Basic Set Theory Research Articles

The Algebraic Structures of Q-Complex Neutrosophic Soft Sets Associated with Groups and Subgroups

Groups and subgroups are rich algebraic structures, and both of them depend on binary operations in their work. The discussion of this paper is organized into two parts. In the first part, we define the notion of Qcomplex neutrosophic soft sets (Q-CNSSs) by amalgamating two previous models of Q-complex neutrosophic set (Q-CNS) and soft set (SS) to address the issues of two-dimensionality (two variables) in a universal set under a parametric environment. Subsequently, the relation between Q-CNSSs and Q- neutrosophic soft sets (Q-NSSs) is verified. A basic set theory for this hybrid model is developed. In particular, null Q-CNSS and absolute Q-CNSS are defined. The basic operators of the complement, subset, equality, union and intersection are advanced and their properties are examined. Further, the notions of the homogeneous and completely homogeneous Q-CNSSs are proposed along with some illustrated examples. In part two, we move to study some algebraic structures of this model when we define the notions of Q-complex neutrosophic soft groups (QCNSG) and Q-complex neutrosophic soft subgroups (Q-CNSSG). Then, the relation between Q-CNSG and Q-neutrosophic soft group (Q-NSG) is scrutinized. Moreover, the algebraic properties of the Q-CNSG and Q-CNSSG are discussed and verified. Finally, some theories that show the relationship between the Q-CNSG and the soft group are proposed.

A Novel Categorical Approach to Semantics of Relational First-Order Logic

We present a categorical formalization of a variant of first-order logic. Unlike other texts on this topic, the goal of this paper is to give a very transparent and self-contained account without requiring more background than basic logic and set theory. Our focus is to show how the semantics of first-order formulas can be derived from their usual deduction rules. For understanding the core ideas, it is not necessary to investigate the internal term structure of atomic formulas, thus we abstract atomic formulas to (syntactically opaque) relations; in this sense, our variant of first-order logic is “relational”. While the derived semantics is based on categorical principles (even the duality that arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other), it is nevertheless “constructive” in that it describes explicit computations of the truth values of formulas. We demonstrate this by modeling the categorical semantics in the RISCAL (RISC Algorithm Language) system which allows us to validate the core propositions by automatically checking them in finite models.

On the relative strengths of fragments of collection

AbstractLet be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to . We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for all , proves that there exists a transitive model of Zermelo Set Theory plus ‐collection, the theory is ‐conservative over the theory . It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus ) that does not include the powerset axiom.

Basic Theorems in the Language of Maximal Intervals

SummaryWe present proofs of several basic set theory and calculus theorems using the concept of maximal intervals. These may be useful in the classroom, as they are simpler and more direct than standard proofs.

Fragments of Kripke–Platek set theory and the metamathematics of $$\alpha $$ α -recursion theory

The foundation scheme in set theory asserts that every nonempty class has an $$\in $$ź-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and $$\alpha $$ź-recursion theory. We take KP set theory without foundation (called KP$$^-$$-) as the base theory. We show that KP$$^-$$- + $$\Pi _1$$ź1-Foundation + $$V=L$$V=L is enough to carry out finite injury arguments in $$\alpha $$ź-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP.

Tổng quan về điều khiển mờ

The basic fuzzy set theory, fuzzy processes and fuzzy model are described in this paper. Following this, the brief of the fuzzy logic control system with different approaches is presented. Finally we discuss the future development of the fuzzy system.

Molecular self-occlusion as a means for accelerating collision cross-section calculations

A recent article by Béla Paizs in the International Journal of Mass Spectrometry represents an emerging class of “data-oriented” strategies that pre-organise coordinate data to enable faster collision cross-section calculations, the latter being central for the structural interpretation of ion-mobility spectra. The lack of a closed-form expression for the collision cross-section mandates numerical integration, which risks drawing focus from mathematical inspection and thereby precluding the conception of new means to augment the integration. It can be shown from basic geometry and set theory that, within the projection approximation, molecular self-occlusion provides a “theory-oriented” avenue to more efficient collision cross-section calculations, especially when combined with data-oriented acceleration strategies. This approach is particularly suited for the study of macromolecular assemblies and for protein–protein docking, and may further structural biology applications of ion-mobility spectrometry.

Cognitive Distortion: Propositions and Possible Worlds

In this article the concept of cognitive distortions is explored. Recent scholarship has established that the concept of cognitive distortions is not clear. However, in order for any scientific body of literature to progress an adequate degree of conceptual clarity is a requirement. Samples concerning cognitive distortions are drawn from the cognitive theory and therapy literatures. First, a small sample is subjected to an analysis where its contents are reduced to a cross-section of propositions. Altogether, 24 propositions concerning the concept of cognitive distortion are extracted from this first sampling. These propositions are then subjected to a preliminary thematic analysis, which is then extended to samples from nine additional articles before basic set theory is used to calculate the number of all possible worlds of semantics present within the literature. 567 possible worlds of cognitive distortion emerge from this calculation. Although a finished concept of cognitive distortions is not suggested here, a working hypothesis is posited in order to assist in the progress of research. Henceforth, until further deliberation has taken place, it is argued that a cognitive distortion is defined as a clinically relevant type of problematic cognition with a specifiable operational status and scope. It would help if practitioners and researchers specify the properties of these categories while utilizing a common vocabulary in order to improve the clarity of communication in the field.

Relating first-order set theories, toposes and categories of classes

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to one carrying a dssi. We prove that the set theory BIST+Coll (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can (up to equivalence) be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation.

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.

Applications of Fuzzy Technology in Business Intelligence

Fuzzy Set Theory has been developed during the last decades to a demanding mathematical theory. There exist more than 50,000 publications in this area by now. Unluckily the number of reports on applications of fuzzy technology has become very scarce. The reasons for that are manifold: Real applications are normally not single-method-applications but rather complex combinations of different techniques, which are not suited for a publication in a journal. Sometimes considerations of competition my play a role, and sometimes the theoretical core of an application is not suited for publication. In this paper we shall focus on applications of fuzzy technology on real problems in business management. Two versions of fuzzy technology will be used: Fuzzy Knowledge based systems and fuzzy clustering. It is assumed that the reader is familiar with basic fuzzy set theory and the goal of the paper is, to show that the potential of applying fuzzy technology in management is still very large and hardly exploited so far.

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All $${\Pi_2}$$ consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus $${\in}$$ -induction scheme. ZFC+ "there is an inaccessible cardinal" proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define ?-Mahlo models and $${\Pi_1^1}$$ -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form $${Loc({\rm ZFC}+\phi)}$$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.

Question-guided stubborn set methods for state properties

This paper presents two stubborn set methods for alleviating the state explosion problem when reasoning about state properties. The first method makes it possible to determine whether a state of the system is reachable in which a given state predicate holds. The second method makes it possible to determine if from all reachable states it is possible to reach a state where a given state predicate holds. The novelty of the two methods is that they rely on so-called up sets and down sets rather than the notion of visible transitions which causes earlier methods to give only limited reduction of the state space, especially when considering state predicates referring to many of the state variables of the system. The suggested stubborn set methods have been implemented in the LoLA tool, and we report on some experimental results obtained with this computer tool together with some general guidance for applying the two question-guided stubborn set methods and their different implementations in verification. The two methods are presented in the context of Petri Nets, but are applicable also to other state and action oriented modelling formalisms for which the basic stubborn set theory is applicable.

The Foundations of Mathematics in the Theory of Sets

Preface Part I. Preliminaries: 1. The idea of foundations of mathematics 2. Simple arithmetic Part II. Basic Set Theory: 3. Semantics, ontology and logic 4. The principal axioms and definitions of set theory Part III. Cantorian Set Theory: 5. Cantorian finitism 6. The axiomatic method 7. Axiomatic set theory Part IV. Euclidean Set Theory: 8. Euclidian finitism 9. The Euclidean theory of cardinality 10. The theory of simply infinite systems 11. Euclidean set theory from the Cantorian standpoint 12. Envoi Appendices Bibliography Index.

Realism, nonstandard set theory, and large cardinals

Mathematicians justify axioms of set theory “intrinsically”, by reference to the universe of sets of their intuition, and “extrinsically”, for example, by considerations of simplicity or usefullness for mathematical practice. Here we apply the same kind of justifications to Nonstandard Analysis and argue for acceptance of BNST + (Basic Nonstandard Set Theory plus additional Idealization axioms). BNST + has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals. We also consider how to practice Nonstandard Analysis in BNST +, and compare it with other existing nonstandard set theories.

A connectionist inference model for pattern-directed knowledge representation

In this paper we propose a connectionist model for variable binding. The model is topology dependent on the graph it builds based on the predicates available. The irregular connections between perceptron-like assemblies facilitate forward and backward chaining. The model treats the symbolic data as a sequence and represents the training set as a partially connected network using basic set and graph theory to form the internal representation. Inference is achieved by opportunistic reasoning via the bidirectional connections. Consequently, such activity stabilizes to a multigraph. This multigraph is composed of isomorphic subgraphs which all represent solutions to the query made. Such a model has a number of advantages over other methods in that irrelevant connections are avoided by superimposing positionally dependent sub-structures that are identical, variable binding can be encoded and multiple solutions can be extracted simultaneously. The model also has the ability to adapt its existing architecture when presented with new clauses and therefore add new relationships/rules to the model explicitly; this is done by some partial retraining of the network due to the superimposition properties.

Utilization of fuzzy techniques in intelligent sensors

Today, several advanced transducer device and emerging signal processing technologies are combined to establish the important area of so called ‘intelligent sensors’. They offer new or improved operational and processing capabilities. However, as modern technical systems get more and more complex with increasing requirements on system reliability and availability, appropriate systemarchitectures have to be designed, which will have strong impacts on kind and quality of signal processing tasks to be performed by intelligent sensors. Regarding this back-ground motivation, this paper will provide to practitions some hints and explanations for processing uncertain information by utilizing basic fuzzy set theory. The proposed applications include uncertainty propagation, self-calibration, human-machine-interfaces, and semantic classification in fuzzy combinatorial networks. The presentation emphasis on the relation between practical issues and appropriate modeling and processing of information with fuzzy techniques. The industrial application to an intelligent sensor for detection of oil pollution in water is described.

The formal semantics of programming languages: an introduction

Formal Semantics of Programming Languages provides the basic mathematical techniques necessary for those who are beginning a study of semantics and logics of programming languages. Although the treatment is elementary, several of the topics covered are drawn from recent research, including the vital area of concurrency. Starting with basic set theory, structural operational semantics is introduced as a way to define the meaning of programming languages along with associated proof techniques. Denotational and axiomatic semantics are illustrated on a simple language of while-programs, and full proofs are given of the equivalence of the operational and denotational semantics and soundness and relative completeness of the axiomatic semantics. A proof of Goedel's incompleteness theorem, which emphasizes the impossibility of achieving a fully complete axiomatic semantics, is included. It is supported by an appendix providing an introduction to the theory of computability based on while-programs. Following a presentation of domain theory, the semantics and methods of proof for several functional languages are treated. The simplest language is that of recursion equations with both call-by-value and call-by-name evaluation. This work is extended to languages with higher and recursive types, including a treatment of the eager and lazy lambda-calculi. Throughout, the relationship between denotational and operational semantics is stressed, and the proofs of the correspondence between the operation and denotational semantics are provided. The treatment of recursive types - one of the more advanced parts of the book - relies on the use of information sytems to represent domains. The book concludes with a chapter on parallel programming languages, accompanied by a discussion of methods for specifying and verifying non-deterministic and parallel programs.

Modelling human operators in control system design : Research Studies Press, Taunton, Somerset, 1990, 212 pp, 232 refs

Man-machine interaction linear models on human operator adaptability simulation and nonlinear models basic fuzzy set theory for man-machine systems applications of fuzzy set theory to man-machine systems.

Sets: an introduction

Introduction: logic and language Collections Relations Basic set theory Numbers Cardinals Ordinals The axiom of choice Lattices The prime ideal property References Index of notation Index of terminology