slide 1: 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
slide 2: Studies in Fuzziness and Soft Computing
Volume 347
Series editor
Janusz Kacprzyk Polish Academy of Sciences Warsaw Poland
e-mail: kacprzykibspan.waw.pl
slide 3: 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
slide 4: 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
slide 5: 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
slide 6: 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.
slide 7: 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
slide 8: Acknowledgment
The authors would like to acknowledge and thank the partial support received from
CNPq.
ix
slide 9: 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
slide 10: 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
slide 11: 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
slide 12: 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
slide 13: 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
slide 14: 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
slide 15: 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
slide 16: 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
slide 17: 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
slide 18: 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-
slide 19: 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.
slide 20: 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.
slide 21: 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.
slide 22: 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
slide 23: 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
slide 24: 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
slide 25: 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
slide 26: 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
slide 27: 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
slide 28: 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
.
slide 29: 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
slide 30: 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