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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Mathematical Foundations of Fuzzy Sets Introduce yourself to the foundations of fuzzy logic with this easy-to-use guide Many fields studied are defined by imprecise information or high degrees of uncertainty. When this uncertainty derives from randomness, traditional probabilistic statistical methods are adequate to address it; more everyday forms of vagueness and imprecision, however, require the toolkit associated with 'fuzzy sets' and 'fuzzy logic'. Engineering and mathematical fields related to artificial intelligence, operations research and decision theory are now strongly driven by fuzzy set theory. Mathematical Foundations of Fuzzy Sets introduces readers to the theoretical background and practical techniques required to apply fuzzy logic to engineering and mathematical problems. It introduces the mathematical foundations of fuzzy sets as well as the current cutting edge of fuzzy-set operations and arithmetic, offering a rounded introduction to this essential field of applied mathematics. The result can be used either as a textbook or as an invaluable reference for working researchers and professionals. Mathematical Foundations of Fuzzy Sets offers thereader: * Detailed coverage of set operations, fuzzification of crisp operations, and more * Logical structure in which each chapter builds carefully on previous results * Intuitive structure, divided into 'basic' and 'advanced' sections, to facilitate use in one- or two-semester courses Mathematical Foundations of Fuzzy Sets is essential for graduate students and academics in engineering and applied mathematics, particularly those doing work in artificial intelligence, decision theory, operations research, and related fields.
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Table of Contents
Cover
Title Page
Copyright
Preface
1 Mathematical Analysis
1.1 Infimum and Supremum
1.2 Limit Inferior and Limit Superior
1.3 Semi‐Continuity
1.4 Miscellaneous
2 Fuzzy Sets
2.1 Membership Functions
2.2
‐level Sets
2.3 Types of Fuzzy Sets
3 Set Operations of Fuzzy Sets
3.1 Complement of Fuzzy Sets
3.2 Intersection of Fuzzy Sets
3.3 Union of Fuzzy Sets
3.4 Inductive and Direct Definitions
3.5
‐Level Sets of Intersection and Union
3.6 Mixed Set Operations
4 Generalized Extension Principle
4.1 Extension Principle Based on the Euclidean Space
4.2 Extension Principle Based on the Product Spaces
4.3 Extension Principle Based on the Triangular Norms
4.4 Generalized Extension Principle
5 Generating Fuzzy Sets
5.1 Families of Sets
5.2 Nested Families
5.3 Generating Fuzzy Sets from Nested Families
5.4 Generating Fuzzy Sets Based on the Expression in the Decomposition Theorem
5.5 Generating Fuzzy Intervals
5.6 Uniqueness of Construction
6 Fuzzification of Crisp Functions
6.1 Fuzzification Using the Extension Principle
6.2 Fuzzification Using the Expression in the Decomposition Theorem
6.3 The Relationships between EP and DT
6.4 Differentiation of Fuzzy Functions
6.5 Integrals of Fuzzy Functions
7 Arithmetics of Fuzzy Sets
7.1 Arithmetics of Fuzzy Sets in
7.2 Arithmetics of Fuzzy Vectors
7.3 Difference of Vectors of Fuzzy Intervals
7.4 Addition of Vectors of Fuzzy Intervals
7.5 Arithmetic Operations Using Compatibility and Associativity
7.6 Binary Operations
7.7 Hausdorff Differences
7.8 Applications and Conclusions
8 Inner Product of Fuzzy Vectors
8.1 The First Type of Inner Product
8.2 The Second Type of Inner Product
9 Gradual Elements and Gradual Sets
9.1 Gradual Elements and Gradual Sets
9.2 Fuzzification Using Gradual Numbers
9.3 Elements and Subsets of Fuzzy Intervals
9.4 Set Operations Using Gradual Elements
9.5 Arithmetics Using Gradual Numbers
10 Duality in Fuzzy Sets
10.1 Lower and Upper Level Sets
10.2 Dual Fuzzy Sets
10.3 Dual Extension Principle
10.4 Dual Arithmetics of Fuzzy Sets
10.5 Representation Theorem for Dual‐Fuzzified Function
Bibliography
Mathematical Notations
Index
End User License Agreement
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Begin Reading
Bibliography
Mathematical Notations
Index
End User License Agreement
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Mathematical Foundations of Fuzzy Sets
Hsien‐Chung WuDepartment of MathematicsNational Kaohsiung Normal UniversityKaohsiungTaiwan
This edition first published 2023© 2023 John Wiley and Sons, Ltd
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Library of Congress Cataloging‐in‐Publication Data Applied for:
Hardback ISBN: 9781119981527
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Preface
The concept of fuzzy set, introduced by L.A. Zadeh in 1965, tried to extend classical set theory. It is well known that a classical set corresponds to an indicator function whose values are only taken to be 0 and 1. With the aid of a membership function associated with a fuzzy set, each element in a set is allowed to take any values between 0 and 1, which can be regarded as the degree of membership. This kind of imprecision draws forth a bunch of applications.
This book is intended to present the mathematical foundations of fuzzy sets, which can rigorously be used as a basic tool to study engineering and economics problems in a fuzzy environment. It may also be used as a graduate level textbook. The main prerequisites for most of the material in this book are mathematical analysis including semi‐continuities, supremum, convexity, and basic topological concepts of Euclidean space, . This book presents the current state of affairs in set operations of fuzzy sets, arithmetic operations of fuzzy intervals and fuzzification of crisp functions that are frequently adopted to model engineering and economics problems with fuzzy uncertainty. Especially, the concepts of gradual sets and gradual elements have been presented in order to cope with the difficulty for considering elements of fuzzy sets such as considering elements of crisp sets.
Chapter 1
presents the mathematical tools that are used to study the essence of fuzzy sets. The concepts of supremum and semi‐continuity and their properties are frequently invoked to establish the equivalences among the different settings of set operations and arithmetic operations of fuzzy sets.
Chapter 2
introduces the basic concepts and properties of fuzzy sets such as membership functions and level sets. The fuzzy intervals are categorized as different types based on the different assumptions of membership functions in order to be used for the different purposes of applications.
Chapter 3
deals with the intersection and union of fuzzy sets including the complement of fuzzy sets. The general settings by considering aggregation functions have been presented to study the intersection and union of fuzzy sets that cover the conventional ones such as using minimum and maximum functions (t‐norm and s‐norm) for intersection and union, respectively.
Chapter 4
extends the conventional extension principle to the so‐called generalized extension principle by using general aggregation functions instead of using minimum function or t‐norm to fuzzify crisp functions. Fuzzifications of real‐valued and vector‐valued functions are frequently adopted in engineering and economics problems that involve fuzzy data, which means that the real‐valued data cannot be exactly collected owing to the fluctuation of an uncertain situation.
Chapter 5
presents the methodology for generating fuzzy sets from a nested family or non‐nested family of subsets of Euclidean space
. Especially, generating fuzzy intervals from a nested family or non‐nested family of bounded closed intervals is useful for fuzzifying the real‐valued data into fuzzy data. Based on a collection of real‐valued data, we can generate a fuzzy set that can essentially represent this collection of real‐valued data.
Chapter 6
deals with the fuzzification of crisp functions. Using the extension principle presented in
Chapter 4
can fuzzify crisp functions. This chapter studies another methodology to fuzzify crisp functions using the mathematical expression in the well‐known decomposition theorem. Their equivalences are also established under some mild assumptions.
Chapter 7
studies the arithmetic operations of fuzzy sets. The conventional arithmetic operations of fuzzy sets are based on the extension principle presented in
Chapter 4
. Many other arithmetic operations using the general aggregation functions haven also been studied. The equivalences among these different settings of arithmetic operations are also established in order to demonstrate the consistent usage in applications.
Chapter 8
gives a comprehensive and accessible study regarding inner product of fuzzy vectors that can be treated as an application using the methodologies presented in
Chapter 7
. The potential applications of inner product of fuzzy vectors are fuzzy linear programming problems and the engineering problems that are formulated using the form of inner product involving fuzzy data.
Chapter 9
introduces the concepts of gradual sets and gradual elements that can be used to propose the concept of elements of fuzzy sets such as the concept of elements of crisp sets. Roughly speaking, a fuzzy set can be treated as a collection of gradual elements. In other words, a fuzzy set consists of gradual elements. In this case, the set operations and arithmetic operations of fuzzy sets can be defined as the operations of gradual elements, like the operations of elements of crisp sets. The equivalences with the conventional set operations and arithmetic operations of fuzzy sets are also established under some mild assumptions.
Chapter 10
deals with the concept of duality of fuzzy sets by considering the lower
‐level sets. The conventional
‐level sets are treated as upper
‐level sets. This chapter considers the lower
‐level sets that can be regarded as the dual of upper
‐level sets. The well‐known extension principle and decomposition theorem are also established based on the lower
‐level sets, and are called the dual extension principle and dual decomposition theorem. The so‐called dual arithmetics of fuzzy sets are also proposed based on the lower
‐level sets, and a duality relation with the conventional arithmetics of fuzzy sets is also established.
Finally, I would like to thank the publisher for their cooperation in the realization of this book.
Hsien‐Chung Wu
Department of MathematicsNational Kaohsiung Normal UniversityKaohsiung, Taiwane‐mail 1: [email protected]‐mail 2: [email protected] site: https://sites.google.com/view/hsien‐chung‐wuApril, 2022
1Mathematical Analysis
We present some materials from mathematical analysis, which will be used throughout this book. More detailed arguments can be found in any mathematical analysis monograph.
1.1 Infimum and Supremum
Let be a subset of . The upper and lower bounds of are defined below.
We say that
is an
upper bound
of
when there exists a real number
satisfying
for every
. In this case, we also say that
is bounded above by
.
We say that
is a
lower bound
of
when there exists a real number
satisfying
for every
. In this case, we also say that
is bounded below by
.
The set is said to be unbounded above when the set has no upper bound. The set is said to be unbounded below when the set has no lower bound. The maximal and minimal elements of are defined below.
We say that
is a
maximal element
of
when there exists a real number
satisfying
for every
. In this case, we write
.
We say that
is a
minimal element
of
when there exists a real number
satisfying
for every
. In this case, we write
.
Example 1.1.1
We provide some concrete examples.
The set
is unbounded above. It has no upper bounds and no maximal element. It is bounded below by 0, but it has no minimal element.
The closed interval
is bounded above by 1 and is bounded below by 0. We also have
and
.
The half‐open interval
is bounded above by 1, but it has no maximal element. However, we have
.
Although the set is bounded above by 1, it has no maximal element. This motivates us to introduce the concepts of supremum and infinum.
Definition 1.1.2
Let be a subset of .
Suppose that
is bounded above. A real number
is called a
least upper bound
or
supremum
of
when the following conditions are satisfied.
is an upper bound of
.
If
is any upper bound of
, then
.
In this case, we write . We say that the supremum is attained when .
Suppose that
is bounded below. A real number
is called a
greatest lower bound
or
infimum
of
when the following conditions are satisfied.
is a lower bound of
.
If
is any lower bound of
, then
.
In this case, we write . We say that the infimum is attained when .
It is clear to see that if the supremum is attained, then . Similarly, if the infimum is attained, then .
Example 1.1.3
Let . Then, we have
If , then does not exists. However, we have .
Proposition 1.1.4
Let be a subset of with . Then, given any , there exists satisfying .
Proof.
We are going to prove it by contradiction. Suppose that we have for all . Then is an upper bound of . According to the definition of supremum, we also have . This contradiction implies that for some , and the proof is complete.
Proposition 1.1.5
Given any two nonempty subsets and of , we define by
Suppose that the supremum and are attained. Then, the supremum is attained, and we have
Proof.
We first have
We write and . Given any , there exist and satisfying . Since and , we have , which says that is an upper bound of . Therefore, the definition of says that . Next, we want to show that . Given any , Proposition 1.1.4 says that there exist and satisfying and . We also see that . Adding these inequalities, we obtain
which says that . Since can be any positive real number, we must have . This completes the proof.
Proposition 1.1.6
Let and be any two nonempty subsets of satisfying for any and . Suppose that the supremum is attained. Then, the supremum is attained and .
Proof.
It is left as an exercise.
1.2 Limit Inferior and Limit Superior
Let be a sequence in . The limit superior of is defined by
and the limit inferior of is defined by
Moreover, we can see that
Let
It is clear to see that is a decreasing sequence and is an increasing sequence. In this case, we have
which also says that
and
Some useful properties are given below.
Proposition 1.2.1
Let be a sequence of real numbers. Then, the following statements hold true.
We have
We have
if and only if
The sequence diverges to
if and only if
The sequence diverges to
if and only if
Let
be another sequence satisfying
for all
. Then, we have
Proof.
To prove part (i), from (1.1), we see that for all . Using (1.2) and (1.3), we obtain
To prove part (ii), suppose that
Then, given any , there exists an integer satisfying
which implies
In other words, we have
which also implies
Therefore, we obtain
which implies, by using (1.2) and (1.3),
For the converse, from (1.1) again, we see that for all . Since
Using the pinching theorem, we obtain the desired limit. The remaining proofs are left as exercise, and the proof is complete.
Proposition 1.2.2
Let and be any two sequences in . Then, we have
and
Proof.
For , we have
which says that
Therefore, we obtain
We similarly have
Therefore, we also obtain
This completes the proof.
Proposition 1.2.3
Let be a sequence of subsets of satisfying for all and , and let be a real‐valued function defined on . Then
and
Proof.
Since
It suffices to prove the case of the supremum. Let
Since for all , we have that is a decreasing sequence of real numbers. We also have for all , which implies
Given any , according to the concept of supremum, there exists satisfying
Let . We consider the subsequence defined by in the sense of
Then and for all . Since for all and , the “last term” of the sequence must be in , a claim that will be proved below. Since for all , we have the subsequence , which also implies
where can be regarded as the “last term” and . Since is the supremum of on , it follows that for each . Since for all , we see that for all . Therefore, we obtain
which implies, by (1.7),
Since is any positive number, we obtain
Combining (1.6) and (1.8), we obtain
Since is a decreasing sequence of real numbers, we conclude that
and the proof is complete.
Proposition 1.2.4
Let be a sequence of subsets of satisfying for all and , and let be a real‐valued function defined on . Then
and
Proof.
It suffices to prove the case of the supremum. Let
Since for all , we have that is an increasing sequence of real numbers. We also have for all , which implies
Given any , according to the concept of supremum, there exists satisfying . Since
we have that for some integer . We construct a sequence satisfying for all and for all . Since for all , it follows that for all . Therefore, the sequence satisfies for all and
which means that is the “last term” of the sequence . We also have
Let . We consider the subsequence defined by in the sense of
Then and for all . Since is the “last term” of the sequence , it follows that is also the “last term” of the sequence . Therefore, we have for all , which implies
Since is any positive number, it follows that
Using (1.10), we obtain
Combining (1.9) and (1.11), we obtain
Since is an increasing sequence of real numbers, we conclude that
and the proof is complete.
Given any and in . The Euclidean distance between and is defined by
Given a point , we consider the open ‐ball
The concept of closure based on open balls will be frequently used throughout this book. For the general concept refer to Kelley and Namioka [55]. In this book, we are going to consider the closure of a subset of , which is given below.
Definition 1.2.5
Let be a subset of . The closure of is denoted and defined by
We say that is a closed subset of when .
Remark 1.2.6
Given any , there exists a sequence in satisfying as . In particular, for , we see that as .
Proposition 1.2.7
Let be a subset of , and let be a continuous function defined on . Then
Proof.
It suffices to prove the case of the supremum, since
It is obvious that
Given any , according to the concept of supremum, there exists satisfying
We also see that there exists a sequence in satisfying . Since is continuous on , we also have as . Therefore, we obtain
Since can be any positive number, it follows that
This completes the proof.
Let be a subset of . For and a sequence in , we write to mean that the sequence is increasing and converges to . We also write to mean that the sequence is decreasing and converges to .
Proposition 1.2.8
Let be a subset of . The following statements hold true.
Let
be a right‐continuous function defined on
. Given any fixed
, suppose that there exists a sequence
in
satisfying
as
and
for all
. Then, we have
Let
be a continuous function defined on
. Given any fixed
, suppose that there exists a sequence
in
satisfying
as
and
for all
. Then, we have
In particular, we can assume .
Proof.
It suffices to prove the case of the supremum. It is obvious that
To prove part (i), given any , according to the concept of supremum , there exists with satisfying
We consider the following two cases.
Suppose that
. Then, we have
Suppose that
. The assumption says that there exists a sequence
in
satisfying
as
and
for all
. Since
is right‐continuous and
, we also have
as
. Therefore, we obtain
Since can be any positive number, it follows that
Part (ii) can be similarly obtained, and the proof is complete.
Proposition 1.2.9
Let and be two sequences of subsets of satisfying
and
Then, we have
and
Proof.
It is obvious that
The results follow immediately from Proposition 1.2.3.
Proposition 1.2.10
Let and be two sequences of sets in satisfying
and
Then, we have
and
Proof.
The results follow immediately from Proposition 1.2.4.
Proposition 1.2.11
Let be a real‐valued function defined on a subset of , and let be a constant. Then, we have
and
Proof.
We have
and
Another equality can be similarly obtained. This completes the proof.
1.3 Semi‐Continuity
Let be a real‐valued function defined on . We say that the supremum is attained when there exists satisfying for all with . Equivalently, the supremum is attained if and only if
Similarly, the infimum is attained when there exists satisfying for all with . Equivalently, the infimum is attained if and only if
Let be an element in . Recall that the Euclidean norm of is given by
Definition 1.3.1
Let be a nonempty set in .
A real‐valued function
defined on
is said to be
upper semi‐continuous
at
when the following condition is satisfied: for each
, there exists
such that
implies
for any
.
A real‐valued function
defined on
is said to be
lower semi‐continuous
at
when the following condition is satisfied: for each
, there exists
such that
implies
for any
.
Remark 1.3.2
We have the following interesting observations.
If
is upper semi‐continuou on
, then
is lower semi‐continuous on
.
If
is lower semi‐continuou on
, then
is upper semi‐continuous on
.
The real‐valued function
is continuous on
if and only if it is both lower and upper semi‐continuous in
.
If
is upper semi‐continuous on
, then
is a closed subset of
for all
.
If
is lower semi‐continuous on
, then
is a closed subset of
for all
.
Proposition 1.3.3
Let be a multi‐variable real‐valued function, and let each real‐valued function be continuous at for . Then, the following statements hold true.
Suppose that
is lower semi‐continuous at
. Then, the composition function
is lower semi‐continuous at
.
Suppose that
is upper semi‐continuous at
. Then, the composition function
is upper semi‐continuous at
.
Proof.
To prove part (i), since is lower semi‐continuous at , given any , there exists such that
Since each is continuous at for , given , there exists such that
Let . Then implies that the inequality (1.13) is satisfied for all . Let . Then
which implies
which says that is lower semi‐continuous at . Part (ii) can be similarly obtained. This completes the proof.
Proposition 1.3.4
Let be a multi‐variable real‐valued function, and let each real‐valued function be left‐continuous at for . Then, the following statements hold true.
Assume that the composition function
is increasing. If
is lower semi‐continuous at
, then
is lower semi‐continuous at
.
Assume that the composition function
is decreasing. If
is upper semi‐continuous at
, then
is upper semi‐continuous at
.
Proof.
To prove part (i), since is lower semi‐continuous at , given any , there exists such that
Since each is left‐continuous at for , given , there exists such that
The argument in the proof of Proposition 1.3.3 is still valid to show that there exists such that
For , since is increasing, it follows that
Therefore, we conclude that
which says that is lower semi‐continuous at .
To prove part (ii), we can similarly show that there exists such that
For , since is decreasing, it follows that
which says that is upper semi‐continuous at . This completes the proof.
Proposition 1.3.5
We have the following properties.
Suppose that the real‐valued functions
and
are lower semi‐continuous on the closed interval
. Then, the addition
is also lower semi‐continuous on the closed interval
.
Suppose that the real‐valued functions
and
are upper semi‐continuous on on the closed interval
. Then, the addition
is also upper semi‐continuous on the closed interval
.
Proof.
To prove part (i), given , there exist such that
and that
Let . Then, for , we have
which shows that is lower semi‐continuous at .
To prove part (ii), given , there exist such that
and that
Let . Then, for , we have
which shows that is upper semi‐continuous at . This completes the proof.
Proposition 1.3.6
We have the following properties.
Suppose that the real‐valued functions
and
are lower semi‐continuous on the closed interval
. Then, the real‐valued functions
and
are also lower semi‐continuous on the closed interval
.
Suppose that the real‐valued functions
and
are upper semi‐continuous on on the closed interval
. Then, the real‐valued functions
and
are also upper semi‐continuous on the closed interval
.
Proof.
To prove part (i), given , there exist such that
and that
Let . Then, for , we have
and
which show that and are lower semi‐continuous at .
To prove part (ii), given , there exist such that
and that
Let . Then, for , we have
and
which show that and are upper semi‐continuous at . This completes the proof.
Proposition 1.3.7
We have the following properties.
Suppose that
is increasing on a subset
of
. Then
is left‐continuous on
if and only if
is lower semi‐continuous on
.
Suppose that
is decreasing on a subset
of
. Then
is left‐continuous on
if and only if
is upper semi‐continuous on
.
Proof.
To prove part (i), we first assume that is left‐continuous at . Then, given any , there exists such that implies , i.e. . For with , since is increasing, we have
Therefore, we conclude that implies , which shows that is lower semi‐continuous at .
Conversely, we assume that is lower semi‐continuous at . Then, given any , there exists such that implies . If then we immediately have by the lower semi‐continuity at . Since is increasing, we also have
2Fuzzy Sets
The main idea of fuzzy sets is to consider the degree of membership. A fuzzy set is described by a membership function that assigns to each member or element a membership degree. Usually, the range of this membership function is from 0 to 1. A degree of 1 represents complete membership to the set, and degree of 0 represents absolutely no membership to the set. A degree between 0 and 1 represents partial membership to the set.
We can define high fever as a temperature higher than 102 . Even if most doctors will agree that the threshold is at about 102 , this does not mean that a patient with a body temperature of 101.9 does not have a high fever while another patient with 102 does indeed have a high fever. Therefore, instead of using this rigid definition, each body temperature is associated with a certain degree. For example, we show a possible description of high fever using membership degree as follows
The degree of membership can also be represented by a continuous function.
2.1 Membership Functions
Let be a subset of . Each element can either belong to or not belong to a set . This kind of set can be defined by the characteristic function
That is to say, the characteristic function maps elements of to elements of the set , which is formally expressed by .
Zadeh [162] proposed a concept of so‐called fuzzy set by extending the range of the characteristic function to the unit interval . A fuzzy set in is defined to be a set of ordered pairs
where is called the membership function of . The value is regarded as the degree of membership of in . In other words, it indicates the degree to which belongs to . Any subset of can also be regarded as a fuzzy set in by taking the membership function as the characteristic function of . In this case, we write by regarding as a fuzzy set in . When is a singleton , we also write .
Example 2.1.1
Let be the set of all real numbers considerably larger than 10. Then can be described as a fuzzy set in with membership function defined by
Let be the set of all real numbers close (but not equal) to 10. Then can also be described as a fuzzy set in with membership function defined by
In this case, we may also write to mean fuzzy real number 10. Therefore, any statements that involve fuzziness can always be represented by a membership function.
2.2 ‐level Sets
An interesting and important concept related to fuzzy sets is the ‐level set. Let be a fuzzy set in with membership function . The range of the membership function is denoted by . Throughout this book, we shall assume that the range contains 1. However, the range is not necessarily equal to the whole unit interval .
For , the ‐level set of is defined by
Since the range is assumed to contain 1, it follows that the ‐level sets are non‐empty for all . Notice that the ‐level set is not defined by (2.1). The ‐level set will be defined in a different way that will be explained afterward.
Given any satisfying , it is easy to see
The strict inclusion can happen.
Proposition 2.2.1
Let be a fuzzy set in with membership function . Suppose that and with . Then where “” means and .
Proof.
Since , there exists satisfying . Suppose that there exists with satisfying . Then, we have which violates . In other words, there exists and for all with . This completes the proof.
Example 2.2.2
The membership function of a fuzzy set is given by
It is clear to see
which also says that . We see that . However, we still have the 0.55‐level set given by
Although is not in the range , we still have
Notice that the expression (2.1) does not include the ‐level set. If we allowed the expression (2.1) taking , the ‐level set of would be the whole ‐dimensional Euclidean space . Defined in this way, the ‐level set would not be helpful for real applications. Therefore, we are going to invoke a topological concept to define the ‐level set. The support of a fuzzy set in is the crisp set defined by
The ‐level set of is defined to be the closure of the support , i.e.
For the concept of closure, refer to Definition 1.2.5.
Proposition 2.2.3
Let be a fuzzy set in with membership function . Then, we have
Proof.
Given any , i.e. , we have . Since , we have the inclusion
For proving the other direction of inclusion, given for some , we have , i.e. . This proves the desired equality.
Let be a subset of . Recall the notation . Then, we see that for any . Also, the ‐level set is given by
Recall that is a closed subset of when . Now, suppose that is a closed subset of . Then, we have for any . In particular, for any , since the singleton is a closed subset of , it follows that for any because of (2.6).
Example 2.2.4
The membership function of a trapezoidal fuzzy interval is given by
which is denoted by . It is clear to see . For , the ‐level set is a closed interval denoted by , where
