Classical Set Theory Research Articles

General vague rough approximation: an extended method of fuzzy knowledge representation

The classical rough set theory has been extended to fuzzy data environments by many authors, resulting in the development of the fuzzy rough set. Recently, vague set is treated as an extension of fuzzy set, but the existing theories and approaches of fuzzy rough set could not be applied directly to data set represented by vague set. In this article, we attempt to establish a theoretical model for vague data by combing both rough set and vague set. We first introduced the basic notions of vague t-norms and t-conorms. Next, we developed a general vague rough approximation set for generalizing to fuzzy rough set. Then, to overcome the inconvenience of dealing vague data, we also introduced a model for transforming a vague set into a fuzzy set to take the place of an interval vague set. Moreover, we also give some perspectives for future research.

ℵn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL

AbstractA module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

Multi-granulation rough sets based on tolerance relations

The original rough set model is primarily concerned with the approximations of sets described by a single equivalence relation on the universe. Some further investigations generalize the classical rough set model to rough set model based on a tolerance relation. From the granular computing point of view, the classical rough set theory is based on a single granulation. For some complicated issues, the classical rough set model was extended to multi-granulation rough set model (MGRS). This paper extends the single-granulation tolerance rough set model (SGTRS) to two types of multi-granulation tolerance rough set models (MGTRS). Some important properties of the two types of MGTRS are investigated. From the properties, it can be found that rough set model based on a single tolerance relation is a special instance of MGTRS. Moreover, the relationship and difference among SGTRS, the first type of MGTRS and the second type of MGTRS are discussed. Furthermore, several important measures are presented in two types of MGTRS, such as rough measure and quality of approximation. Several examples are considered to illustrate the two types of multi-granulation tolerance rough set models. The results from this research are both theoretically and practically meaningful for data reduction.

Incremental updating approximations in dominance-based rough sets approach under the variation of the attribute set

Dominance-based Rough Sets Approach (DRSA) is a generalized model of the classical Rough Sets Theory (RST) which may handle information with preference-ordered attribute domain. The attribute set in the information system may evolve over time. Approximations of DRSA used to induce decision rules need updating for knowledge discovery and other related tasks. We firstly introduce a kind of dominance matrix to calculate P-dominating sets and P-dominated sets in DRSA. Then we discuss the principles of updating P-dominating sets and P-dominated sets when some attributes are added into or deleted from the attribute set P. Furthermore, we propose incremental approaches and algorithms for updating approximations in DRSA. The proposed incremental approaches effectively reduce the computational time in comparison with the non-incremental approach are validated by experimental evaluations on different data sets from UCI.

Quasi-Polish spaces

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is Π20 in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework of Type-2 Theory of Effectivity as precisely the countably based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an ω-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every ω-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff–Kuratowski theorem generalizes to all quasi-Polish spaces.

A Comparative Analysis of Multigranular Approaches and on Topoligical Properties of Incomplete Pessimistic Multigranular Rough Fuzzy Sets

Rough sets, introduced by Pawlak as a model to capture impreciseness in data have been a very useful tool in several applications. These basic rough sets are defined by taking equivalence relations over a universe. In order to enhance the modeling powers of rough sets, several extensions to the basic definition has been introduced over the past few years. Extending the single granular structure of research in classical rough set theory two notions of Multigranular approaches; Optimistic Multigranulation and Pessimistic Multigranulation have been introduced so far. Topological properties of rough sets along with accuracy measures are two important features of rough sets from the application point of view. Topological properties of Optimistic Multigranular rough sets Optimistic Multigranular rough fuzzy sets and Pessimistic Multigranular rough sets have been studied. Incomplete information systems take care of missing values for items in data tables. Optimistic and pessimistic MGRS have also been extended to such type of incomplete information systems. In this paper we provide a comparative study of the two types of Multigranular approaches along with other related notions. Also, we extend the study to topological properties of incomplete pessimistic MGRFS. These results hold both for complete and incomplete information systems.

Rough set analysis of relational structures

In classical rough set theory, objects are partitioned into equivalence classes based on their attribute values, which essentially represent the functional information associated with the objects. Therefore, rough set theory can be viewed as a theory of functional granulation. In contrast, relational information systems (RISs) specify the relationships between objects, instead of their properties. This study presents a rough set analysis of relational structures, which are more general than functional information systems (FISs) and RISs. Unlike classical rough set theory, in which the attribute values of objects fully determine the indiscernibility relation, the rough set analysis of relational structures must account for the relationships between objects. This study considers three important concepts of indiscernibility with respect to relational structures: congruence, bisimulation, and exact equivalence. Using these indiscernibility relations, we investigate rough approximations and knowledge reduction. This study extends the application scope of rough set analysis from table-style information systems to relational structures. This extension is important because relational structures play a crucial role in the processing of complex data, such as graph mining or social network analysis.

Using a rough set model to extract rules in dominance-based interval-valued intuitionistic fuzzy information systems

Interval-valued intuitionistic fuzzy information systems are generalizations of conventional fuzzy-valued information systems. We introduce a dominance relation in the framework of interval-valued intuitionistic fuzzy information systems to come up with the concept we call a dominance-based interval-valued intuitionistic fuzzy information system (DIIFIS). This system is used to establish a dominance-based rough set model, which is grounded primarily on the substitution of the indiscernibility relation in the classic rough set theory with the aforementioned dominance-based relation. This relation is defined by the score and accuracy of interval-valued intuitionistic fuzzy value. To simplify knowledge representation and extract useful and simple dominance-based interval-valued intuitionistic fuzzy rules, we present two attribute reduction approaches to eliminating redundant information. To demonstrate the potential of these approaches, we apply them to computer auditing risk assessment, decision-making problems in wealth management, and pattern classification. Our findings confirm that the proposed rough set model is an effective means of extracting knowledge from dominance-based interval-valued intuitionistic fuzzy information systems.

Reliability analysis of repairable systems using Petri nets and vague Lambda-Tau methodology

The main objective of the paper is to developed a methodology, named as vague Lambda-Tau, for reliability analysis of repairable systems. Petri net tool is applied to represent the asynchronous and concurrent processing of the system instead of fault tree analysis. To enhance the relevance of the reliability study, vague set theory is used for representing the failure rate and repair times instead of classical(crisp) or fuzzy set theory because vague sets are characterized by a truth membership function and false membership functions (non-membership functions) so that sum of both values is less than 1. The proposed methodology involves qualitative modeling using PN and quantitative analysis using Lambda-Tau method of solution with the basic events represented by intuitionistic fuzzy numbers of triangular membership functions. Sensitivity analysis has also been performed and the effects on system MTBF are addressed. The methodology improves the shortcomings of the existing probabilistic approaches and gives a better understanding of the system behavior through its graphical representation. The washing unit of a paper mill situated in a northern part of India, producing approximately 200ton of paper per day, has been considered to demonstrate the proposed approach. The results may be helpful for the plant personnel for analyzing the systems' behavior and to improve their performance by adopting suitable maintenance strategies.

On Some Topological Properties of Pessimistic Multigranular Rough Sets

Rough set theory was introduced by Pawlak as a model to capture impreciseness in data and since then it has been established to be a very efficient tool for this purpose. The definition of basic rough sets depends upon a single equivalence relation defined on the universe or several equivalence relations taken one each at a time. There have been several extensions to the basic rough sets introduced since then in the literature. From the granular computing point of view, research in classical rough set theory is done by taking a single granulation. It has been extended to multigranular rough set (MGRS) model, where the set approximations are defined by taking multiple equivalence relations on the universe simultaneously. Multigranular rough sets are of two types; namely optimistic MGRS and pessimistic MGRS. Topological properties of rough sets introduced by Pawlak in terms of their types were studied by Tripathy and Mitra to find the types of the union, intersection and complement of such sets. Tripathy and Raghavan have extended the topological properties of basic single granular rough sets to the optimistic MGRS context. Incomplete information systems take care of missing values for items in data tables. MGRS has also been extended to such type of incomplete information systems. In this paper we have carried out the study of topological properties of pessimistic MGRS by finding out the types of the union, intersection and complement of such sets. Also, we have provided proofs and examples to illustrate that the multiple entries in the table can actually occur in practice. Our results hold for both complete and incomplete information systems. The multiple entries in the tables occur due to impreciseness and ambiguity in the information. This is very common in many of the real life situations and needed to be addressed to handle such situations in efficient manner.

Model of Covering Rough Sets in the Machine Intelligence

Rough set theory has been proposed by Pawlak as a useful tool for dealing with the vagueness and granularity in information systems. Classical rough set theory is based on equivalence relation. The covering rough sets are an improvement of Pawlak rough set to deal with complex practical problems which the latter one can not handle. This paper studies covering-based generalized rough sets. In this setting, we investigate common properties of classical lower and upper approximation operations hold for the covering-based lower and upper approximation operations and relationships among some type of covering rough sets.

Polish Set Theorists and Large Cardinals

Polish Set Theorists and Large Cardinals

Jourdain, Russell and the Axiom of Choice: A New Document

June 25, 2012 (9:21 pm) E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1 060 red.wpd 1 Jourdain’s principal historical writings are gathered together in Selected Essays on the History of Set Theory and Logics (1906–1918), ed. I. Grattan-Guinness (Bologna: clueb, 1991) and his introduction to G. Cantor, Contributions to the Founding of the Theory of TransWnite Numbers, trans. and ed. Jourdain (La Salle, Ill.: Open Court, 1915; repr. New York: Dover, 1955). 2 I. Grattan-Guinness, Dear Russellz—zDear Jourdain: a Commentary on Russelly’s Logic, Based on His Correspondence with Philip Jourdain (London: Duckworth; New York: Columbia U. P., 1977); hereafter “RJy”. russell: the Journal of Bertrand Russell Studies n.s. 32 (summer 2012): 69–74 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 ocuments JOURDAIN, RUSSELL AND THE AXIOM OF CHOICE: A NEW DOCUMENT I. Grattan-Guinness Middlesex U. Business School Hendon, London nw4 4bt, uk Centre for Philosophy of Natural and Social Science, l.s.e. London wc2a 2ae, uk ivor2@mdx.ac.uk i.wcareer P hilip Edward Bertrand Jourdain (1879–1919) went up to Trinity College Cambridge in 1899 with a scholarship in mathematics. Two years later he attended a special College course in mathematical logic oTered by Bertrand Russell, the Wrst of its kind ever taught in a British university. This contact with Russell was of major consequence for his intellectual career, for he focused upon set theory and its history, and also on the histories of mathematical analysis and of logic, and aspects of the history and philosophy of mechanics and of science.1 He corresponded at length with Russell until his death in 1919; the exchange forms the core of my book on their relationship.2 From his youth Jourdain suTered from a creeping paralysis called “Friedreich ’s ataxia”. It prevented him from taking the Part 2 Tripos; thus he did not graduate as a Wrangler and so could not compete for a fellowship at Trinity. So he made his living from a few scholarships and from freelance writing as an June 25, 2012 (9:21 pm) E:\CPBR\RUSSJOUR\TYPE3201\russell 32,1 060 red.wpd 70 i. grattan-guinness 3 E. Zermelo, “Beweis, dass jede Menge wohlgeordnet werden kann”, Mathematische Annalen 59 (1904): 514–16. 4 See especially H. Rubin and J.yE Rubin, Equivalents of the Axiom of Choice, 2nd ed. (Amsterdam: North-Holland, 1985). 5 The history of this/these axioms has been charted in much detail. See especially J. Cassinet and M. Guillemot, “L’axiome du choix dans les mathématiques de Cauchy (1821) à Gödel (1940)”, 2 vols. (U. of Toulouse double docteur d’état des sciences, 1983); F.yA. MedvedeT, Rannyaya istoriya aksiomi vibora (Moscow: Nauka, 1982); and G.yH. Moore, Zermelo’s Axiom of Choice … (New York: Springer, 1982). RJ discusses the axioms in the contexts of the correspondence. independent scholar. In particular, the American house The Open Court Publishing Company employed him, especially from 1912 on their principal journal The Monist, commissioning papers (and also some books) from colleagues, in particular Russell. He seems to have served as its general editor soon after the death in February 1919 of its founder editor, Paul Carus, but he died himself early in October. Those last months form the time-frame of this paper. 2.wobsession Jourdain’s principal research interest in the foundations of mathematics came to focus upon the denumerable “axiom of choice” in set theory and mathematics , as Ernst Zermelo named it a few years after introducing it in 1904.3 Take an inWnite ensemble of non-empty and pairwise disjoint sets and choose a member from each set in independent actions; then the axiom asserts that the collection of chosen members can always be regarded as a genuine set. But this manner of forming a set roused considerable doubts and opposition among many (though not all) set-theorists and logicians. Some of them rejected the axiom altogether; others tried to reprove theorems that used the axiom by proofs that avoided it; several sought hopefully more congenial assumptions that were logically equivalent to the axiom, and eventually many came to light.4 Logicists such...

A Logical Framework for Set Theories

Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for Mathematical Knowledge Management. However, in order to be used for this task it is necessary to overcome serious gaps that exist between the "official" formulations of set theory (as given e.g. by formal set theory ZF) and actual mathematical practice. In this work we present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity.

The Application of Mechanical Failure Diagnosis Expert System for the Improvement of Vibration Fault Monitor and Diagnosis Based on Rough Set Theory

Focuses on rough set theory and the genetic algorithms and theories on their integration mechanism based on the study of the failure diagnosis on gas turbine-set, A new algorithm were put forward in the discretization of continuous attribute, attribute reduction and value reduction methods based on the classical set theory and rough set theory and solved the difficult problem which expert system to obtain the rules in the paper. Knowledge system attribute table is regard as a major tool to reduce the rules of expert system in which unnecessary attributes are eliminated. The redundancy of failure diagnosis information and the complexity of structure in failure diagnosis expert system are revealed. It brings forth the reliable and practical method, which have further enriched and improved the theories on the vibration fault monitor and diagnosis of a large gas turbine-set by testing and verifying

Generalized approximations defined by non-equivalence relations

The classical rough set theory, based on the conventional indiscernibility relation, is not much useful for analyzing incomplete information. Some successful generalized rough set models based on different non-equivalence relations have been proposed. Nowadays the generalized definitions of approximations have become one of the important research issues of the generalized rough set models. Although some different definitions of approximations have been presented, the relationships among them have not been analyzed adequately. In this paper, we investigate the relationships among 12 different basic definitions of approximations and suggest the suitable generalized definitions of approximations for each class of generalized indiscernibility relations. We also review eight classes of generalized indiscernibility relations and 12 different basic definitions of approximations as our foundations for discussion. Three examples are given to illustrate the relationships among the different generalized definitions of approximations.

On the Need for Fractal Logic in High Energy Quantum Physics

Modern advances in pure mathematics and particularly in transfinite set theory have introduced into the fundamentals of theoretical physics many novel concepts and devices such as fractal quasi manifolds with non-integer (Hausdorff) dimension for its geometry as well as infinite dimensional wild topology and non classical fuzzy logic. In the present work transfinite fractal sets and fuzzy logic are combined to enable the introduction of a new theory termed fractal logic to the foundation of high energy particle physics. This leads naturally to a new look at quantum gravity. In particular we will show that to understand and develop quantum gravity we have to bring various fields together, particularly fractals and nonlinear dynamics as well as sphere packing, fuzzy set theory, number theory and quantum entanglement and irrationally q-deformed algebra.

More Parts Than Elements: How Databases Multiply

Databases organise, configure, and perform thing-and-people multiples in sets. Belonging, inclusion, participation, and membership: many of the relations that make up the material-social life of people and things can be formally apprehended in terms of set-like multiples stored in databases. Mid-20th century database design derived different ways of gathering, sorting, ordering, and searching data from mathematical set theory. The dominant database architecture, the relational database management system, can be seen as a specific technological enactment of the mathematics of set theory. More recent developments such as grids, clouds, and other data-intensive architectures apprehend ever greater quantities of data. Arguably, in emerging data architectures databases themselves are subsumed by even more intensive set-like operations. Whole databases undergo set-like transformations as they are aggregated, divided, filtered, and sorted. At least at a theoretical level, the mathematics of set theory, as philosophically rendered by Alain Badiou, can also suggest some explanations of how multiples expand, ramify, and split in databases. Badiou's account locates forms of instability or incoherence inherent in any set-based doing of multiples in the relation between parts and elements, between inclusion and belonging. Against the grain of Badiou's resolutely philosophical project, a set-theoretical account of databases might also point towards some vertiginous elements that elude regulation, norms, and representation.

A New Time-Invariant Fuzzy Time Series Forecasting Method Based on Genetic Algorithm

In recent years, many fuzzy time series methods have been proposed in the literature. Some of these methods use the classical fuzzy set theory, which needs complex matricial operations in fuzzy time series methods. Because of this problem, many studies in the literature use fuzzy group relationship tables. Since the fuzzy relationship tables use order of fuzzy sets, the membership functions of fuzzy sets have not been taken into consideration. In this study, a new method that employs membership functions of fuzzy sets is proposed. The new method determines elements of fuzzy relation matrix based on genetic algorithms. The proposed method uses first-order fuzzy time series forecasting model, and it is applied to the several data sets. As a result of implementation, it is obtained that the proposed method outperforms some methods in the literature.

FUZZY SETS APPLICATION TO HEALTHCARE SYSTEMS

The fuzzy set concept was introduced by LotfiA. Zadeh in 1965 to provide tool to describe vague predicates or classes with imprecise boundaries and which enable uncertainty nonadditive calculation, based on softoperators.Zadeh recognises that conventional quantitative techniques of systems analysis are intrinsically inappropriate to address humanistic systems. The medical field is appropriate to apply the fuzzy set theory, as vague predicates and subjectivity are important part of it.In this paper, the fuzzy set concept is used to provide more flexible definition of certain parameters which state individual's health condition, and which will allow to formulate diagnosis model for the risk of suffering certain disease, and it is characterized as case of cardiovascular risk. This model is applied to the development of expansion strategy for private health insurance company, which considers sustainability of the healthcare system.Keywords: fuzzy sets, health parameters, healthcare systemsJEL Classification: I19, P46, C65(ProQuest: ... denotes formulae omitted.)1. INTRODUCTIONLotfiA. Zadeh (1965) introduced the fuzzy set concept to promote tool which aim is to describe vague predicates or classes with imprecise boundaries and which enables uncertainty non-additive calculation, based on softoperators.A predicate is adjective that qualifies certain known objects. It linguistically formulates certain object's property.A crisp predicate is accurate property of the elements of set that may or may not check, for example, even number. It allows to classify discourse objects into two crisp sets and it does not accept linguistic modifiers.A vague predicate is inaccurate property of set of elements that admits different degrees of truth. For example, the high predicate may be used in different domains, an item's price is high, Alice's blood pressure is high, and it allows for linguistic modifiers, as in a product's performance is very high, an individual's cholesterol is too high, or George's weight is quite high. Given that the properties expressed as vague predicates are not only true or false, but admit different degrees of truth, they do not classify discourse objects into two sets (Trillas, 1998).Zadeh extended the classical sets theory to be able to operate with classes defined by vague predicates, generalizing set membership concept, for which there were only two possibilities: It belongs or it does not belong. Membership has ceased being dichotomic to become graded. Objects belong to several classes with different membership degrees in each of them.Zadeh's calculation enables way of representing and managing reasoning with vague predicates and it has crisp predicate calculations, as particular case, which defines classical sets. Crispness or accuracy appear as fuzziness or inaccuracy boundary case (Trillas, 1998).Zadeh recognises that conventional quantitative techniques of systems analysis are intrinsically inappropriate to address humanistic systems.Fuzzy models have turned into appropriate tool to treat uncertainty in different fields. One of them is the medical field, as vague predicates and subjectivity are important part of it (Mordeson et al., 2000).An issue posed is assessing the admissible variability range of certain parameters which indicate individual's health condition. Basic health parameters are simple, accessible and good indicators of the functioning of the body.Some of them are:* Weight and volume parameters: BMI (Body Mass Index) and WHP (Waist to Hip Ratio).* Systolic blood pressure (maximum) and diastolic blood pressure (minimum).* Heart rate.* Glucose.* Total cholesterol, LDL (bad) cholesterol, HDL (good) cholesterol and triglycerides.* Respiratory parameters: VC (Vital Capacity) and FEV1 (Forced Expiratory Volume in 1 second). …