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

2020 ◽  
Author(s):  
Mohadaseh Mahdiyan ◽  
Ali Barahmand
Keyword(s):  
Explicit Formula ◽  
Set Theory ◽  
Mathematics Teachers ◽  
Mathematical Concept ◽  
Apos Theory ◽  
Infinite Sets ◽  
One To One ◽  
University Lecturers ◽  
Metaphorical Concept ◽  
Hierarchical Manner

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.

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

scholarly journals Small sets with large power sets

1973 ◽  
Vol 8 (3) ◽  
pp. 413-421 ◽  
Author(s):  
G.P. Monro
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Large Power ◽  
Present Evidence ◽  
Power Set ◽  
Infinite Sets ◽  
The Axiom Of Choice

One problem in set theory without the axiom of choice is to find a reasonable way of estimating the size of a non-well-orderable set; in this paper we present evidence which suggests that this may be very hard. Given an arbitrary fixed aleph κ we construct a model of set theory which contains a set X such that if Y ⊆ X then either Y or X - Y is finite, but such that κ can be mapped into S(S(S(X))). So in one sense X is large and in another X is one of the smallest possible infinite sets. (Here S(X) is the power set of X.)

scholarly journals Counting Derangements, Non Bijective Functions and the Birthday Problem

2010 ◽  
Vol 18 (4) ◽  
pp. 197-200
Author(s):  
Cezary Kaliszyk
Keyword(s):  
Explicit Formula ◽  
Recursive Formula ◽  
Finite Sets ◽  
Birthday Problem ◽  
Finite Set ◽  
One To One

Counting Derangements, Non Bijective Functions and the Birthday Problem The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of [10].

scholarly journals IMPULSIVE AND REFLECTIVE STUDENTS’ UNDERSTANDING TO LINEAR EQUATIONS SYSTEM: AN ANALYSIS THROUGH APOS THEORY

MATHEdunesa ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 128-135
Author(s):  
Dinda Ayu Rachmawati ◽  
Tatag Yuli Eko Siswono
Keyword(s):  
Linear Equation ◽  
Cognitive Style ◽  
Junior High School ◽  
Linear Equations ◽  
Equation System ◽  
Mathematical Concept ◽  
Apos Theory ◽  
Process Stage ◽  
Linear Equation System ◽  
Action Process

Understanding is constructed or reconstructed by students actively. APOS theory (action, process, object, schema) is a theory that states that individuals construct or reconstruct a concept through four stages, namely: action, process, object, and scheme. APOS theory can be used to analyze understanding of a mathematical concept. This research is a qualitative research which aims to describe impulsive and reflective students’ understanding to linear equations system based on APOS theory. Data collection techniques were carried out by giving Matching Familiar Figure Test (MFFT) and concept understanding tests to 32 students of 8th grade in junior high school, then selected one subject with impulsive cognitive style and one subject with reflective cognitive style that can determine solutions set and solve story questions of linear equation system of two variables correctly, then the subjects were interviewed. The results show that there were differences between impulsive and reflective subjects at the stage of action in explaining the definition and giving non-examples of linear equation system of two variables, show the differences in initial scheme of two subjects. At the process stage, impulsive and reflective subjects determine solutions set of linear equation system of two variables. At the object stage, impulsive and reflective subjects determine characteristics of linear equation system of two variables. At the schema stage, impulsive and reflective subjects solve story questions of of linear equation system of two variables, show the final schematic similarity of two subjects.Keywords: understanding, APOS theory, linear equations system of two variables, impulsive cognitive style, reflective cognitive style.

scholarly journals A Need Analysis For Developing Integrated Stem Course Training Module For Pre-Service Mathematics Teachers

2019 ◽  
Vol 8 (5C) ◽  
pp. 47-52
Keyword(s):  
Mathematics Teachers ◽  
Instructional Strategy ◽  
Structured Interview ◽  
Training And Development ◽  
Training Module ◽  
Knowledge For Teaching ◽  
Integrated Stem ◽  
Need Analysis ◽  
And Mathematics ◽  
University Lecturers

The purpose of this paper is to present the need to develop the integrated Science, Technology, Engineering and Mathematics (iSTEM) course training module for preparing the pre-service mathematics teachers’ readiness in teaching the iSTEM course in their future classroom instruction. In this paper, the researchers present the data for the study collected qualitatively using semi-structured interview questions. Purposive sampling was used in choosing 6 university lecturers in Nigeria for conducting the interview to identify and seek their insights regarding the need for iSTEM course training and module for teaching iSTEM course. An interview data collected was transcribed and inductively analysed using an exploratory content analysis approach in which five main themes have been identified, namely: (i) pedagogical knowledge; (ii) content knowledge; (iii) conception of the iSTEM; (iv) instructional strategy and (v) curricula knowledge for teaching iSTEM course. The results of this study indicated that all the six interview participants responded that they have no idea and had insufficient knowledge about the teaching of integrated STEM course. Moreover, the lecturers indicated that they were not exposed to the iSTEM course at all. The findings revealed that there is a need for the iSTEM course training and development of the iSTEM teaching module in helping the pre-service mathematics teachers’ readiness in teaching the iSTEM course. Based on the findings, the researchers recommended suitable iSTEM course training module to prepare the pre-service mathematics teachers’ readiness in teaching iSTEM course could be integrated into the teacher education curriculum in Nigeria.

scholarly journals ENUMERABILITY OF THE SET OF ALGEBRAIC NUMBERS

2021 ◽  
Vol 12 (3) ◽  
pp. 71-77
Author(s):  
Fernanda Loureiro Honorio ◽  
Fernando Pereira de Souza
Keyword(s):  
Set Theory ◽  
Algebraic Numbers ◽  
Basic Concepts ◽  
Infinite Sets

Through the studies of the Set Theory, the cardinality of the infinite sets is classified according to their enumerability, thus making the importance of the theme remarkable. Therefore, in this work, the enumerability of the Set of Algebraic Numbers will be approached, through studies that were carried out as part of the scientific initiation project linked as activities of the group PET Connections of Knowledge Mathematics at UFMS, Campus of Três Lagoas. The concepts defined for the study of the AlgebraicNumbers Set will be conceptualized, exposing their definitions and properties, with examples and accounts of their theorems. This work, besides aiming to deepen the studies on the theme, also aims, in the presentation of the same, a way to make its understanding more accessible. In this way, it is found in the results of this research the exposure of basic concepts about the enumerability of sets, which facilitate the understanding of the most complex counts of the propositions that follow.

scholarly journals The enhancement of pre-service mathematics teachers’ mathematical understanding ability through ACE teaching cyclic

2019 ◽  
Vol 9 (2) ◽  
pp. 153 ◽  
Author(s):  
Muhammad Win Afgani ◽  
Didi Suryadi ◽  
Jarnawi Afgani Dahlan
Keyword(s):  
Data Analysis ◽  
Mathematics Teachers ◽  
Mathematical Understanding ◽  
Class Discussion ◽  
Control Group ◽  
Apos Theory ◽  
Activity Class ◽  
Ability Test ◽  
Significant Difference ◽  
Direct Learning

The aim of this study was to investigate the enhancement of the mathematical understanding ability of pre-service mathematics teachers through Activity-Class Discussion-Exercise (ACE) teaching cyclic based on APOS theory. This study used a quasi-experiment method with non-equivalent pre-post test control group design. The subjects of this study were 120 pre-service mathematics teachers from two universities in Palembang, Indonesia. The subjects were divided into two class, that is, experiment and control class. Experiment class was a class that is applied ADE teaching cyclic based on APOS theory, whereas control class was a class that is applied direct learning. The subjects were also divided into three groups of mathematical initial ability, that is, high, average, and low. The Instruments used in this study were mathematical initial ability test, mathematical understanding ability test, observation, and interview. Data analysis tests used in this study were statistic test of parametric and non-parametric. The results of data analysis showed that 1) there is no significant difference between the improvement of mathematical understanding ability of pre-service mathematics teachers applied ACE teaching cyclic based on APOS theory and direct learning in terms of overall and the group of mathematical initial ability, 2) there is no interaction between learning factors (APOS and direct learning) and the group of mathematical initial ability (high, average, and low) to the improvement of mathematical understanding ability of pre-service mathematics teachers.

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