A First Course in Fuzzy Logic, Fuzzy Dynamical Systems

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This book provides an essential introduction to the field of dynamical models. Starting from classical theories such as set theory and probability, it allows readers to draw near to the fuzzy case. On one hand, the book equips readers with a fundamental understanding of the theoretical underpinnings of fuzzy sets and fuzzy dynamical systems. On the other, it demonstrates how these theories are used to solve modeling problems in biomathematics, and presents existing derivatives and integrals applied to the context of fuzzy functions. Each of the major topics is accompanied by examples, worked-out exercises, and exercises to be completed. Moreover, many applications to real problems are presented.

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Studies in Fuzziness and Soft Computing Laécio Carvalho de Barros Rodney Carlos Bassanezi W eldon  Alexander  L odwick A First Course in Fuzzy Logic Fuzzy Dynamical Systems and Biomathematics Theory and Applications

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Studies in Fuzziness and Soft Computing Volume 347 Series editor Janusz Kacprzyk Polish Academy of Sciences Warsaw Poland e-mail: kacprzykibspan.waw.pl

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About this Series The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing which include fuzzy sets rough sets neural networks evolutionary computation probabilistic and evidential reasoning multi-valued logic and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. More information about this series at http://www.springer.com/series/2941

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Laécio Carvalho de Barros Rodney Carlos Bassanezi Weldon Alexander Lodwick A First Course in Fuzzy Logic Fuzzy Dynamical Systems and Biomathematics Theory and Applications 123

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Laécio Carvalho de Barros Departamento de Matemática Aplicada Universidade Estadual de Campinas São Paulo Brazil Rodney Carlos Bassanezi Centro de Matemática e Computação Universidade Federal do ABC Santo AndréSão Paulo Brazil Weldon Alexander Lodwick Department of Mathematical and Statistical Sciences University of Colorado Denver Denver CO USA ISSN 1434-9922 ISSN 1860-0808 electronic Studies in Fuzziness and Soft Computing ISBN 978-3-662-53322-2 ISBN 978-3-662-53324-6 eBook DOI 10.1007/978-3-662-53324-6 Library of Congress Control Number: 2016948236 © Springer-Verlag Berlin Heidelberg 2017 This work is subject to copyright. All rights are reserved by the Publisher whether the whole or part of the material is concerned specifically the rights of translation reprinting reuse of illustrations recitation broadcasting reproduction on microfilms or in any other physical way and transmission or information storage and retrieval electronic adaptation computer software or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names registered names trademarks service marks etc. in this publicationdoesnotimplyevenin theabsenceofa specific statementthatsuch namesare exemptfrom the relevant protective laws and regulations and therefore free for general use. The publisher the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarrantyexpressorimpliedwithrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Germany The registered company address is: Heidelberger Platz 3 14197 Berlin Germany

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The authors wish to thank all their students who over the years graced our presence and with whom we share a space in time devoted to our mutual love of mathematics and the pursuit of truth. In particular we would like to thank Estevão Esmi Laurenao for the assistance in reading editing early versions and with the layout of the text and figures.

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Preface This book is the result of courses we have given for more than a decade to upper level undergraduate students and to graduate students majoring in mathematics applied mathematics statistics and engineering. In this book the reader will encounter the basic concepts that span the initial notions of fuzzy sets to more advanced notions offuzzy differential equation and dynamical systems. We follow in our ordering of topics a pedagogical unfolding beginning with classical theory such as set theory and probability in such a way that these serve as an opening into the fuzzy case. Moreover the classical differential and integral calculus is the beginning step from which fuzzy differential and integral analysis are developed. Therearevariousderivativesandintegralsthatexistandappliedinthecontextof fuzzy functions. These are clearly delineated and interpreted in our presentation of fuzzy integral and differential equations. Each of the major topics is accompanied with examples worked exercises and exercises to be completed. Many applications of our concept to real problems are found throughout the book. Even though this book may be and has been used as a textbook for various courses in it are sufficient ideas for beginning the research projects in fuzzy mathematics. It is the hope of the authors that our joy passion and respect for all who seriously the study offuzzy mathematics modeling and applications emerges through the written page. Laécio Carvalho de Barros Rodney Carlos Bassanezi Weldon Alexander Lodwick vii

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Acknowledgment The authors would like to acknowledge and thank the partial support received from CNPq. ix

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Contents 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling ... 1 1.1 Uncertainty in Modeling and Analysis.................... 1 1.2 Fuzzy Subset ....................................... 6 1.3 Operations with Fuzzy Subsets.......................... 11 1.4 Concept of α-Level................................... 17 1.5 Summary .......................................... 21 References............................................... 21 2 The Extension Principle of Zadeh and Fuzzy Numbers.......... 23 2.1 Zadeh’s Extension Principle............................ 23 2.2 Fuzzy Numbers...................................... 28 2.2.1 Arithmetic Operations with Fuzzy Numbers......... 31 References............................................... 41 3 Fuzzy Relations.......................................... 43 3.1 Fuzzy Relations ..................................... 43 3.1.1 Forms of Representation and Properties of the Binary Relations......................... 46 3.2 Composition Between Binary Fuzzy Relations.............. 48 References............................................... 51 4 Notions of Fuzzy Logic.................................... 53 4.1 Basic Connectives of Classical Logic..................... 54 4.2 Basic Connectives of Fuzzy Logic....................... 58 4.2.1 Operations T-Norm and T-Conorm................ 58 4.3 Approximate Reasoning and Linguistic Variables............ 63 4.4 Modus Ponens and Generalized Modus Ponens............. 65 4.5 Linguistic Modifiers.................................. 69 4.6 Independence and Non-interactivity ...................... 73 4.6.1 Probabilistic Independence and Non-interactivity ..... 74 4.6.2 Possibilistic Independence and Non-interactivity...... 75 4.6.3 The Conditional Distributions and Modus Ponens..... 76 References............................................... 78 xi

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5 Fuzzy Rule-Based Systems................................. 79 5.1 Fuzzy Rule Bases.................................... 81 5.2 Fuzzy Controller..................................... 82 5.2.1 Fuzzification Module........................... 82 5.2.2 Base-Rule Module............................. 82 5.2.3 Fuzzy Inference Module........................ 82 5.2.4 Defuzzification Module......................... 83 5.3 Mamdani Inference Method............................ 84 5.4 Defuzzification Methods............................... 87 5.4.1 Centroid or Center of Mass or Center of Gravity GB.............................. 88 5.4.2 Center of Maximum CB...................... 89 5.4.3 Mean of Maximum MB...................... 89 5.5 Takagi–Sugeno–Kang Inference Method TSK............. 90 5.6 Applications........................................ 95 5.6.1 Model 1 – Forecasting the Salinity of an Estuary in Cananeia and Ilha Comprida................... 95 5.6.2 Model 2 – Rate of Seropositive Transfer HIV þ ..... 101 5.6.3 Model 3 – Pharmacological Decay................ 107 References............................................... 111 6 Fuzzy Relational Equations and Universal Approximation ....... 113 6.1 Generalized Compositions of Fuzzy Relations .............. 114 6.2 Fuzzy Relational Equations............................. 116 6.2.1 Fuzzy Relational Equations with the max–min Composition ................................. 116 6.2.2 Fuzzy Relational Equations with the sup–t Composition ................................. 118 6.2.3 Mathematical Modelling: Medical Diagnosis......... 120 6.3 Fuzzy Relational Equation and Bayesian Inference........... 123 6.3.1 Possibility Distribution and Bayesian Inference....... 125 6.3.2 Possibilistic Rule of Bayes ...................... 126 6.4 Universal Approximation.............................. 127 6.4.1 Approximating Capability....................... 128 6.5 Applications of Fuzzy Controllers in Dynamic Systems....... 131 References............................................... 133 7 Measure Integrals and Fuzzy Events ........................ 135 7.1 Classic Measure and Fuzzy Measure ..................... 136 7.1.1 Probability Measure............................ 136 7.2 Fuzzy Measure...................................... 138 7.3 Possibility Measure................................... 140 7.4 Probability/Possibility Transformations.................... 144 xii Contents

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7.5 Fuzzy Integrals...................................... 146 7.5.1 Lebesgue Integral ............................. 147 7.5.2 Choquet Integral.............................. 149 7.5.3 Sugeno Integral............................... 150 7.6 Fuzzy Events ....................................... 159 7.6.1 Probability of Fuzzy Events ..................... 160 7.6.2 Independence Between Fuzzy Events .............. 164 7.6.3 Random Linguistic Variable and Fuzzy Random Variable.............................. 166 References............................................... 172 8 Fuzzy Dynamical Systems.................................. 175 8.1 Continuous Fuzzy Dynamical Systems.................... 175 8.1.1 Integration and Differentiation of Fuzzy Functions.... 177 8.1.2 Fuzzy Initial Value Problem FIVP ............... 180 8.1.3 Generalized Fuzzy Initial Value Problem GFIVP.... 184 8.2 Discrete Fuzzy Dynamic System ........................ 195 8.2.1 Discrete Fuzzy Malthusian Model................. 195 8.2.2 Discrete Fuzzy Logistic Model................... 198 References............................................... 201 9 Modeling in Biomathematics: Demographic Fuzziness........... 205 9.1 Demographic Fuzziness: Discrete Modeling................ 207 9.1.1 Fuzzy Rules with Opposite Semantics.............. 208 9.1.2 Equilibrium and Stability of One-Dimensional Discrete p-Fuzzy Systems....................... 211 9.1.3 Discrete p-Fuzzy Predator-Prey Model ............. 216 9.2 Demographic Fuzziness: Continuous Modeling ............. 219 9.2.1 Characteristics of a Continuous p-Fuzzy Systems..... 219 9.2.2 Numerical Methods for the Solution of the Continuous p-Fuzzy System IVP.................. 220 9.2.3 A Study of Montroll’s p-Fuzzy Model ............. 223 9.3 Bi-Dimensional Models: Predator-Prey and p-Fuzzy Lotka–Volterra...................................... 227 9.3.1 Predator-Prey p-Fuzzy Model .................... 228 References............................................... 235 10 Biomathematical Modeling in a Fuzzy Environment ............ 237 10.1 Life Expectancy and Poverty ........................... 238 10.1.1 The Model................................... 238 10.1.2 Statistical Expectation: E½nðtÞ.................... 239 10.1.3 Fuzzy Expectation Value: FEV nðtÞ nð0Þ hi ............... 241 Contents xiii

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10.1.4 Application: Life Expectancy of a Group of Metal Workers in Recife Pernambuco - Brazil..... 242 10.1.5 Comparisons of the Statistical Expected Value and the Fuzzy Expected Value ................... 245 10.2 The SI Epidemiological Model.......................... 247 10.2.1 The Fuzzy SI Model........................... 248 10.2.2 Expected Value of the Number of Infected Individuals................................... 250 10.2.3 Statistical Expected Values of the Number in Infected................................... 253 10.2.4 IðFEV½V tÞ Versus FEV½IðVtÞ ................. 256 10.2.5 Control of Epidemics and the Basic Reproductive Number.......................... 257 10.3 A Fuzzy Model of the Transference from Asymptomatic to Symptomatic in HIV þ Patients ....................... 258 10.3.1 The Classical Model........................... 259 10.3.2 The Fuzzy Model ............................. 260 10.3.3 The Fuzzy Expectation of the Asymptomatic Individuals................................... 261 10.4 Population Dynamics and Migration of Blow Flies .......... 264 References............................................... 268 11 End Notes .............................................. 271 11.1 Subtration of Interactive Fuzzy Numbers.................. 272 11.1.1 Difference Between Fuzzy Numbers............... 274 11.2 Prey-Predator ....................................... 277 11.2.1 Prey-Predator with the Minimum t-Norm ........... 278 11.2.2 Prey-Predator with the Hamacher t-Norm........... 278 11.3 Epidemiological Model................................ 280 11.3.1 SI Model with Minimum t-Norm ................. 281 11.3.2 SI Model with Hamacher t-Norm ................. 282 11.4 Takagi–Sugeno Method to Study the Risk of Dengue ........ 284 11.4.1 Takagi–Sugeno Model.......................... 284 11.4.2 Dengue Risk Model............................ 285 11.4.3 Simulations.................................. 288 11.4.4 Final Considerations ........................... 289 11.5 The SI-model with Completely Correlated Initial Conditions .................................... 290 References............................................... 293 Index ...................................................... 297 xiv Contents

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About the Authors Laécio Carvalho de Barros is Professor of Applied Mathematics at the Institute of Mathematics Statistics and Computational Sciences the University of CampinasandholdsaPh.D.degreeinAppliedMathematicsfromtheUniversityof Campinas São Paulo Brazil in 1997. He is the co-author of the book Fuzzy Logic in Action: Applications in Epidemiology and Beyond Studies in Fuzziness andSoft Computing Vol. 232 2008 Springer-Verlag Berlin Heidelberg and of the book Fuzzy Differential Equations in Various Approaches SpringerBriefs in Mathematics Number 1 2015 Springer International Publishing. His current research interests include modeling of biological phenomena fuzzy sets theory and fuzzy dynamical systems. Moreover he has taught fuzzy mathematical modeling and fuzzy set theory classes for over 15 years to both undergraduate and graduate students. Rodney Carlos Bassanezi is Professor Emeritus of Applied Mathematics at the Institute ofMathematicsStatistics andComputational Sciences at the Universityof Campinas starting his university teaching career there in 1969. He received a Ph.D. degree in Mathematics from the University of Campinas in 1977. He held post- doctoral and research positions at the Libera Universitad di Trento Italy 1981 19851990and1993.Hisresearchactivitiescovermathematical analysisminimal surfaces biomathematics and fuzzy dynamical systems. He has published some books in Portuguese notably one textbook on differential equations 1988 one textbook on mathematical modeling 2002 as well as an introduction to calculus and applications 2015. He has been the president of the Sociedade Latino-Americano de Biomatemática 1999–2001 and the coordinator of the graduate program in mathematics at the Federal University ABC in São Paulo. He has directed 55 Masters and 21 Ph.D. theses and his students have been teaching throughout Latin American. xv

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Weldon Alexander Lodwick is Professor of Mathematics at the University of Colorado Denver. He holds a Ph.D. degree in Mathematics 1980 from Oregon State University He is the co-editor of the book Fuzzy Optimization: Recent Developments and Applications Studies in Fuzziness and Soft Computing Vol. 254 Springer-Verlag Berlin Heidelberg 2010 and the author of the monograph Interval and Fuzzy Analysis: A Unified Approach in Advances in Imaging and Electronic Physics Vol. 148 pp. 76–192 Elsevier 2007. His current research interests include interval analysis distance geometry as well as flexible and gen- eralized uncertainty optimization. Over the last 30 years he has taught applied mathematical modeling classes to undergraduate and graduate students on topics such as radiation therapy of tumor fuzzy and possibilistic optimization modeling molecular distance geometry problems and neural networks applied to control problems. xvi About the Authors

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Chapter 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling Man is the measure of all things: of things which are that they are and of things which are not that they are not. Protagoras – 5 th Century BCE Abstract This chapter presents a brief discussion about uncertainty based on philo- sophical principles mainly from the point of view of the pre-Socratic philosophers. Next the notions of fuzzy sets and operations on fuzzy sets are presented. Lastly the concepts of alpha-level and the statement of the well-known Negoita-Ralescu Representation Theorem the representation of a fuzzy set by its alpha-levels are discussed. 1.1 Uncertainty in Modeling and Analysis The fundamental entity of analysis for this book is set a collection of objects. A second fundamental entity for this book is variable. The variable represents what one wishes to investigate by a mathematical modeling process that aims to quantify it. In this context the variable is a symbolic receptacle of what one wishes to know. The quantification process involves a set of values which is ascribed a-priori. Thus when one talks about a variable being fuzzy a real-number a random number and so on one is ascribing to the variable its attribution. A set also has an existence or context. That is when one is in the process of creating a mathematical model one ascribes to sets attributions associated with the model or problem at hand. One speaks of a set being a classical set a fuzzy set a set of distributions a random sets and so on. Given that models of existent problems or conditions are far from ideal deterministic mathematical entities we are interested in dealing directly with associated inexactitudes and so ascribe to our fundamental objects of modeling and analysis properties of determinism exactness and non- determinism inexactness. This book is about processes in which uncertainty both in the input or data side and in the relational structure is inherent to the problem at hand. Social and biological © Springer-Verlag Berlin Heidelberg 2017 L.C. de Barros et al. A First Course in Fuzzy Logic Fuzzy Dynamical Systems and Biomathematics Studies in Fuzziness and Soft Computing 347 DOI 10.1007/978-3-662-53324-6_1 1

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2 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling the modeling are characterized by such uncertainties. The mathematical theory on which we focus to enable modeling with uncertainty occurring in biological and social systems is fuzzy set theory first developed by L. Zadeh 1. Uncertainty has long been a concern of researchers and philosophers alike throughout the ages as it is to us in this present book. The pursuit of the truth of what is of what exists which is one aspect of uncertainty if we characterize truth or existence certainty has been debated since the dawn of thinking. In ancient Greece individuals and schools explicitly asked the question: “What exists Is everything in transformation or is there permanence” These are two dimensions of thought and can be considered completely separate issues and even contradictory issues. The pre-Socratic philosophers tried to make statements summarizing their thoughts about the Universe in an attempt to explain what is existent in the uni- verse. In the words of Heraclitus of Ephesus 6th to 5th Century BCE “panta hei” which means “everything flows everything changes”. By way of illustration con- sider a situation in which a river is never the same one cannot bathe in the same river twice. Cratylus his disciple took Heraclitus’ thoughts to the extreme by saying that we cannot bathe in the river even once because if we assign an identity to things or give them names we are also giving stability to these things which in his view are undergoing constant change. The Eleatic school in contrast to Heraclitus questions the existence of motion or change itself. According to Parmenides of Elea 6th to 5th Century BCE: “the only thing that exists is the being - which is the same as thinking”. Zeno his main follower denies that there is motion as this was understood at the time by giving his famous paradox of Achilles and the turtle 2. The Sophists interpret what Parmenides said as the impossibility of false rhetoric. According to Protagoras 5th Century BCE: “Man is the measure of all things”. There is no absolute truth or falsehood. In the Sophists’ view humankind must seek solutions in the practical. The criterion of true or false is related to the theoretical and must therefore be replaced by more practical patterns related to the concepts of better or worse. Rhetoric is the way to find such patterns. Most of the pre-Socratic philosophers with the exception of Heraclitus believed there was something eternal and unchanging behind the coming-to-be that which is in the process of being of becoming that was the eternal source the foundation of all beings. According to Thales it was water in the opinion of Anaximenes the air Pythagoras thought it was numbers and Democritus believed that this source lay in the atoms and in the void. This eternal something which was unchangeable and which held all things was called by the Greeks arche. Certainty and uncertainty were widely discussed by Greek philosophers. The Sophists a term derived from sophistes sages were known to teach the art of rhetoric. Protagoras the most important Sophist along with Górgias taught students how to turn weaknesses of argument into strengths. Rhetoric for the Sophists is a posture or attitude with respect to knowledge that has a total skepticism in relation to any kind of absolute knowledge. This no matter how things are is because everything is relative and also depends on who gives judgement about them. Górgias said that rhetoric surpasses all other arts being the best because it makes all things submit

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1.1 Uncertainty in Modeling and Analysis 3 to spontaneity rather than to violence. As is well known Socrates confronted the Sophists of his day with the question: “What is” That is if everything is relative what exists Plato a disciple of Socrates initially shared the ideas of Heraclitus that every- thing is changing the flow of coming-to-be everything was in process. However if everything was in motion then knowledge would not be possible. To avoid falling back into skepticism Plato thought of a “world of ideas”. Around this world there would be changes and things would be eternal beyond the space-time dimension. The so-called “sensory world” which is the world as perceived by the five senses would then come into being. It would be true that the “world of ideas” would be behind the coming-to-be of this “sensory world”. For Plato the most important thing was not the final concept but the path taken to reach it. The “world of ideas” is not accessible by the senses but rather just by intuition while intellectual dialectics is the movement of asceticism in pursuit of the truth. Therefore Plato promotes a synthesis between Heraclitus and Parmenides. On the other hand for Aristotle the world of ideas and essences is not contained in things themselves. Universal knowledge is linked to its underlying logic the Logos the same reason the principle of order and study of the consequences and also the syllogism which is the formal mechanism for deduction. Based on certain general assumptions knowledge must strictly follow an order using the concept of the demonstrative syllogism. In short and perhaps naively we think that the most important difference between Aristotle and the Sophists is the fact that for Aristotle there is an eternal an immutable independent of human beings while the Sophists consider that there is no eternal and absolute truth but rather just the knowledge obtained from our senses. For Plato and Aristotle respectively dialectics and syllogisms are to be used in the quest for the truth. The Sophists consider that rhetoric the art of persuasion is convincing in relation to the search for the truth because truth does not exist as an absolute. Understanding that subjectivity imprecision uncertainty are inherent to certain terms of language Górgias denied the existence of absolute truth: even if absolute truth existed it would be incomprehensible to man even if it were comprehensible to one man it would not be communicable to others. In order to stimulate our thought about this aspect of the uncertainty of language we will try to reach a compromise between the positions of the Sophists on the one hand and Plato and Aristotle on the other by means of a simple example. It is common practice to propose a meeting with another person by saying some- thing like “Let’s meet at four o’clock”. Well the abstract concept of “four o’clock” indicating a measurement of time shows a need to establish communication in the abstract and also enable the holding of the event our meeting. If this were not the case how should we then communicate our meeting - A point for Plato. On the other hand if we take this at face value the meeting would never take place as our respective clocks would never reach four o’clock simultaneously even if they had been synchronized as we could not get to the point marked in hours minutes seconds and millionths of seconds. A point for Górgias. Admitting that we often

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4 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling carry out our commitments at the appointed time and place it looks like we equally need abstract truths and practical standards of a sensible world. We articulated the thoughts above in order to point out the difficulty of talking about certainty or uncertainty and of fuzzy or determinism. If we look in a dictionary for terms synonymous with uncertainty we find for example: subjectivity inaccu- racy randomness doubt ambiguity and unpredictably among others. Historically researchers from what we have noticed have in their quantitative treatment made distinctions between the different types of uncertainty. The uncertainty arising from the randomness of events has been well documented and now occupies a prominent position in the gallery of mathematics in probability theory. Quantum physics has used stochastic theories and a series of formulae now try to explain the “relation- ships of uncertainty”. One of the most widely known of these is the Uncertainty Principle devised by the physicist W. Heisenberg 1927 which relates the position and the velocity momentum of a particle. In a nutshell Heisenberg’s Uncertainty Principle says that one cannot know simultaneous for certain the exact position and speed momentum of a subatomic particle. One can know one or the other but not both. Unlike randomness some variables used in our daily lives and which are perfectly understood when transmitted linguistically between partners have always remained outside the scope of traditional mathematical treatment. This is the case of some lin- guistic variables that have arisen from the need to distinguish between qualifications through a grading system. To describe certain phenomena within the sensible world we have used degrees that represent qualities or partial truths or “better standards” to use Sophist language. This is the case for example with such concepts as tall heavy smoker or infections. This kind ambiguity in language which is a type of uncertainty in terms of its precise meaning since these terms are by their very nature imprecise is from a linguistic point of view a flexibility regarding what elements belong to the category/set tall heavy smoker. Moreover the main contribution that fuzzy logic made and is making is to the mathematical analysis of fuzzy sets these vague flexible open concepts. Fuzzy logic gives precision to imprecise linguistic terms so that mathematical analysis of these flexible categories is meaningful. In language usage we could refer to the sets of tall people smokers or infections. These are typical examples of “sets” whose boundaries can be considered transitional flexible vague since they are defined through subjective or flexible properties or attributes. Let’s consider the example of tall people. To make a formal mathematical repre- sentation of this set we could approach it in at least two different ways. The first is the classical approach establishing a height above which a person could be considered tall. In this case the set is well-defined. The second and less conventional approach to this issue would be that of considering all people as being tall with greater or less extent that is there are people who are more or less tall or not tall at all. This means that the less tall the individual the lower the degree of relevance to this class. We can therefore say that all people belong to the set of tall people with greater or less extent. This latter approach is what we intend to discuss in our book. It was from such notions where the defining characteristics or properties of the set is flexible transitional open that fuzzy theory appeared. Fuzzy set theory has grown consider-

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1.1 Uncertainty in Modeling and Analysis 5 ably since it was introduced in 1965 both theoretically and in diverse applications especially in the field of technology - microchips. The word “fuzzy” is of English origin and means see Concise Oxford English Dictionary11 th edition indistinct or vague. Other meanings include blurred hav- ing the nature or characteristic of fuzzy. Fuzzy set theory was introduced in 1965 by Lotfi Asker Zadeh 1 an electrical engineer and researcher in mathematics com- puter science artificial intelligence who initially intended to impart a mathematical treatment on certain subjective terms of language such as “about” and “around” among others. This would be the first step in working towards programming and storing concepts that are vague on computers making it possible to perform calcu- lations on vague or flexible entities as do human beings. For example we are all unanimous in agreeing that the doubling of a quantity “around 3” results in another “around 6 ”. The formal mathematical representation of a fuzzy set is based on the fact that any classic subset can be characterized by a function its characteristic function as follows. Definition 1.1 Let U be a non-empty set and A a subset of U. The characteristic function of A is given by: χ A x 1if x ∈ A 0if x / ∈ A for all x ∈ U. In this context a classical subset A of U can uniquely be associated with its characteristic function. So in the classical case we may opt for using the language from either “set theory” or “function theory” depending on the problem at hand. Note that the characteristic function χ A : U →0 1 of the subset A shows which elements of the universal set U are also elements of A where χ A x 1 meaning that the element x ∈ A while χ A x 0 means that x is not element of A. However there are cases where an element is partially in a set which means we cannot always say that an element completely belongs to a given set or not. For example consider the subset of real numbers “near 2”: Ax ∈ R : x is near 2. Question. Does the number 7 and the number 2.001 belong to A The answer to this question is not no/yes so is uncertain from this point of view because we do not know to what extent we can objectively say when a number is near 2. The only reasonable information in this case is that 2.001 is nearer 2 than 7. We now start the mathematical formalization of fuzzy set theory that shall be addressed in this text starting with the concept of fuzzy subsets.

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6 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling 1.2 Fuzzy Subset Allowing leeway in the image or range set of the characteristic function of a set from the Boolean set 0 1 to the interval 0 1 Zadeh suggested the formalization of the mathematics behind vague concepts such as the case of “near 2” using fuzzy subsets. Definition 1.2 Let U be a classic universal set. A fuzzy subset F of U is defined by a function ϕ F called the membership function of F ϕ F : U −→ 0 1. The subscript F on ϕ identifies the subset F in this case and the function ϕ F is the analogue of the characteristic function of the classical subset as defined in Definition 1.1 above. The value of ϕ F x ∈ 0 1 indicates the degree to which the element x of U belongs to the fuzzy set F ϕ F x 0 and ϕ F x 1 respectively mean x for sure does not belong to fuzzy subset F and x for sure belongs to the fuzzy subset F. From a formal point of view the definition of a fuzzy subset is obtained simply by increasing the range of the characteristic function from 0 1 to the whole interval 0 1. We can therefore say that a classical set is a special case of a fuzzy set when the range of the membership function ϕ F is restricted to 0 1⊆0 1 that is the membership function ϕ F retracts to the characteristic function χ F . In fuzzy language a subset in the classic sense is usually called a crisp subset. A fuzzy subset F of U can be seen as a standard classic subset of the Cartesian product U ×0 1. Moreover we can identify a fuzzy subset F of U with the set of ordered pairs i.e. the graph of ϕ F : xϕ F x : with x ∈ U. The classic subset of U defined below supp F x ∈ U : ϕ F x 0 is called the support of F and has a fundamental role in the interrelation between classical and fuzzy set theory. Interestingly unlike fuzzy subsets a support is a crisp set. Figure 1.1 illustrates this fact. It is common to denote a fuzzy subset say F in fuzzy set literature not by its membership function ϕ F but simply by the letter F. In this text we have decided to distinguish between F and ϕ F . In classical set theory whenever we refer to a particular set A we are actually considering a subset of a universal set U but for the sake of simplicity or convenience we say set A even though set A is actually a subset. The fuzzy set literature also uses of these terms. This text will use both terms interchangeably. We now present some examples of fuzzy subsets.

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1.2 Fuzzy Subset 7 ϕ F F χ F F supp U 11 b crisp subset U F supp a fuzzy subset Fig. 1.1 Illustration of subsets fuzzy and crisp Example 1.1 Even numbers Consider the set of natural even numbers: E n ∈ N : n is even. This set E has characteristic function which assigns to any natural number n the value χ E n 1if n is even and χ E n 0if n odd. This means that the set of even numbers is a particular fuzzy set of the set of natural numbers N since χ E n ∈ 0 1 in particular χ E n ϕ E n 1if n is even 0 otherwise. In this case it was possible to determine all the elements of E in the domain of the universal set N of natural numbers because every natural number is either even or odd. However this is not the case for other sets with imprecise boundaries. Example 1.2 Numbers near 2 Consider the following subset F of the real num- bers near of 2: F x ∈ R : x is near 2. We can define the function ϕ F : R −→ 0 1 which associates each real value x proximity to point 2 using the expression ϕ F x 1−|x − 2| if 1 ≤ x ≤ 3 0if x / ∈ 1 3 x ∈ R. In this case the fuzzy subset F of points near 2 characterized in ϕ F is such that ϕ F 2.001 0.999 and ϕ F 7 0. We say that x 2.001 is near to 2 with proximity degree 0.999 x 7 is not near 2.

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8 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling On the other hand in the example above one may suggest a different membership function to show proximity to the value 2. For example if the closeness proximity function were defined by ν F x exp −x − 2 2 with x ∈ R then the elements of the fuzzy set F characterized by the function ν F as above have different degrees of belonging from ϕ F : ν F 2.001 0.99999 and ν F 7 1.388 × 10 −11 . We can see that the notion of proximity is subjective and also depends on the membership function which can be expressed in countless different ways depending on how we wish to evaluate the idea of a “nearness”. Note that we could also define the concept “numbers near 2” by a classic set with membership functionϕ F considering for example a sufficiently small value of and the characteristic function for the interval 2 − 2 + by following expression: ϕ F x 1if |x − 2| 0if |x − 2|≥ . Note that being close to 2 means being within a preset neighborhood of 2. The element of subjectivity lies in the choice of the radius of the neighborhood consid- ered.In this specific case all the values within the neighborhood are close to 2 with the same degree of belonging which is 1. Example 1.3 Small natural numbers Consider the fuzzy subset F containing the small natural numbers F n ∈ N : n is small. Does the number 0 zero belong to this set What about the number 1.000 In the spirit of fuzzy set theory it could be said that both do indeed belong to Fbut with different degrees depending on the membership function ϕ F with respect to the fuzzy set F. The membership function associated with F must be built in a way that is consistent with the term “small” the context of the problem and the application mathematical model. One possibility for the membership function of F would be ϕ F n 1 n + 1 n ∈ N. Therefore we could say that the number 0 zero belongs to F withadegreeof belonging equal to ϕ F 0 1 while 999 also belongs to F albeit with a degree of belonging equal to ϕ F 999 0.001. It is clear that in this case the choice of the function ϕ F was made in a some- what arbitrary fashion only taking into account the meaning of “small”. To make a mathematical model of the “small natural number” notion and thus to link F to a

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1.2 Fuzzy Subset 9 membership function we could for example choose any monotonically decreasing sequence starting at 1 one and converging to 0 zero as ϕ n n∈N with ϕ 0 1. For example ϕ F n e −n ϕ F n 1 n 2 + 1 ϕ F n 1 ln n + e . The function to be selected to represent the fuzzy set considered depends on several factors related to the context of the problem under study. From the standpoint of strict fuzzy set theory any of the previous membership functions can represent the subjective concept in question. However what should indeed be noted is that each of the above functions produces a different fuzzy set according to Definition 1.2. The examples we have presented above possess a universal set U for each fuzzy set that is clearly articulated. However this is not always the case. In most cases of interest for mathematical modeling the universal set needs to be delineated and in most instances the support set as well. Let’s illustrate this point with a few more examples. Example 1.4 Fuzzy set of young people Y Consider the inhabitants of a specific city. Each individual in this population can be associated to a real number corre- sponding to their age. Consider the whole universe as the ages within the interval U 0 120 where x ∈ U is interpreted as the age of a given individual. A fuzzy subset Y of young people of this city could be characterized by the following two membership functions for young Y 1 Y 2 according different experts: ϕ Y 1 x ⎧ ⎪ ⎨ ⎪ ⎩ 1if x ≤ 10 80 − x 70 if 10 x ≤ 80 0if x 80 ϕ Y 2 x ⎧ ⎨ ⎩ 40 − x 40 2 if 0 ≤ x ≤ 40 0if40 x ≤ 120 . In the first case Y 1 the support is the interval 0 80 and in the second case Y 2 the support is 0 40. The choice of which function to be used to represent the concept of young people relies heavily on the context or analysis. Undoubtedly about to retire professors would choose Y 1 . Note that the choice of U 0 120 as the interval for the universal set is linked to the fact that we have chosen to show how much

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10 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling an individual is young and our knowledge that statistically in the world no one has lived beyond 120. If another characteristic were to be adopted such as the number of grey hairs to indicate the degree of youth the universe would be different as well as the support. The next example shows a bit more about fuzzy set theory in the mathematical modeling of “fuzzy concepts”. In this example we shall present a mathematical modeling treatment that allows the quantification and exploration of a theme of important social concern poverty. This concept could be modeled based on a variety of appropriate variables calorie intake consumption of vitamins iron intake the volume of waste produced or even the income of every individual among many other features that are possible. However we have chosen to represent poverty assuming that the only variable is income level. A possible mathematical model for poverty is shown below. Example 1.5 Fuzzy subset of the poor Consider that the concept of poor is based on the income level r. Hence it is reasonable to assume that when you lower the income level you raise the level of poverty of the individual. This means that the fuzzy subset A k of the poor in a given location can be given by the following membership function: ϕ A k r ⎧ ⎨ ⎩ 1 − r r 0 2 k if r ≤ r 0 0if r r 0 . The parameter k indicates a characteristic of the group we are considering. This parameter might indicate such things as the environment in which the individual people are situated. The parameter value r 0 is the minimum income level believed to be required to be out of poverty. As illustrated in Fig. 1.2 above we have that if k 1 ≥ k 2 then ϕ A k 1 r ≤ ϕ A k 2 r which means that an individual group in k 1 with an income level ¯ r would be poorer for this income level were the individual in group k 2 . We can also say that in terms of income it is easier to live in the places where k is greatest. So intuitively k shows Fig. 1.2 The membership function of the fuzzy subset of “poor” r r A k 1 ϕ r 0 r A k 2 ϕ r

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1.2 Fuzzy Subset 11 whether the environment where the group lives is less or more favorable to life. The parameter k may give an idea of the degree of saturation that a group has on the environment and therefore can be considered as an environmental parameter. 1.3 Operations with Fuzzy Subsets This section presents the typical operations on fuzzy sets such as union intersection and complementation. Each one of these operations is obtained from membership functions. Let A and B be two fuzzy subsets of U with their respective membership functions ϕ A and ϕ B . We say that A is a fuzzy subset of B and write A ⊂ B if ϕ A x ≤ ϕ B x for all x ∈ U. Remember that the membership function of the empty set ∅ is given by ϕ ∅ x 0 while the universal set U has membership function ϕ U x 1 for all x ∈ U. Hence we can say that∅⊂ A and A ⊂ U for all A. Definition 1.3 Union The union between A and B is the fuzzy subset of U whose membership function is given by: ϕ A∪B x maxϕ A xϕ B x x ∈ U. We note that this definition is an extension of the classic case. In fact when A and B are classics subsets of U have: maxχ A xχ B x 1if x ∈ A or x ∈ B 0if x / ∈ A and x / ∈ B 1if x ∈ A ∪ B 0if x / ∈ A ∪ B χ A∪B x x ∈ U. Definition 1.4 Intersection The intersection between A and B is the fuzzy subset of U whose membership function is given by the following equation: ϕ A∩B x minϕ A xϕ B x x ∈ U. Definition 1.5 Complement The complement of A is the fuzzy subset A in U whose membership function is given by: ϕ A x 1 − ϕ A x x ∈ U. Exercise 1.1 Suppose that A and B are classic subsets of U. 1. Check that

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12 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling minχ A xχ B x 1if x ∈ A ∩ B 0if x / ∈ A ∩ B. 2. Check that χ A∩B x χ A x χ B x. Note that this identity is does not hold in cases where A and B are fuzzy subsets. 3. Check that χ A∩A x 0 A ∩ A ∅ and that χ A∪A x 1 A ∪ A U for all x ∈ U. Unlike the classical situation in the fuzzy context see Fig. 1.3 we can have: • ϕ A∩A x 0 ϕ ∅ x which means that we may not have A ∩ A ∅ • ϕ A∪A x 1 ϕ U x which means that we may not have A ∪ A U. In the following example we intend to exploit the special features presented by the concept of the complement of a fuzzy set. Example 1.6 Fuzzy set of the elderly The fuzzy set O of the elderly the old should reflect a situation opposite of young people given above when considering the ages that belong to O. While youth membership functions should decrease with age the elderly should be increase with age. One possibility for the membership function of O is: ϕ O x 1 − ϕ Y x where ϕ Y is the membership function of the fuzzy subset “young”. Therefore the fuzzy set O is the complement of fuzzy Y . In this example if we take the set of young people Y 1 as having the membership function mentioned in the first part of Example 1.4 then: ’ A U B U B A ∩ U B ∪ A U A U c b a Fig. 1.3 Operations with fuzzy subsets: a union b intersection and c complement

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1.3 Operations with Fuzzy Subsets 13 Fig. 1.4 Fuzzy subsets of young and elderly ϕ J ϕ age young: elderly: I 10 20 30 40 50 60 70 80 90 100 110 120 1 ϕ O x 1 − ϕ Y 1 x ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0if x ≤ 10 x − 10 70 if 10 x ≤ 80 1if x 80 . A graphical representation for O and Y 1 is shown in Fig. 1.4. Note that this operation complement exchanges degrees of belonging for the fuzzy subsets of O and Y 1 . This property characterizes the fuzzy complement which means that whileϕ A x represents the degree of compatibility of x with the linguistic concept in question ϕ A x shows the incompatibility of x with the same concept. One consequence of the imprecision of fuzzy sets is that there is a certain overlap of a fuzzy set with its complement. In Example 1.6 an individual who belongs to the of fuzzy set young with grade 0.8 also belongs to its complement O with grade 0.2. Also note that it is quite possible for a member to belong to one set and also its complementary set with the same degree of belonging in Fig. 1.4 this value is 45 showing that the more doubt we have about an element belonging to the set the nearer to 0.5 is the degree of belong to this set. That is the closer to a 0.5 membership value an element is the greater the doubt of whether or not this element belongs to the set. The degree 0.5 is the maximum doubt greatest entropy. This is a major difference from classical set theory in which an element either belongs to a set or to its complement these being mutually exclusive and there is absolutely no doubt. Here it must also be noted that we have defined young and elderly old which are admittedly linguistic terms of opposite meanings through the use of fuzzy sets that are not necessarily complementary. For example we could have used ϕ Y 1 : ϕ Y 2 x ⎧ ⎪ ⎨ ⎪ ⎩ 40 − x 40 2 if 0 ≤ x ≤ 40 0if40 x ≤ 120

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14 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling in which case we could have obtained ϕ O x ⎧ ⎪ ⎨ ⎪ ⎩ x − 40 80 2 if 40 x ≤ 120 0if x ≤ 40. Exercise 1.2 Assume that the fuzzy set for young people Y is given by ϕ Y x ⎧ ⎪ ⎨ ⎪ ⎩ 1 − x 120 2 4 if x ∈ 10 120 1if x / ∈ 10 120 . 1. Define a fuzzy set for the elderly. 2. Determine the age of an individual considered of middle age which means grade 0.5 both in terms of youth and of elderly old age assuming that the fuzzy set of the elderly is the complement to that of the young. 3. Draw the graph of the young and elderly old for part 2 and then compare it with Example 1.6. We will next extend the concept to the complement for A ⊆ B where A is a fuzzy subset of fuzzy set B and both in relation to the universe U. In this case the complement of A in relation to B is denoted by the fuzzy set A B which has the following membership function: ϕ A B x ϕ B x − ϕ A x x ∈ U. Note also that the complement of A in relation to U is a particular case of the complement of A in B since ϕ U x 1. In the following example we shall try to further exploit the concept of the ideas of complements with fuzzy subsets as defined in Example 1.5. Example 1.7 Fuzzy set of the poor revisited If the environment in which a group lives suffers any kind of degradation from what we saw in Example 1.5 this results in a decreased environmental parameter declining from k 1 to a lower value k 2 so that the individual having income level r in k 1 has degree of poverty ϕ A k 1 r less than that of another ϕ A k 2 r with the same income r in k 2 . That is ϕ A k 1 r ϕ A k 2 r ⇔ A k 1 ⊂ A k 2 . Such a change could lead to the poverty level of a pauper represented by A k 2 . The fuzzy complement of A k 1 in A k 2 is the fuzzy subset given by A A k 2 .

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1.3 Operations with Fuzzy Subsets 15 This set is not empty and its membership function is given by ϕ A A k 2 ϕ A k 2 r − ϕ A k 1 r r ∈ U. A recompense to the group that has suffered such a fall should be that of the same status of poverty as before. That is given an income of r 1 the group should have an income of r 2 after the fall which means that ϕ A k 2 r 2 − ϕ A k 1 r 1 0. Therefore r 2 − r 1 0 and the recompense should be r 2 − r 1 see Fig. 1.5. We shall now make some brief comments and also look at the consequences of the major operations between fuzzy sets. If A and B are sets in the classical sense then the characteristic functions of their operations also satisfy the definitions defined for the fuzzy case showing coher- ence between such concepts. For example if A is a classic subset of U then the characteristic function χ A x of its complement is such that χ A x 0if χ A x 1 ⇔ x ∈ A χ A x 1if χ A x 0 ⇔ x / ∈ A. In this case either x ∈ A or x / ∈ A while the theory of fuzzy sets does not necessarily have this dichotomy. As seen in the Example 1.6 it is not always true that A∩ A ∅ for fuzzy sets and it may not even true that A ∪ A U. The following example reinforces these facts. Example 1.8 Fuzzy sets of fever and/or myalgia muscular rheumatism Let’s sup- pose that the universal set U is the set of all patients within a clinic identified by numbers 1 2 3 4 and 5. Let A and B be fuzzy subsets that represent patients with fever and myalgia respectively. Table 1.1 shows the operations union intersection and complement. 2 r 1 r r 0 r 1 ϕ A k 1 ϕ A k 2 ϕ A k 1 r 1 ϕ A k 2 r 2 Fig. 1.5 Recompense for changing in environment

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16 1 Fuzzy Sets Theory and Uncertainty in Mathematical Modeling Table 1.1 Illustration of operations between fuzzy subsets Patient Fever: A Myalgia: B A ∪ B A ∩ B A A ∩ A A ∪ A 1 0.7 0.6 0.7 0.6 0.3 0.3 0.7 2 1.0 1.0 1.0 1.0 0.0 0.0 1.0 3 0.4 0.2 0.4 0.2 0.6 0.4 0.6 4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 5 1.0 0.2 1.0 0.2 0.0 0.0 1.0 The values in all columns except the first show the degree to which each patient belongs to the fuzzy sets A B A∪ B A∩ B A A∩ A A∪ A respectively where A and B are hypothetical data. In the column A ∩ A the valueof0.3 shows that patient number 1 is both in the first group of patients with a fever as well as in the group with non-fever. As we have seen this is a fact that would not be possible in classical set theory in which there is the exclusion law by which any set and its complement are mutually exclusive A ∩ A ∅. The fuzzy subsets A and B of U are equal if their membership functions are identical that is if ϕ A x ϕ B x for all x ∈ U. Below is listed the main properties of the operations as defined in this section. Proposition 1.1 The operations between fuzzy subsets satisfying the following prop- erties: • A ∪ B B ∪ A • A ∩ B B ∩ A • A ∪ B ∪ C A ∪ B ∪ C • A ∩ B ∩ C A ∩ B ∩ C • A ∪ A A • A ∩ A A • A ∪ B ∩ C A ∪ B ∩ A ∪ C • A ∩ B ∪ C A ∩ B ∪ A ∩ C • A∩∅ ∅ and A∪∅ A • A ∩ U A and A ∪ U U • A ∪ B A ∩ B and A ∩ B A ∪ B DeMorgan sLaw . Proof The proof of each property is an immediate application of the properties between maximum and minimum functions which means ⎧ ⎪ ⎨ ⎪ ⎩ max ϕxψ x 1 2 ϕx + ψ x+|ϕx − ψ x | min ϕxψ x 1 2 ϕx + ψ x−|ϕx − ψ x |. where ϕ and ψ are functions with image in R. We will only prove one of De Morgan’s laws because the other properties have similar proofs. If we consider that ϕ A is the

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