Classical Set Theory Research Articles

Fuzzy-rough set models and fuzzy-rough data reduction

Rough set theory is a powerful tool to analysis the information systems. Fuzzy rough set is introduced as a fuzzy generalization of rough sets. This paper reviewed the most important contributions to the rough set theory, fuzzy rough set theory and their applications. In many real world situations, some of the attribute values for an object may be in the set-valued form. In this paper, to handle this problem, we present a more general approach to the fuzzification of rough sets. Specially, we define a broad family of fuzzy rough sets. This paper presents a new development for the rough set theory by incorporating the classical rough set theory and the interval-valued fuzzy sets. The proposed methods are illustrated by an numerical example on the real case.

Topological vacuum fluctuation and Dvoretzky's theorem – Mathematical proofs in the context of the dark energy density of the universe

Starting from the initial triality of physics, namely mathematical philosophy, transfinite set theory and number theory we drive the inevitability of a topological quantum vacuum fluctuation of spacetime resulting in the fundamental reality of pair creation and annihilation. Subsequently we give a simple but strong mathematical proof of Dvoretzky‘s marvellous theorem on measure concentration, thus making dark energy and accelerated cosmic expansion not only an astrophysical measurement and observational reality, but also a plausible topological-geometrical fact of a pointless Cantorian actual universe akin to the Penrose fractal tiling space. This space is described accurately via the von Neumann-Conne noncommutative geometry using their golden mean dimensional function and the corresponding bijection of E-infinity theory. The said theory was developed by the authors of the present paper and their group and is based on and starts from the pioneering efforts of the Canadian physicist G. Ord and the French astrophysicist L. Nottale.

Platonic Golden Network Theory for High Energy Physics and Quantum Cosmology

In the spirit of Pythagorean cosmic mathematical music of numbers and following the fundamental tenets of platonic transfinite set theory and the methodology of E-infinity Cantorian spacetime, we present a golden section based network with deep physical meaning confirming experimental results in high energy physics as well as cosmological measurements and observations related to the phenomenon of cosmic dark energy and dark matter.

A new approach to the degree of greyness

PurposeThe degree of greyness may be regarded as a measure of cognitive uncertainty. Therefore, it is a part of the epistemological core of the grey systems theory. The theoretical importance of the degree of greyness concept is also due to its application in a range of uncertainty modelling methods: predictive, relational and decision-making methods. Greyness, being a result of cognitive uncertainty, was recently subjected to axiomatization in the form of grey space with the use of the classical sets theory. The purpose of this article is to develop a new approach to the degree of greyness, being consistent with the grey space concept.Design/methodology/approachIn order to realise the article’s goals, the research is divided into three stages described in particular sections. The first section of the article presents a theoretical framework of the degree of greyness and the grey space. The second part includes the assumptions of the new degree of greyness concept, along with the mathematical models for the first, the second and the third degree of greyness. The third section contains numerical examples for each degree of greyness.FindingsAs a result of the research, a concept of a degree of greyness was created and it was linked with a concept of grey space. This new approach to the issue of the degree of greyness has allowed the analysing of this category in three dimensions dependent on an accepted reference base. As a result, a concept of concrete and abstractive grey numbers was introduced and relationships between these categories of numbers and the degree of greyness were determined.Originality/valueThe proposed approach to the issue of the degree of greyness is a theoretical unification of the previous considerations in this area. The proposed three dimensions of greyness degree will be derived from the grey space, so they will also be a function of quantity. Thus, the degree of greyness was linked with a classical set theory. An original input in this article is also a differentiation of concrete and abstractive grey numbers, which give a basis for deliberations connected with interpretation of grey numbers in the context of real applications.

Does Choice Really Imply Excluded Middle? Part I: Regimentation of the Goodman–Myhill Result, and Its Immediate Reception†

ABSTRACTThe one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general ($a\!=\!b\vee\neg a\!=\!b$) is derived; then, of sentences in general ($\psi\vee\neg\psi$). Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the constructive set theorist with a dispositive reason for not adopting (full) Choice.

Data meaning and knowledge discovery: Semantical aspects of information systems

Data tables provide the standard means of representation of qualitative or quantitative information about objects of interest. They also form a starting point for the task of information processing: an operation of passing from raw data or information to semantically processed knowledge. The fundamental issue here is the question about the meaning of data: What do entries in the data table actually tell us about objects? It entails another question: How should the meaning be further processed? The primary aim of the article is an attempt to answer these two questions. To this end we are going to employ conceptual scales from formal concept analysis, an important theory of data processing introduced by Rudolf Wille, and apply them to data tables so as to obtain multivalued information systems, which were introduced and developed within the conceptual framework of rough set theory by Zdzisław Pawlak and Ewa Orłowska. Our main idea is to regard multivalued information systems as the semantics or meaning of the original tables. This idea allows us to describe and combine classical rough set theory, dominance-based rough set approach, and formal concept analysis within the single framework of multivalued information systems, which is rich enough to cope with a number of semantical nuances that may occur during the process of data analysis.

ВОЗМОЖНОСТИ СИСТЕМЫ КОМПЬЮТЕРНОЙ АЛГЕБРЫ MAPLE ДЛЯ РЕШЕНИЯ ЗАДАЧ ПО ТЕОРИИ МНОЖЕСТВ

Abstract. Introduction. The paper substantiates the relevance of studying set theory in mathematics and computer science lessons using modern technologies. The research aims to shed light on the capabilities of the computer algebra system Maple for specifying sets in various ways, demonstrating the validity of the laws (properties) of sets, and solving set theory word problems. Materials and methods. The methods applied in the research include analysis of school educational literature with a focus on identifying the types of tasks on set theory, analysis of methodological literature and the experience of teachers in teaching set theory, analysis of the functional and software capabilities of Maple, and synthesis of Maple capabilities to perform set theory assignments. Results. The paper has described possible directions of Maple application in teaching set theory. The examples of using the computer algebra system to specify sets, and demonstrate the validity of their laws (properties) are presented. The possibilities of visualizing educational material in Maple are illustrated by the example of solving set theory word problems of various degrees of complexity. Conclusion. The results of the study on the use of Maple in teaching set-theoretic transformations give an idea of ways to improve the learning process by visualizing educational material, getting time free from routine calculations, and imparting the study a research character. Keywords: computer algebra systems, Maple, set theory, membership tables.

Rough set models of interval rough number information system

Classical rough set theory (RST) is based on equivalence relations, and does not have an effective mechanism when the attribute value of the objects is uncertain information. However, the information in actual problems is often uncertain, and an accurate or too vague description of the information can no longer fully meet the actual needs. Interval rough number (IRN) can reflect a certain degree of certainty in the uncertainty of the data when describing the uncertainty of the data, and can enable decision makers to make decisions more in line with actual needs according to their risk preferences. However, the current research on rough set models (RSMs) whose attribute values are interval rough numbers is still very scarce, and they cannot analyze the interval rough number information system (IRNIS) from the perspective of similar relation. therefore, three new interval rough number rough set models (IRNRSMs) based on similar relation are proposed in this paper. Firstly, aiming at the limitations of the existing interval similarity degree (ISD), new interval similarity degree and interval rough number similarity degree (IRNSD) are proposed, and their properties are discussed. Secondly, in the IRNIS, based on the newly proposed IRNSD, three IRNRSMs based on similar class, β-maximal consistent class and β-equivalent class are proposed, and their properties are discussed. And then, the relationships between these three IRNRSMs and those between their corresponding approximation accuracies are researched. Finally, it can be found that the IRNRSM based on the β-equivalent classes has the highest approximation accuracy. Proposing new IRNRSMs based on similar relation is a meaningful contribution to extending the application range of RST.

Effects of think-pair-share on senior school students’ performance in mathematics in Ilorin, Nigeria

The performance of Nigerian students in the Senior School Certificate Mathematics Examinations over the years has not been encouraging. Studies have indicated that this is partly due to students’ lack of in-depth knowledge of some selected topics in mathematics. Pieces of evidence abound in literature and WAEC Chief examiner’s reports indicating that students performed poorly in set theory with Venn diagram. Therefore, this study examined the effects of think-pair-share on senior school students’ performance in mathematics in Ilorin, Nigeria. The objectives of this study were to examine: (i) the performance of students’ taught set theory using think-pair-share cooperative instructional strategy; (ii) the difference in the performance of students taught using think-pair-share instructional strategy based on gender and score level; (iii) the influence of think-pair-share instructional strategy on students’ retention in set theory. The researcher employed a quasi-experimental design for the study. The sample consisted of 118 SS II students. The instruments used for the study was Mathematics Performance Test (MPT). The instruments yielded reliability values of 0.78 using Pearson Product Moment Correlation procedure. The data collected were analyzed using independent sample t-test, and Analysis of Covariance (ANCOVA). Findings of the study showed: (i) a statistically significant difference in the performance of students taught set theory using think-pair-share compared with their counterparts in the control group in favour of think-pair-share group, (ii) no statistically significant difference in the performance of male and female students taught set theory in Mathematics using think-pair-share; (iii) no statistically significant difference in the performance of students taught set theory in Mathematics using think-pair-share based on scoring levels; and (iv) a statistically significant difference in the knowledge retained by students taught set theory in Mathematics using think-pair-share compared with their counterparts in the control group in favour of think-pair-share group. The study concluded that the use of think-pair-share improved students’ performance in Mathematics, gender of a student does not affect his or her performance in Mathematics, and the use of think-pair-share improved the retention ability of the students. It was recommended among others that the use of think-pair-share be encouraged for teaching and learning of Mathematics at all levels of education.

M-Polar Neutrosophic Topology with Applications to Multi-criteria Decision-Making in Medical Diagnosis and Clustering Analysis

In this paper, we first introduce novel concepts of m-polar neutrosophic set (MPNS) and topological structure on m-polar neutrosophic set by combining the m-polar fuzzy set (MPFS) and neutrosophic set. Then, we investigate several characterizations of m-polar neutrosophic set and establish its various operations with the help of examples. We propose score functions for the comparison of m-polar neutrosophic numbers (MPNNs). We establish m-polar neutrosophic topology and define interior, closure, exterior, and frontier for m-polar neutrosophic sets (MPNSs) with illustrative examples. We discuss some results with counter examples, which hold for classical set theory, but do not hold for m-polar neutrosophic set theory. We introduce a cosine similarity measure and a set theoretic similarity measure for m-polar neutrosophic sets (MPNSs). Furthermore, we present three algorithms for multi-criteria decision-making (MCDM) in medical diagnosis and clustering analysis under uncertainty by using m-polar neutrosophic sets (MPNSs) and m-polar neutrosophic topology. Lastly, we present advantages, validity, flexibility, and comparison of our proposed algorithms with the existing techniques.

A novel learning cloud Bayesian network for risk measurement

Bayesian network (BN) is a popularly used approach for risk analysis. Because it is a graphic model being able to deal with randomness yet unable to model ambiguity, the fuzzy set theory is often combined with it to create a so-called fuzzy BN. Instead of using the classical fuzzy set theory, this paper intends to combine a normal Cloud model with the BN. In the normal Cloud model, an element belonging to a certain qualitative concept is not certain and precise as well. The Cloud BN is a generalization of the fuzzy BN. It is more adaptive for the uncertainty description of linguistic concepts, for example, the risks. Using the normal Cloud model, the following numerical characteristics of the variables can be estimated: the expectation, the dispersion degree compared with the expectation, and the dispersion degree of entropy. Consequently, the risk assessment contains a richer set of analytical information. Cloud BNs attract growing research interests. Compared to its precedents, the Cloud BN in this paper has a learning capability. Since the risk factors may have a combined effect, the causal relationships among the variables can be very complex, and hidden variables may exist. The learning mechanism allows for automatic structure discovery from data, giving rise to a dynamically evolving network. The proposed learning Cloud BN is able to represent the real risk situation better than its precedents. Its effectiveness and applicability are demonstrated by an illustrative case for risk prediction of the face instability in an underground tunnel construction project.

Fuzzy Rough Attribute Reduction for Categorical Data

Classical rough set theory is considered a useful tool for dealing with the uncertainty of categorical data. The major deficiency of this model is that the classical rough set model is sensitive to noise in classification learning due to the stringent condition of equivalence relation. Thus, a class of fuzzy similarity relations was introduced to describe the similarity between samples with categorical attributes. However, these kinds of similarity relations also have deficiencies when they are used in fuzzy rough computation. In this article, we propose a new fuzzy-rough-set model for categorical data by introducing a variable parameter to control the similarity of samples. This model employs the iterative computation strategy to define fuzzy rough approximations and dependence functions. It is proved that the proposed rough dependence function is monotonic. Finally, the proposed model is applied to the attribute reduction of categorical data. The experimental results indicate that the proposed model is more effective for categorical data than some existing algorithms.

AN Application of Neutrosophic Vague Max-Min Composition Technique

The theory of fuzzy set was introduced by Zadeh, has a wider scope in terms of applications than the classical set theory in solving various problems. Extensive applications of the fuzzy set theory have been found in various fields such as Computer Sciences, Artificial Intelligence, Medical Sciences, Economics, Statistics, Neural Networks, etc. In fuzzy set each element had a degree of membership. Smarandache defined the neutrosophic set theory, which was one of the most important new mathematical tools for handling problems involving imprecise, indeterminacy, and inconsistent data. The theory of vague sets was first proposed by Gau and Buehrer as an extension of fuzzy set theory and vague sets are regarded as a special case of context-dependent fuzzy sets. Shawkat Alkhazaleh introduced the concept of neutrosophic vague set as a combination of neutrosophic set and vague set. Neutrosophic vague theory is an effective tool to process incomplete, indeterminate and inconsistent information. This paper attempts to develop a new model known as neutrosophic vague max-min composition technique to examine the daily life problems. One of the major problems that everyone comes across is their children’s studies. Education is a very important aspect of the lives of all people all over the world. What we learn, not just in the classroom, shapes who we are. We take our education everywhere we go. Student life is not as easier as it seems. Now a days the studies of the student is deviated due to various reasons. Here we identify those problems of students and by using neutrosophic vague max-min composition technique we analyze the problem faced by the students that affect their lack of concentration in studies.

Finite Dimensional Imprecise Numbers

We discuss difference between the definition of fuzzy number and imprecise number in term of complement. Later using the definition of imprecise number we study the multi-dimensional imprecise numbers. Various classical set theory properties under the intersection and unions are discussed in terms of multi-dimensional imprecise numbers.

Introducing fuzzy set theory to evaluate risk of misclassification of land cover maps to land mapping applications: Testing on coastal watersheds

In land cover maps, categories represent a continuum of variation and for this reason, fuzzy set theory, which accepts degrees of membership, has been suggested for land classification. Nevertheless, classical set theory, which only assumes single map categories, is still widely used. The purpose of this study is to develop a methodology to reduce the weakness of land cover maps in which classical theory has been applied. To do so, we propose adding an error relevance step after accuracy assessment, which evaluates how relevant are the classification errors to selected land applications. First, a membership matrix is built based on a linguistic scale associated to land cover rates obtained from literature. Then, two fuzzy measures are calculated and the frequency of categories, that do not pose a problem to the user in light of the land application, is determined. The methodology is demonstrated using two Brazilian tropical coastal regions and two land applications relevant for coastal watershed management. The study presents land cover maps of the Mamanguape and the Paraíba estuarine regions, their full accuracy assessment, and the relevance of the classification errors to the land applications.The accuracy assessment step has demonstrated that the land cover maps are reliable. The error relevance step has shown that the map weakness can be reduced. Both steps show that the land cover maps produced are suitable for further land mapping applications. The results on land cover composition point to the importance of future work focused on the environmental sustainability of the studied regions. The new procedure has proven useful to decrease the degree of distrust with which land cover maps are regarded. The framework provided is suitable for virtually any land mapping application.

Intuitionistic fuzzy $$\beta $$-covering-based rough sets

Covering-based rough set is an important extended type of classical rough set model. In this model, concepts are approximated through substitution of a partition in classical rough set theory with a covering in covering-based rough set theory. Various generalized covering-based rough sets have been investigated, however, little work has been done on extending four classical covering-based rough set to intuitionistic fuzzy (IF) settings. In this study, four novel IF covering-based rough set models are developed by combining an IF $$\beta $$-covering with four classical covering-based rough set models. First, we present the concept of IF $$\beta $$-minimal description, and then construct four order relations on IF $$\beta $$ approximation space. Second, we propose four IF $$\beta $$-covering-based rough set models and derive that they are generalizations of four existing covering-based rough sets in IF settings. We also discuss the properties of these IF $$\beta $$-covering-based rough sets and reveal their relationships. We use the existing distance between two IF sets to characterize the uncertainty of the presented IF $$\beta $$-covering-based rough sets. Third, we define the reducts of IF $$\beta $$-covering decision systems and examine their discernibility-function-based reduction methods for these IF $$\beta $$-covering-based rough sets. Fourth, we present four optimistic and pessimistic multi-granulation IF $$\beta $$-covering-based rough sets and analyze their properties and uncertainty measures from multi-granulation perspective. Fifth, we study the discernibility-function-based reduction methods for the presented multi-granulation IF $$\beta $$-covering-based rough sets. Finally, we discuss another two neighborhood-based IF covering-based rough sets. This study can provide a covering-based rough set method for acquiring knowledge from IF decision systems.

The Noble Family as “Singular Multiplicity”? Redefining the Smoczynski–Zarycki’s Totemic Definition of Nobility through the Lenses of Alain Badiou’s Mathematical Ontology

AbstractIn our paper, we redefine the category of “family” denoting the relationship of selected members of a post-noble/post-aristocratic milieu in Poland using Alain Badiou’s terminology. Badiou’s ontology based on a mathematical set theory and a generic theory is the most developed, complex, and revolutionary ontology of the 20thand 21stcenturies. However, it is rarely adapted to new empirical studies probably because of its novelty and complexity. We do not intend to use the empirical case study made by Smoczynski–Zarycki to inform our argument but instead perform a translation of the Durkheim–Lacanian theoretical standpoint from “Totem…” into the category of “singularity” [singularité] in its relation to “the state of situation” [état de la situation] from “Being and Event” (Badiou 2005). This approach seeks to find a universalizing potential of nobility that will allow it to become a relevant subject for truth procedure analysis.

On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions

Intuitionistic fuzzy set theory is a generalized version of classical fuzzy set theory, where membership degree of acceptance and membership degree of rejection are both measured. In the present study, we formulate a balanced transportation problem with multiple objectives having all parameters and variables as intuitionistic fuzzy numbers. The problem is first reduced to a crisp multiobjective transportation problem using accuracy function on each objective function and then an algorithm is proposed to solve the problem. In the solution procedure, linear, exponential and hyperbolic membership functions are used to tackle intuitionistic fuzzy constraints related with each objective. To show the relations among the intuitionistic fuzzy transportation problem, its equivalent crisp formulation and the problems obtained by applying different membership functions, various theorems are established. Finally, two numerical examples are illustrated to clarify the steps involved in the proposed algorithm and to draw the comparison among linear, exponential and hyperbolic membership functions.

Can Quantum Mechanics Correlate All Natural Forces — An Experimental and Observational Approach

According to a previous study, mathematical set theory in analogous terms of symbols can be used to relate the material, body, and spiritual world to develop artificial intelligence. This paper tries to study how the natural forces — electromagnetic, gravitational, and strong–weak — can be unified by applying quantum mechanics theory. Indeed, Strong–weak forces correspond to materials (the basis of element formation), electromagnetic force refers to the body (human cells generate electricity for mankind activities), and the gravitational force corresponds to the spiritual one (referring to the nature and universe). In this paper, different quantum experiments were performed to verify the above hypotheses. For the electromagnetic, and the Casimir effects were considered. A tabletop experiment was used for the gravitational force. Finally, for the strong–weak forces, then the Baryons mass spectrum was considered. The results show that the “set symbols connection” mentioned in the special rationalisation of Penrose’s philosophy is not only a theory but can also be tested experimentally w.r.t. quantum mechanics. Thus, the implementation of real machine reasonings becomes feasible. In other words, the quantum theories — quantum chromodynamics and electrodynamics together with the quantum gravity can be sublimed into another form of Roger’s three world philosophy. Hence, the QCD, ED and QG give birth to the unified field theory. This is because these theories unify all natural forces and relate the elementary particles in a single theoretical framework — the Roger’s theory through the quantum mechanics behind. The outcome is one may investigate more about early universe and black hole.

Soft dominance based rough sets with applications in information systems

Classical rough set theory is considered as a useful mathematical tool to deal with the uncertainty of categorical data; a major deficiency of this model is its sensitivity to noise in classification learning due to the strong condition of equivalence relation. Meanwhile decision making can be considered as a mental process in which human beings make a choice among several alternatives. However, with the increasing complexity of real decision making problems, the decision makers frequently face the challenge of characterizing their preferences in an uncertain context. In the present paper, we initiate multi-attribute group decision making problems in the presence of multi-attribute and multi-decision decision making with preferences. We present the concept of soft preference relation and soft dominance relation corresponding to a decision attribute in multi-criteria and multi-decision information system. Further we present the idea of soft dominance rough sets and their applications in solving a multi-agent conflict analysis decision problem. The proposed method addresses the limitations of the Pawlak model and Sun's conflict analysis model and thus improve these models. Finally, the experimental results on labor management negotiation problems show that the proposed algorithms are more effective and efficient for feasible consensus strategy when compared with Sun's technique.