as a generalization of fuzzy set, where besides the degree of membership and the degree of non- membership of each element. Salama et al [11] defined intuitionistic fuzzy ideal and neutrosophic ideal for a set and generalized the concept of fuzzy ideal concepts, first initiated by Sarker [19]. Smarandache [16, 17, 18] defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. Neutrosophic sets have been investigated by Salama et al. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In this paper is to introduce and study some new neutrosophic crisp notions via neutrosophic crisp ideals. Also, neutrosophic crisp L-openness and neutrosophic crisp L- continuity are considered. Relationships between the above new neutrosophic crisp notions and the other relevant classes are investigated. Recently, we define and study two different types of neutrosophic crisp functions. The paper unfolds as follows. The next section briefly introduces some definitions related to neutrosophic set theory and some terminologies of neutrosophic crisp set and neutrosophic crisp ideal. Section 3 presents neutrosophic crisp L- open and neutrosophic crisp Lclosed sets. Section 4 presents neutrosophic crisp L– continuous functions. Conclusions appear in the last section.

The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets

In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of λ-cut sets of IFS and show that λ-cut sets of IFS have the same properties as the cut sets of Zadeh fuzzy set. Thirdly, by using the concepts of λ-cut set of IFS and INS, we build up the decomposition theorem, representation theorem and extension principle of IFS. Finally, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup.

Some results for dual hesitant fuzzy sets

Dual hesitant fuzzy sets DHFSs, originally introduced by Zhu et al. [23], are a new tool to handle fuzziness considering the membership and nonmembership degrees by a set of possible values respectively. As a more comprehensive set, DHFSs have close relationship with the existing fuzzy sets, including the Zadeh's fuzzy set, Atanassov's intuitionistic fuzzy set, and hesitant fuzzy set, etc. And under certain conditions, DHFSs can also reduce to some existing fuzzy sets. However, sometimes the related operations of DHFSs do not have uniform expressions by considering different conditions. To better understand DHFSs, distinguish them from other fuzzy sets, and make further research of DHFSs, in this paper we develop typical DHFSs T-DHFSs, and present a good number of operations and properties, which shall provide the good theory foundation of DHFSs.

Several Similarity Measures of Neutrosophic Sets

Smarandache (1995) defined the notion of neutrosophic sets, which is a generalization of Zadeh's fuzzy set and Atanassov's intuitionistic fuzzy set. In this paper, we first develop some similarity measures of neutrosophic sets. We will present a method to calculate the distance between neutrosophic sets (NS) on the basis of the Hausdorff distance. Then we will use this distance to generate a new similarity measure to calculate the degree of similarity between NS. Finally we will prove some properties of the proposed similarity measures.

A Method to My Madness: What Counts as Innovation in Social Science?

This article summarizes a presentation made to an ESRC-sponsored workshop on social science methodological innovation. First, the notion of methodological innovation is explored on the way to an argument that innovation in social sciences occurs at a conceptual level rather than at the level of techniques for data gathering. Four historical examples are briefly described to exemplify the argument: Zadeh's fuzzy set theory, Goffman's presentation of self, Bartlett's schema, and Glaser and Strauss' grounded theory. In response to questions posed by the organizers, it is noted that the four examples were produced by individuals with a considerable time lag before uptake of their concepts in the diffusion curve. The article ends with speculation on the possible nature of a social science center for conceptual innovation.

Fully implicational methods for interval-valued fuzzy reasoning with multi-antecedent rules

Based on the fully implicational idea, we investigate the interval-valued fuzzy reasoning with multi-antecedent rules. First, we construct a class of interval-valued fuzzy implications by means of a type of implications and a parameter on the unit interval, then use them to establish three kinds of fully implicational reasoning methods for the interval-valued fuzzy reasoning with multi-antecedent rules which are called triple I method, hierarchical triple I method and URC triple I method respectively. At the same time, we analyze the Modus-Ponens (MP for short) and continuity properties of these methods and investigate the equivalence between the first method and the latter two methods. We also restrict our discussion to Zadeh fuzzy set theory and obtain a series of corresponding results.

Interval-valued Level Cut Sets of Fuzzy Set

The connections between Zadeh fuzzy set and three-valued fuzzy set are established in this paper. The concepts of interval-valued level cut sets on Zadeh fuzzy set are presented and new decomposition theorems and representation theorems of Zadeh fuzzy set are established based on new cut sets. Firstly, four interval-valued level cut sets on Zadeh fuzzy set are defined as three-valued fuzzy sets and it is shown that the interval-valued level cut sets of Zadeh fuzzy set are generalizations of normal cut sets on Zadeh fuzzy set, and have the same properties as those of normal cut sets of Zadeh fuzzy set. Secondly, the new decomposition theorems are established based on these new cut sets. It is pointed out that each kind of interval-valued level cut sets corresponds to two decomposition theorems. Thus eight decomposition theorems are obtained. Finally, the definitions of three-valued inverse order nested sets and three-valued order nested sets are presented with eight representation theorems based on new nested sets.

Multi-fuzzy Sets: An Extension of Fuzzy Sets

In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T, S, M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multi-fuzzy sets, a method for obtaining Atanassov's intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov's intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of the T and S operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh's fuzzy set theory, Atanassov's intuitionsitic fuzzy set theory and L-fuzzy set theory.