Fuzzy Set Theory: Fundamentals and Applications

By Fouad Sabry

What Is Fuzzy Set Theory

In the field of mathematics, fuzzy sets are defined as sets with constituents that have varying degrees of membership. Lotfi A. Zadeh independently developed the concept of fuzzy sets in 1965 and presented it to the world as an expansion of the traditional concept of set.During this same time period, Salii (1965) defined a more broad sort of structure that he referred to as an L-relation. He examined this structure in the framework of abstract algebra. Fuzzy relations, which are currently utilized across fuzzy mathematics and have applications in fields such as linguistics, decision-making, and clustering, are special examples of L-relations when L is the unit interval [0, 1]. Fuzzy relations have applications in areas such as linguistics, decision-making, and clustering.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Fuzzy set

Chapter 2: Kaluza-Klein theory

Chapter 3: Dirac equation

Chapter 4: Stress-energy tensor

Chapter 5: Fuzzy control system

Chapter 6: Measurable cardinal

Chapter 7: Radon-Nikodym theorem

Chapter 8: Stable distribution

Chapter 9: Four-gradient

Chapter 10: Pearson distribution

(II) Answering the public top questions about fuzzy set theory.

(III) Real world examples for the usage of fuzzy set theory in many fields.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of fuzzy set theory.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 25, 2023

Book preview

Chapter 1: Fuzzy set

In mathematics, Known as fuzzy sets,.

Those sets whose members have varying degrees of membership are called uncertain sets..

Lotfi A. Introducing Fuzzy Sets on His Own.

It was developed by Zadeh in 1965 as an expansion of the traditional idea of set.

While doing so, An L-relation, as defined by Salii (1965), is a more broad type of structure, which he investigated on a purely mathematical level.

Fuzzy relations, which have found widespread use in recent years in fields like linguistics and fuzzy mathematics (De Cock), Bodenhofer and Kerre (2000), decision-making (Kuzmin 1982), and grouping (Bezdek 1978), are special cases of L-relations when L is the unit interval [0, 1].

The basic concepts of set theory, Element membership in a set is evaluated using a binary condition: either the element belongs to the set or it does not.

By contrast, Membership in a set can be tentatively determined using fuzzy set theory; this is described with the aid of a membership function valued in the real unit interval [0, 1].

Classical sets are a special case of fuzzy sets, The indicator functions of classical sets are special examples of the membership functions of fuzzy sets, hence it follows that, If the latter can only be either 0 or 1, then.

A fuzzy set is a pair (U,m) where U is a set (often required to be non-empty) and {\displaystyle m\colon U\rightarrow [0,1]} a membership function.

The reference set U (sometimes denoted by \Omega or X ) is called universe of discourse, and for each x\in U, the value m(x) is called the grade of membership of x in (U,m) .

The function {\displaystyle m=\mu _{A}} is called the membership function of the fuzzy set {\displaystyle A=(U,m)} .

For a finite set U=\{x_{1},\dots ,x_{n}\}, the fuzzy set (U,m) is often denoted by

\{m(x_{1})/x_{1},\dots ,m(x_{n})/x_{n}\}.

Let x\in U .

Then x is called

not included in the fuzzy set (U,m) if m(x)=0 (no member), fully included if m(x)=1 (full member), partially included if 0

The (crisp) set of all fuzzy sets on a universe U is denoted with {\displaystyle SF(U)} (or sometimes just F(U) ).

For any fuzzy set {\displaystyle A=(U,m)} and \alpha \in [0,1] the following crisp sets are defined:

{\displaystyle A^{\geq \alpha }=A_{\alpha }=\{x\in U\mid m(x)\geq \alpha \}}

is called its α-cut (aka α-level set)

{\displaystyle A^{>\alpha }=A'_{\alpha }=\{x\in U\mid m(x)>\alpha \}}

is called its strong α-cut (aka strong α-level set)

{\displaystyle S(A)=\operatorname {Supp} (A)=A^{>0}=\{x\in U\mid m(x)>0\}}

is called its support

{\displaystyle C(A)=\operatorname {Core} (A)=A^{=1}=\{x\in U\mid m(x)=1\}}

is called its core (or sometimes kernel {\displaystyle \operatorname {Kern} (A)} ).

It's important to keep in mind that different authors have varied interpretations of the word kernel; for examples, see below.

A fuzzy set {\displaystyle A=(U,m)} is empty ( {\displaystyle A=\varnothing } ) iff (if and only if)

\forall

{\displaystyle x\in U:\mu _{A}(x)=m(x)=0}

Two fuzzy sets A and B are equal ( A=B ) iff

{\displaystyle \forall x\in U:\mu _{A}(x)=\mu _{B}(x)}

A fuzzy set A is included in a fuzzy set B ( A\subseteq B ) iff

{\displaystyle \forall x\in U:\mu _{A}(x)\leq \mu _{B}(x)}

For any fuzzy set A , any element x\in U that satisfies

{\displaystyle \mu _{A}(x)=0.5}

is what we refer to as a crossover.

Given a fuzzy set A , any \alpha \in [0,1] , for which

{\displaystyle A^{=\alpha }=\{x\in U\mid \mu _{A}(x)=\alpha \}}

is not empty, is classified as A level.

The level set of A is the set of all levels \alpha \in [0,1] representing distinct cuts.

It is the image of \mu _{A} :

{\displaystyle \Lambda _{A}=\{\alpha \in [0,1]:A^{=\alpha }\neq \varnothing \}=\{\alpha \in [0,1]:{}}

\exist

{\displaystyle x\in U(\mu _{A}(x)=\alpha )\}=\mu _{A}(U)}

For a fuzzy set A , Specifically, its height is

{\displaystyle \operatorname {Hgt} (A)=\sup\{\mu _{A}(x)\mid x\in U\}=\sup(\mu _{A}(U))}

where \sup denotes the supremum, which exists because {\displaystyle \mu _{A}(U)} is non-empty and bounded above by 1.

Assuming a finite U, We may easily substitute maximum for supremum.

A fuzzy set A is said to be normalized iff

{\displaystyle \operatorname {Hgt} (A)=1}

Regarding the finite case, in which the highest point is the supremum, that at least one member of the fuzzy set meets all the criteria for membership.

A non-empty fuzzy set A may be normalized with result {\tilde {A}} by dividing the membership function of the fuzzy set by its height:

{\displaystyle \forall x\in U:\mu _{\tilde {A}}(x)=\mu _{A}(x)/\operatorname {Hgt} (A)}

While the normalizing constant is still a number, unlike standard normalization, it is not a sum.

For fuzzy sets A of real numbers (U ⊆ ℝ) with bounded support, definition of width:

{\displaystyle \operatorname {Width} (A)=\sup(\operatorname {Supp} (A))-\inf(\operatorname {Supp} (A))}

In the case when {\displaystyle \operatorname {Supp} (A)} is a finite set, or a complete set in a broader sense, The breadth is ideal

{\displaystyle \operatorname {Width} (A)=\max(\operatorname {Supp} (A))-\min(\operatorname {Supp} (A))}

In the n-dimensional case (U ⊆ ℝn) the above can be replaced by the n-dimensional volume of {\displaystyle \operatorname {Supp} (A)} .

In general, Any metric based on U can be used to define this, for example, by incorporating (like.

Lebesgue integration) of {\displaystyle \operatorname {Supp} (A)} .

A real fuzzy set A (U ⊆ ℝ) is said to be convex (in the fuzzy sense, (to differentiate it from a sharp convex set), iff

{\displaystyle \forall x,y\in U,\forall \lambda \in [0,1]:\mu _{A}(\lambda {x}+(1-\lambda )y)\geq \min(\mu _{A}(x),\mu _{A}(y))}

.

Without narrowing the scope, we may take x ≤ y, This provides the corresponding formula

{\displaystyle \forall z\in [x,y]:\mu _{A}(z)\geq \min(\mu _{A}(x),\mu _{A}(y))}

.

This definition can be extended to one for a general topological space U: we say the fuzzy set A is convex when, in case U has a subset Z, the condition

{\displaystyle \forall z\in Z:\mu _{A}(z)\geq \inf(\mu _{A}(\partial Z))}

holds, where {\displaystyle \partial Z} denotes the boundary of Z and {\displaystyle f(X)=\{f(x)\mid x\in X\}} denotes the image of a set X (here {\displaystyle \partial Z} ) under a function f (here \mu _{A} ).

While there is consensus on how to define a fuzzy set's complement, the union and intersection operations are more open to interpretation.

For a given fuzzy set A , its complement {\displaystyle \neg {A}} (sometimes denoted as A^{c} or {\displaystyle cA} ) is defined by the following membership function:

{\displaystyle \forall x\in U:\mu _{\neg {A}}(x)=1-\mu _{A}(x)}

.

Accept t as a t-norm, and s is the s-norm that goes with it (aka t-conorm).

Given a pair of fuzzy sets A,B , their intersection {\displaystyle A\cap {B}} is defined by:

{\displaystyle \forall x\in U:\mu _{A\cap {B}}(x)=t(\mu _{A}(x),\mu _{B}(x))}

, and their union {\displaystyle A\cup {B}} is defined by:

{\displaystyle \forall x\in U:\mu _{A\cup {B}}(x)=s(\mu _{A}(x),\mu _{B}(x))}

.

The t-norm is defined as the, Union and intersection are shown to be commutative, monotonic, associative, They include both a blank and a self-referential component.

For the crossroads, these are ∅ and U, respectively, while union members, This is backwards.

However, U may not be the union of a fuzzy set and its complement, and the intersection of them may not give the empty set ∅.

Due to the associativity between union and intersection, It is intuitive to recursively define intersection and union for a finite family of fuzzy sets.

If the standard negator {\displaystyle n(\alpha )=1-\alpha ,\alpha \in [0,1]} is replaced by another strong negator, It is possible to generalize the fuzzy set difference by

{\displaystyle \forall x\in U:\mu _{\neg {A}}(x)=n(\mu _{A}(x)).}

The intersection, union, and complement of fuzzy sets constitute a De Morgan Triplet. This triple integral is covered by the laws of De Morgan,.

The samples in the t-norms article can be used to construct examples of fuzzy intersection/union pairs with a standard negator.

Since only the regular t-norm minimum satisfies the idempotent property, the fuzzy intersection cannot be used in general. True, the resulting fuzzy intersection operation is not idempotent if the arithmetic multiplication is employed as the t-norm. Taking the intersection of a fuzzy set with itself iteratively is not an easy task. Instead, it gives us a definition of the m-th power of a fuzzy set that may be extended to non-integer exponents in the canonical fashion:

For any fuzzy set A and {\displaystyle \nu \in \mathbb {R} ^{+}} the ν-th power of A is defined by the membership function:

{\displaystyle \forall x\in U:\mu _{A^{\nu }}(x)=\mu _{A}(x)^{\nu }.}

There is a particular name for the situation where exponent 2 occurs.

For any fuzzy set A the concentration {\displaystyle CON(A)=A^{2}} is defined

{\displaystyle \forall x\in U:\mu _{CON(A)}(x)=\mu _{A^{2}}(x)=\mu _{A}(x)^{2}.}

Taking 0^{0}=1 , we have {\displaystyle A^{0}=U} and {\displaystyle A^{1}=A.}

Given fuzzy sets A,B , the fuzzy set difference A\setminus B , also denoted {\displaystyle A-B} , can be easily characterized by the membership function:

{\displaystyle \forall x\in U:\mu _{A\setminus {B}}(x)=t(\mu _{A}(x),n(\mu _{B}(x))),}

which means {\displaystyle A\setminus B=A\cap \neg {B}} , e.

g:

{\displaystyle \forall x\in U:\mu _{A\setminus {B}}(x)=\min(\mu _{A}(x),1-\mu _{B}(x)).}

A further set-difference idea could be:

{\displaystyle \forall x\in U:\mu _{A-{B}}(x)=\mu _{A}(x)-t(\mu _{A}(x),\mu _{B}(x)).}

Dubois and Prade (1980) propose an absolute value, a difference, or both as methods for achieving symmetric fuzzy set differences.

{\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=|\mu _{A}(x)-\mu _{B}(x)|,}

maximum, minimum, and the conventional negation would yield

{\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=\max(\min(\mu _{A}(x),1-\mu _{B}(x)),\min(\mu _{B}(x),1-\mu _{A}(x))).}

Averaging operations can be defined for fuzzy sets, unlike crisp sets.

Whereas intersection and union operations are typically vague,, there is clearness for disjoint fuzzy sets: Two fuzzy sets A,B are disjoint iff

{\displaystyle \forall x\in U:\mu _{A}(x)=0\lor \mu _{B}(x)=0}

corresponding to

\nexists

{\displaystyle x\in U:\mu _{A}(x)>0\land \mu _{B}(x)>0}

Similarly to, or the same as

{\displaystyle \forall x\in U:\min(\mu _{A}(x),\mu _{B}(x))=0}

We recall that min/max is a t/s-norm combination, and that any other will do.

In contrast to the usual concept of disjointness for crisp sets, fuzzy sets are only disjoint if and only if their supports are disjoint.

For disjoint fuzzy sets A,B any intersection will give ∅, and any combination will provide the same outcome, for which the symbol is

{\displaystyle A\,{\dot {\cup }}\,B=A\cup B}

and its membership function can be written as

{\displaystyle \forall x\in U:\mu _{A{\dot {\cup }}B}(x)=\mu _{A}(x)+\mu _{B}(x)}

Take into account that only one of the two summands is positive.

For disjoint fuzzy sets A,B the following holds true:

{\displaystyle \operatorname {Supp} (A\,{\dot {\cup }}\,B)=\operatorname {Supp} (A)\cup \operatorname {Supp} (B)}

This can be generalized to finite families of fuzzy sets as follows: Given a family {\displaystyle A=(A_{i})_{i\in I}} of fuzzy sets with index set I (e.g.

I = {1,2,3,...,n}).

This is not a monophyletic family (pairwise) if

{\displaystyle {\text{for all }}x\in U{\text{ there exists at most one }}i\in I{\text{ such that }}\mu _{A_{i}}(x)>0.}

A family of fuzzy sets {\displaystyle A=(A_{i})_{i\in I}} is disjoint, iff the family of underlying supports

{\displaystyle \operatorname {Supp} \circ A=(\operatorname {Supp} (A_{i}))_{i\in I}}

is disjoint in the standard sense for families of crisp sets.

Separate from the t and s norm, intersection of a disjoint family of fuzzy sets will give ∅ again, while there is no doubt about the union:

{\displaystyle {\dot {\bigcup \limits _{i\in I}}}\,A_{i}=\bigcup _{i\in I}A_{i}}

and its membership function can be written as

{\displaystyle \forall x\in U:\mu _{{\dot {\bigcup \limits _{i\in I}}}A_{i}}(x)=\sum _{i\in I}\mu _{A_{i}}(x)}

Once again, none of the totals add up to more than zero.

For disjoint families of fuzzy sets {\displaystyle A=(A_{i})_{i\in I}} the following holds true:

{\displaystyle \operatorname {Supp} \left({\dot {\bigcup \limits _{i\in I}}}\,A_{i}\right)=\bigcup \limits _{i\in I}\operatorname {Supp} (A_{i})}

For a fuzzy set A with finite support {\displaystyle \operatorname {Supp} (A)} (i.e.

a infinite fuzzy set, its sigma-count (also known as its cardinality) is

{\displaystyle \operatorname {Card} (A)=\operatorname {sc} (A)=|A|=\sum _{x\in U}\mu _{A}(x)}

.

The relative cardinality is given by if and only if U is a finite set.

{\displaystyle \operatorname {RelCard} (A)=\|A\|=\operatorname {sc} (A)/|U|=|A|/|U|}

.

This can be generalized for the divisor to be a non-empty fuzzy set: For fuzzy sets {\displaystyle A,G} with G ≠ ∅, related cardinality can be defined as:

{\displaystyle \operatorname {RelCard} (A,G)=\operatorname {sc} (A|G)=\operatorname {sc} (A\cap {G})/\operatorname {sc} (G)}

, This looks a lot like the conditional probability expression. Note:

{\displaystyle \operatorname {sc} (G)>0} here.

The answer could be different depending on which intersection (t-norm) is used.

For {\displaystyle G=U} the result is unambiguous and resembles the prior definition.

For any fuzzy set A the membership function {\displaystyle \mu _{A}:U\to [0,1]} can be regarded as a family

{\displaystyle \mu _{A}=(\mu _{A}(x))_{x\in U}\in [0,1]^{U}}

.

The latter is a metric space with several metrics d known.

A metric can be derived from a norm (vector norm) {\displaystyle \|\,\|} via