DOCSLIB.ORG

Assessment of E-Commerce Web Security Using Ahp - Demster Shafer Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Assessment of E-Commerce Web Security Using Ahp - Demster Shafer Theory International Conference on Mathematics, Science, and Education 2014 (ICMSE 2014) ASSESSMENT OF E-COMMERCE WEB SECURITY USING AHP - DEMSTER SHAFER THEORY Bambang Suhartono*, Joseph Teguh Santoso, Setiyo Prihatmoko STEKOM, Semarang *Email : [email protected] ABSTRACT As well as an increase in the volume, nature Ecommerce business to consumer (B2C) has grown since the beginning of the web. The earliest use commercial web sites as effectively as a "window shopping" electronic, with visitors able to see the products and services available, but can not buy directly. This site can now do, moving away. Than serves to improve the security requirements, because of personal and sensitive financial details are given on a regular basis when conducting a transaction. A number of security issues should be considered to improve the security of the entire E-Commerce, such as: security law (de Lamberterie, 2003), physical security (Furnell, 2004), security management (Tomlinson, 2000), and so on (Oosthuizen, 1999; Tsiakis & Stephanides , 2005). In addition, some security research website E Commerce focused on security solutions E-Commerce web site. Zuccato (2004, 2005) proposed an approach for obtaining security requirements and then develop a security management framework to improve the security of E-Commerce web site. Meanwhile, the adaptive security methodology has been proposed by Tak and Park, for support services without rejection in E-Commerce and E-Commerce transactions, with high quality security services (Tak and Park, 2004). Many factors determine the security of e-commerce web site that determine this problem is Multiple Criteria Decision-Making (MCDM) (Vincke, 1992). There are many approaches to classify MCDM methods. Belton and Stewart (2002) provides a classification: value measurement models such as Multi-Attribute Utility Theory (DEATH) and Analytical Hierarchy Process (AHP). Elimination and outranking models such as (Et) Choice Translating Reality (ELECTRE) and Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE). The basis of the theory above that the decision makers choose an alternative value is the maximum expected utility (Keeney & Raiffa, 1976). Fuzzy sets have the ability to present this problem and AHP is widely used to present the MCDM problems in real situations (Chan & Kumar, 2007). However, due to technological limitations of security and complexity of security issues, it is difficult to find security solutions Ecommerce website complete and absolute safety. Therefore, some qualitative and quantitative analysis and safety assessment of E-Commerce website is a must. Safety assessment of E-Commerce web site using AHP with the Dempster-Shafer theory is that the original title was taken from the literature and summarized in accordance with the fact that in the world of E-Commerce, we need a certain tool which is used in decision-making routine. Keywords: AHP, Dempster-Shafer, e-commerce, security and sensitive financial details are given on a regular basis INTRODUCTION when transaction. A number of security issues should be As well as an increase in volume, the nature of considered to improve the security of the entire E- e-commerce business to consumer (B2C) has grown Commerce, such as: security law (de Lamberterie, 2003), since the beginning of the web. The earliest use physical security (Furnell, 2004), security management commercial web sites as effectively as a "window (Tomlinson, 2000), and so forth (Oosthuizen, 1999; shopping" electronic, with visitors able to see the Tsiakis & Stephanides, 2005). In addition, some security products and services available, but can not buy directly. research website E-Commerce focused on security This site can now do, moving far beyond. Further work solutions E-Commerce web site. Zuccato (2004, 2005) to improve the security requirements, because of personal proposed an approach for obtaining security requirements M - 294 International Conference on Mathematics, Science, and Education 2014 (ICMSE 2014) and then develop a security management framework to explaining complex aspects observed in the improve the security of e-commerce websites. implementation of services via the internet. Meanwhile, the adaptive security methodology has been Web Site E-Commerce in Indonesia has never done proposed by Tak and Park, for support services without research on security aspects. This study builds a penolakandalam E-Commerce and E-Commerce comprehensive model to solve problems in Ecommerce transactions, with high quality security services (Tak and security assessment using AHP with the Dempster-Shafer Park, 2004). theory. AHP with Dempster-Shafer theory, and fuzzy Many factors determine the safety of E- sets are combined for the safety assessment of E- Commerce web sites, this is a decisive problem of Commerce web sites, utilizing the Analytic Hierarchy Multiple Criteria Decision-Making (MCDM) (Vincke, Process (AHP) which is integrated with the Dempster- 1992). There are many approaches to classify MCDM Shafer theory of evidence (DS) used in the model to get a methods. Belton and Stewart (2002) provides a web site's security criteria weights E commerce to assist classification: value measurement models such as Multi- the safety assessment of E-Commerce web site, fuzzy Attribute Utility Theory (DEATH) and Analytical sets to describe the uncertainty with the values of HierarchyProcess (AHP). linguistic security E-commerce web site. Elimination and outranking models such as (Et) How to build a comprehensive model to solve problems Choice Translating Reality (ELECTRE) and Preference in the assessment of E-Commerce security using AHP Ranking Organization Method for Enrichment with the Dempster-Shafer theory. Evaluation (PROMETHEE) and finally, aspirations and Research on E-Commerce security, is not a new objectives level models such as Technique for Order thing anymore, some studies have mentioned the use of Preference by Similarity to Ideal Solutions (TOPSIS). AHP as a device that helps the analysis, including The basis of the theory above that the decision makers assessment of E-Commerce security using AHP and choose an alternative value is the maximum expected evidential reasoning (Zhang et al., 2012). About safety utility (Wang & Lee, 2009). TOPSIS often dikritikkarena assessment of E-Commerce web site using AHP and inability to deal with issues that are unclear and Dempster Shafer. Fuzzy set theory and the concept of uncertain. However, fuzzy sets have the ability to present entropy weights applied to the integration of evaluation this problem and AHP is widely used to present the into the decision-making process (Wang et al., 2009). MCDM problems in real situations (Chan & Kumar, About formulate evaluation software outsourcing 2007). company as a MCDM models, and then present an However, due to technological limitations of effective approach by modifying TOPSIS to solve the security and complexity of security issues, it is difficult problem. to find a website security solution E-Commerce Unlike the previous application, this study will yanglengkap and absolutely safe. Therefore, some conduct an assessment of input variables is a safety qualitative and quantitative analysis and safety factor of E-Commerce web site using AHP for weighting assessment of E-Commerce web site becomes expert assessment of the Dempster-Shafer theory and in imperative. Safety assessment of E-Commerce web site the assessment of each web site which carried a using AHP with the Dempster-Shafer theory is that the combination of variables AHP results, to obtain security original title was taken from the literature and information E-Commerce website. summarized in accordance with the fact that in the world The model provides a tool for practitioners and of E-Commerce, we need a certain tool which is used in E-Commerce has shown its potential in the security decision-making routine. assessment. Although the process of assessing and ranking variables may be influenced by contingent Research on E-Commerce security, is not a new thing variables in various situations, this model can be easily anymore, some studies have mentioned the use of AHP adapted to these variables for the purposes of assessment as a device that helps analysis (Zhang et al., 2012). of E-Commerce security. This model is very potential in About safety assessment of E-Commerce web site. addressing the problem of decision-making, such as the Completeness and fairness of the e-service quality assessment of E-Commerce security. measurement by applying AHP to obtain the weight of the criteria and the TOPSIS ranking (Buyukozkan et al., Analytical Hierarchy Process (AHP) 2012). On the application of e-services framework in Analytical Hierarchy Process developed by Saaty (1980) is a powerful tool to deal with quantitative M - 295 International Conference on Mathematics, Science, and Education 2014 (ICMSE 2014) and qualitative factors in a multi-criteria decision-making the evaluation of sensor reliability in classification problems. With this method, a complicated problem can problems (Guo et al., 2006). Meanwhile, Khokhar and be converted into a regular hierarchy structure. AHP his friends, has been applying the theory in the decision- method has been widely used for multi-criteria decision- making system for risk assessment project E-Commerce making situations, such as: the selection of web sites (Khokhar et al., 2006). Furthermore, a method based on (Ngai, 2003), an evaluation tool (Ngai & Chan, 2005), DS theory introduced by
Load more

Recommended publications
  • 1 Elementary Set Theory
    1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection).
    [Show full text]
  • Mathematics 144 Set Theory Fall 2012 Version
    MATHEMATICS 144 SET THEORY FALL 2012 VERSION Table of Contents I. General considerations.……………………………………………………………………………………………………….1 1. Overview of the course…………………………………………………………………………………………………1 2. Historical background and motivation………………………………………………………….………………4 3. Selected problems………………………………………………………………………………………………………13 I I. Basic concepts. ………………………………………………………………………………………………………………….15 1. Topics from logic…………………………………………………………………………………………………………16 2. Notation and first steps………………………………………………………………………………………………26 3. Simple examples…………………………………………………………………………………………………………30 I I I. Constructions in set theory.………………………………………………………………………………..……….34 1. Boolean algebra operations.……………………………………………………………………………………….34 2. Ordered pairs and Cartesian products……………………………………………………………………… ….40 3. Larger constructions………………………………………………………………………………………………..….42 4. A convenient assumption………………………………………………………………………………………… ….45 I V. Relations and functions ……………………………………………………………………………………………….49 1.Binary relations………………………………………………………………………………………………………… ….49 2. Partial and linear orderings……………………………..………………………………………………… ………… 56 3. Functions…………………………………………………………………………………………………………… ….…….. 61 4. Composite and inverse function.…………………………………………………………………………… …….. 70 5. Constructions involving functions ………………………………………………………………………… ……… 77 6. Order types……………………………………………………………………………………………………… …………… 80 i V. Number systems and set theory …………………………………………………………………………………. 84 1. The Natural Numbers and Integers…………………………………………………………………………….83 2. Finite induction
    [Show full text]
  • A Taste of Set Theory for Philosophers
    Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • Surviving Set Theory: a Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory
    Surviving Set Theory: A Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Angela N. Ripley, M.M. Graduate Program in Music The Ohio State University 2015 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Johanna Devaney Copyright by Angela N. Ripley 2015 Abstract Undergraduate music students often experience a high learning curve when they first encounter pitch-class set theory, an analytical system very different from those they have studied previously. Students sometimes find the abstractions of integer notation and the mathematical orientation of set theory foreign or even frightening (Kleppinger 2010), and the dissonance of the atonal repertoire studied often engenders their resistance (Root 2010). Pedagogical games can help mitigate student resistance and trepidation. Table games like Bingo (Gillespie 2000) and Poker (Gingerich 1991) have been adapted to suit college-level classes in music theory. Familiar television shows provide another source of pedagogical games; for example, Berry (2008; 2015) adapts the show Survivor to frame a unit on theory fundamentals. However, none of these pedagogical games engage pitch- class set theory during a multi-week unit of study. In my dissertation, I adapt the show Survivor to frame a four-week unit on pitch- class set theory (introducing topics ranging from pitch-class sets to twelve-tone rows) during a sophomore-level theory course. As on the show, students of different achievement levels work together in small groups, or “tribes,” to complete worksheets called “challenges”; however, in an important modification to the structure of the show, no students are voted out of their tribes.
    [Show full text]
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show full text]
  • Applied Mathematics for Database Professionals
    7451FM.qxd 5/17/07 10:41 AM Page i Applied Mathematics for Database Professionals Lex de Haan and Toon Koppelaars 7451FM.qxd 5/17/07 10:41 AM Page ii Applied Mathematics for Database Professionals Copyright © 2007 by Lex de Haan and Toon Koppelaars All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the copyright owner and the publisher. ISBN-13: 978-1-59059-745-3 ISBN-10: 1-59059-745-1 Printed and bound in the United States of America 9 8 7 6 5 4 3 2 1 Trademarked names may appear in this book. Rather than use a trademark symbol with every occurrence of a trademarked name, we use the names only in an editorial fashion and to the benefit of the trademark owner, with no intention of infringement of the trademark. Lead Editor: Jonathan Gennick Technical Reviewers: Chris Date, Cary Millsap Editorial Board: Steve Anglin, Ewan Buckingham, Gary Cornell, Jonathan Gennick, Jason Gilmore, Jonathan Hassell, Chris Mills, Matthew Moodie, Jeffrey Pepper, Ben Renow-Clarke, Dominic Shakeshaft, Matt Wade, Tom Welsh Project Manager: Tracy Brown Collins Copy Edit Manager: Nicole Flores Copy Editor: Susannah Davidson Pfalzer Assistant Production Director: Kari Brooks-Copony Production Editor: Kelly Winquist Compositor: Dina Quan Proofreader: April Eddy Indexer: Brenda Miller Artist: April Milne Cover Designer: Kurt Krames Manufacturing Director: Tom Debolski Distributed to the book trade worldwide by Springer-Verlag New York, Inc., 233 Spring Street, 6th Floor, New York, NY 10013.
    [Show full text]
  • The Axiom of Choice and Its Implications
    THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial.
    [Show full text]
  • Equivalents to the Axiom of Choice and Their Uses A
    EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper.
    [Show full text]
  • Set-Theoretic Foundations
    Contemporary Mathematics Volume 690, 2017 http://dx.doi.org/10.1090/conm/690/13872 Set-theoretic foundations Penelope Maddy Contents 1. Foundational uses of set theory 2. Foundational uses of category theory 3. The multiverse 4. Inconclusive conclusion References It’s more or less standard orthodoxy these days that set theory – ZFC, extended by large cardinals – provides a foundation for classical mathematics. Oddly enough, it’s less clear what ‘providing a foundation’ comes to. Still, there are those who argue strenuously that category theory would do this job better than set theory does, or even that set theory can’t do it at all, and that category theory can. There are also those who insist that set theory should be understood, not as the study of a single universe, V, purportedly described by ZFC + LCs, but as the study of a so-called ‘multiverse’ of set-theoretic universes – while retaining its foundational role. I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand. 1. Foundational uses of set theory The most common characterization of set theory’s foundational role, the char- acterization found in textbooks, is illustrated in the opening sentences of Kunen’s classic book on forcing: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate . axioms 2010 Mathematics Subject Classification. Primary 03A05; Secondary 00A30, 03Exx, 03B30, 18A15.
    [Show full text]
  • Set Theory in Computer Science a Gentle Introduction to Mathematical Modeling I
    Set Theory in Computer Science A Gentle Introduction to Mathematical Modeling I Jose´ Meseguer University of Illinois at Urbana-Champaign Urbana, IL 61801, USA c Jose´ Meseguer, 2008–2010; all rights reserved. February 28, 2011 2 Contents 1 Motivation 7 2 Set Theory as an Axiomatic Theory 11 3 The Empty Set, Extensionality, and Separation 15 3.1 The Empty Set . 15 3.2 Extensionality . 15 3.3 The Failed Attempt of Comprehension . 16 3.4 Separation . 17 4 Pairing, Unions, Powersets, and Infinity 19 4.1 Pairing . 19 4.2 Unions . 21 4.3 Powersets . 24 4.4 Infinity . 26 5 Case Study: A Computable Model of Hereditarily Finite Sets 29 5.1 HF-Sets in Maude . 30 5.2 Terms, Equations, and Term Rewriting . 33 5.3 Confluence, Termination, and Sufficient Completeness . 36 5.4 A Computable Model of HF-Sets . 39 5.5 HF-Sets as a Universe for Finitary Mathematics . 43 5.6 HF-Sets with Atoms . 47 6 Relations, Functions, and Function Sets 51 6.1 Relations and Functions . 51 6.2 Formula, Assignment, and Lambda Notations . 52 6.3 Images . 54 6.4 Composing Relations and Functions . 56 6.5 Abstract Products and Disjoint Unions . 59 6.6 Relating Function Sets . 62 7 Simple and Primitive Recursion, and the Peano Axioms 65 7.1 Simple Recursion . 65 7.2 Primitive Recursion . 67 7.3 The Peano Axioms . 69 8 Case Study: The Peano Language 71 9 Binary Relations on a Set 73 9.1 Directed and Undirected Graphs . 73 9.2 Transition Systems and Automata .
    [Show full text]
  • Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
    Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite.
    [Show full text]