Concept Of Cardinality Research Articles

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

This essay delves deep into one of the most intriguing mathematical puzzles of all time: the Continuum Hypothesis. Beginning with a robust foundational exploration, it sheds light on the key concepts of cardinality and power sets, which are pivotal to the realm of set theory. These foundational ideas set the stage for a deeper investigation into the relationship that the Continuum Hypothesis shares with real numbers and natural numbers. Historically, the Continuum Hypothesis has tantalized mathematicians. This paper takes a journey through time, highlighting the various endeavors to either prove or refute this hypothesis. Some of the most brilliant minds have grappled with its complexities, leaving behind a rich tapestry of mathematical thought. Furthermore, a significant portion of our discussion is centered on situating the Continuum Hypothesis within the context of Zermelo-Fraenkel Set Theory (ZFC). The intricate interplay between the hypothesis and ZFC offers profound insights and raises thought-provoking questions about the very nature of mathematical truth.

Avoiding flatness in factoring ordinal data

Factorization of classical, two-valued Boolean data became a widely studied topic in the past decade due to its role in analyzing relational data as well as its significance for other fields. Recently, various extensions to factorization of ordinal data, or data with graded (fuzzy) attributes, have been proposed. We identify and describe a fundamental problem regarding quality of factors, which is non-existent in the Boolean case, but naturally appears in the more general setting of ordinal data. As we demonstrate, the problem gets more significant with growing size of the factorized data. We analyze the problem, propose a method to alleviate it, and evaluate experimentally our solution to the problem. We also provide a discussion regarding ramifications of our findings for the concept of cardinality of fuzzy sets.

The Numerals Dva, Tri, Četyre in the Novgorod Birch Bark Letters: A Diachronic Perspective

Abstract: This article investigates the emergence of the cardinal numerals dva 'two', tri 'three', and četyre 'four' in nominal phrases in the Novgorod dialect during the 11th–15th centuries. An innovative approach presented here brings together three productive lines of inquiry—corpus analysis, historical linguistics, and diachronic generative syntax. A corpus analysis was conducted to identify 301 tokens of numeral-containing NPs and to trace the patterns of their diachronic development. The cardinal numerals 2, 3, 4 are shown to evolve from the adjectival "number" words through the process of grammaticalization, more specifically, numeralization. After the loss of dual number, the lower adjectival "number" words for 2, 3, and 4 turned into the cardinal numerals, as their lexically encoded numerosity became functionally encoded countability. The diachrony of the cardinal numerals in the Old Novgorod dialect has shown that the cognitive concepts of cardinality, individuality, and countability are inextricably connected. These concepts are grammatically encoded and subject to diachronic change. This study has demonstrated that a change in grammatical number (loss of the dual) led to the restructuring of countability and the rise of cardinal numerals.

Musical activities for enhancing number concepts in preschool children

A concept is the basic unit of all types of learning. Young children generally form concepts through the use of senses. Number concept is defined by McIntosh et al. (1992), as a person’s general understanding of numbers and operations. Research studies suggest that children construct basic number and arithmetic concepts gradually from experiences that involve numbers and difficulties in counting might hinder in mathematical progress (Baroody and Coslick, 1998). Therefore, attention should be given to assess and understand the number concepts formed during preschool age. Music is considered to play an important role in stimulating young children in learning. Hetland (2000) described that using songs and different music components can help in teaching numbers because of the high association between music and spatial-temporal reasoning skills. The study was carried out on twenty four numbers of pre-school children belonging to the age of 3- 4 years, selected through purposive sampling method. Pre and post-intervention design was used to assess the influence of musical activities on the development of number concepts in children. The number concepts were assessed in the categories of Number sense, Number representation and Number relation based on the principles of counting proposed by Gelman and Galistell (1978). It was found from the study that most children could name numbers but the concept of cardinality was not attained by many of the children. Findings revealed that children improved in all the areas of number concepts after the musical intervention. However, it was also observed that all the concepts are not mastered at the same time by all the children.

Strategies on admissible total orders over typical hesitant fuzzy implications applied to decision making problems

Multi Expert-Multi Criteria Decision Making (ME-MCDM) problems have been well explored on hesitant fuzzy environments dealing with membership degrees as subsets which do not necessarily have the same cardinality. Admissible (total) orders collaborate by reducing the collapse in the ranking of alternatives related to preference relations. In such context, three classes of admissible orders are presented in a non-restrictive way, integrating the concepts of linearity and cardinality and providing support to the comparison between two typical hesitant fuzzy elements. In the sense of hesitant fuzzy logic, the study of fuzzy operators and their main properties is extended considering the admissible linear orders. Namely, the typical hesitant fuzzy aggregation functions, negations and implication functions are discussed mainly related to their (iso/anti)tonicity properties w.r.t. these admissible orders. An algorithmic procedure is introduced illustrating our strategy to solve an ME-MCDM problem, by selecting a Computer Integrating Manufacturing software and making use of the achieved theoretical results.

Numerosities are not ersatz numbers.

In describing numerosity as "a kind of ersatz number," Clarke and Beck fail to consider a familiar and compelling definition of numerosity, which conceptualizes numerosity as the cognitive counterpart of the mathematical concept of cardinality; numerosity is the magnitude, whereas number is a scale through which numerosity/cardinality is measured. We argue that these distinctions should be considered.

Construction of the concept of pairability in comparing infinite sets in the individuals’ mind: an APOS approach

Abstract This study focuses on the concept of pairability as a fundamental metaphorical concept in the Cantorian set theory regarding comparison of infinite sets. In this theory, pairability appears in a hierarchical manner of generating and developing as a one-to-one correspondence by arrows in finite, and then in infinite states, a bijection map through an explicit formula and the concept of cardinality. Adapting these hierarchical components of pairability and the APOS theory, about description of constructing and understanding a mathematical concept in a hierarchical manner, this study examines the construction of the concept of pairability in the individuals’ mind. In this way, their imaginations of pairability and practical performances in different situations will also be surveyed. In so doing, a total of 20 mathematics teachers and university lecturers holding at least an M.Sc. degree in mathematics were chosen. To collect the data, interviews were conducted in which the participants not only answered questions about the concept of the various types of pairability but also compared infinite countable sets in different situations. Our findings revealed that a bijection map via an explicit formula was the individuals’ dominant conception of pairability and most of the incorrect answers were related to unsuccessful attempts to recall a formula or method as the only possible way, and the encapsulated concept of cardinality was used less frequently in practice. Therefore, there was not a total schema of actions, processes and objects of the concept of pairability in the individuals’ mind.

An outlier detection approach in large-scale data stream using rough set

Outlier detection has become an important research area in the field of stream data mining due to its vast applications. In the literature, many methods have been proposed, but they work well for simple and positive regions of outliers, where boundary regions are not given much importance. Moreover, an algorithm which processes stream data must be effective and able to compute infinite data in one pass or limited number of passes. These problems have motivated us to propose an outlier detection approach for large-scale data stream. The proposed algorithm employs the concept of relative cardinality, entropy outlier factor theory of information-based system, and size-variant sliding window in stream data. In addition, we propose a new methodology for concept drift adaptation on evolving data streams. The proposed method is executed on nine benchmark datasets and compared with six existing methods that are EXPoSE, iForest, OC-SVM, LOF, KDE, and FastAbod. Experimental results show that the proposed method outperforms six existing methods in terms of receiver operating characteristic curve, precision recall, and computational time for positive regions as well as for boundary regions.

Core Cognition and its Aftermath

A current and very influential theory in psychology holds that infants have innate, perceptually informed systems that endow them with surprisingly high-level concepts—for example, concepts of cardinality and causality. Proponents of core cognition hold that these initial concepts then provide the building blocks for later adult ideas within these domains. This paper reviews the evidence for core cognition and argues that these systems aren't sufficient to explain how children learn their way to adult thoughts about language, number, or cause.

On the minimum probability of classification error through effective cardinality comparison

A straightforward ''collision'' (quadratic) entropy estimator is used to give support, in a very pragmatic approach, to data analysis based on the concept of effective cardinality of sets. Thus, by using basic concepts of probability and set theories, a method is proposed to estimate the minimum probability of classification error, in two-class problems, without the deployment of any particular classifier. The usefulness as well as some limitations of the analysis based on effective cardinality are exemplified throughout the text.

Domination in vague graphs and its applications

The concept of vague graph introduced by Ramakrishna in [8]. The main purpose of this paper is to introduce the concepts of cardinality, dominating set, independent set, total dominating number and independent dominating number of a vague graph. The notion of irredundance number of a vague graph is discussed, too. Finally we give an application of domination in vague graphs.

Introduction to Image Processing via Neutrosophic Techniques

This paper is an attempt of proposing the processing approach of neutrosophic technique in image processing. As neutrosophic sets is a suitable tool to cope with imperfectly defined images, the properties, basic operations distance measure, entropy measures, of the neutrosophic sets method are presented here. In this paper we, introduce the distances between neutrosophic sets: the Hamming distance, the normalized Hamming distance, the Euclidean distance and normalized Euclidean distance. We will extend the concepts of distances to the case of neutrosophic hesitancy degree. Entropy plays an important role in image processing. In our further considertions on entropy for neutrosophic sets the concept of cardinality of a neutrosophic set will also be useful. Possible applications to image processing are touched upon.

Development of numerical concepts

The development of numerical concepts is described from infancy to preschool age. Infants a few days old exhibit an early sensitivity for numerosities. In the course of development, nonverbal mental models allow for the exact representation of small quantities as well as changes in these quantities. Subitising, as the accurate recognition of small numerosities (without counting), plays an important role. It can be assumed that numerical concepts and procedures start with insights about small numerosities. Protoquantitative schemata comprise fundamental knowledge about quantities. One-to-one-correspondence connects elements and numbers, and, for this reason, both quantitative and numerical knowledge. If children understand that they can determine the numerosity of a collection of elements by enumerating the elements, they have acquired the concept of cardinality. Protoquantitative knowledge becomes quantitative if it can be applied to numerosities and sequential numbers. The concepts of cardinality and part-part-whole are key to numerical development. Developmentally appropriate learning and teaching should focus on cardinality and part-part-whole concepts.

Young Students Exploring Cardinality by Constructing Infinite Processes

In this paper, we describe the design and implementation of computer programming activities aimed at introducing young students (9–13 years old) to the idea of infinity, and in particular, to the cardinality of infinite sets. This research was part of the WebLabs project where students from several European countries explored topics in mathematics and science by building computational models and programs, which they shared and discussed. We focus on a subset of the work in which students explored concepts of cardinality of infinite sets by interpreting and constructing computer programs in ToonTalk, a programming language and environment that is especially well-suited for young students. Our hypothesis is that via carefully designed computational explorations within an appropriately constructed medium, infinity can be approached in a learnable way that does not sacrifice the rigour necessary for mathematical understanding of the concept, and at the same time contributes to introducing the real spirit of mathematics to the school classroom.

Fuzzy sets in data protection: strategies and cardinalities

In this article, we analyze how classical techniques of data privacy, as k-anonymity, can be developed within a framework of fuzzy information. In particular, the main notion that has to be studied is the concept of cardinality, in order to obtain a proper way to count how many elements belong to a given equivalence class.

Against recall

The concept of recall has been one of the key elements of system measurement throughout the history of information retrieval, despite the fact that there are many unanswered questions as to its value. In this essay, we review those questions and explore several further issues that affect the usefulness of recall. In particular, we ask whether it is reasonable to expect to be able to measure recall; whether some researchers are conflating the concepts of recall and answer set cardinality; and whether it is plausible that a user would rely on a belief that a system is "high recall" to deeply explore an answer list. Combined with earlier observations about the unknowability of recall, and the lack of a plausible user model in which recall is a measure of satisfaction, we conclude that use of recall as a measure of the effectiveness of ranked querying is indefensible.

On the cardinalities of interval-valued fuzzy sets

Wygralak has developed an axiomatic theory of scalar cardinalities of fuzzy sets with finite support which covers as particular cases all standard (historical) concepts of the scalar cardinality. In this paper we present a possible extension of this theory to interval-valued fuzzy set theory. We introduce the cardinality of interval-valued fuzzy sets as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [ 0 , + ∞ [ . We study some properties of these cardinalities using t-norms, t-conorms and negations on the lattice L I (the underlying lattice of interval-valued fuzzy set theory). In particular, the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule will be discussed.

Utility of income as a random function: Behavioral characterization and empirical evidence

The paper proposes a particular approach to model the utility of income. We develop a theoretical framework that restricts the class of admissible functional forms and distributions of the random components of the model. The theoretical approach is based on theories of probabilistic choice and ideas that are used in modern psychophysical research. From our theoretical framework, we obtain the empirical model and the corresponding likelihood function. The empirical analysis is based on a “Stated Preference” survey. The model fits the data quite well. Finally, we discuss the concept of cardinality and the implications for consumer demand relations.

On the number of non-isomorphic subspaces of a Banach space

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following: let X be a Banach space with an unconditional basis {ei}i∈N; then either there exists a perfect set P of infinite subsets of N such that for any two distinct A, B ∈ P, [ei]i∈A ≇ [ei]i∈B, or for a residual set of infinite subsets A of N, [ei]i∈A is isomorphic to X, and in that case, X is isomorphic to its square, to its hyperplanes, uniformly isomorphic to X ⊕ [ei]i∈D for any D ⊂ N and to a denumerable Schauder decomposition into uniformly isomorphic copies of itself. The starting point of this article is the so called ”homogeneous space problem”, due to S. Banach. It has been solved at the end of last century by W.T. Gowers ([2],[3]), R. Komorowski- N. Tomczak ([7],[8]) and W.T. Gowers- B. Maurey ([4]); see e.g. [10] for a survey. Recall that a Banach space is said to be homogeneous if it is isomorphic to all its (infinite-dimensional closed) subspaces. The previously named authors showed that l2 is the only homogeneous Banach space. A very natural question was posed to us by G. Godefroy: if a Banach space is not isomorphic to l2, then how many mutually non-isomorphic subspaces must it contain (obviously, at least 2). In this article, we concentrate on spaces with a basis and subspaces of it spanned by subsequences. We shall also be interested in the relation of equivalence of basic sequences. By ”many” we shall mean the Cantor concept of cardinality, and sometimes finer concepts from the theory of classification of equivalence relations in descriptive set theory.

Exponentiation and Euler measure

Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with n elements (which can be done in 2 n ways), and the notion of choosing a k-element subset of a set with n elements (which can be done in $$ n \choose k $$ ways). In this article I sketch the beginnings of a theory that would import these notions into the category of hedral sets in the sense of Morelli and the category of polyhedral sets in the sense of Schanuel. Both of these theories can be viewed as extensions of the theory of finite sets and mappings between finite sets, with the concept of cardinality being replaced by the more general notion of Euler measure (sometimes called combinatorial Euler characteristic). I prove a “functoriality” theorem (Theorem 1) for subset-selection in the context of polyhedral sets, which provides quasi-combinatorial interpretations of assertions such as $$ 2^{-1} = {1 \over 2} \quad\mathrm{and}\quad{{1/2} \choose {2}} = -{1 \over 8} $$ . Furthermore, the operation of forming a power set can be viewed as a special case of the operation of forming the set of all mappings from one set to another; I conclude the article with a polyhedral analogue of the set of all mappings between two finite sets, and a restrictive but suggestive result (Theorem 2) that offers a hint of what a general theory of exponentiation in the polyhedral category might look like. (Other glimpses into the theory may be found in [11].)