Fuzzy set

In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh^[1] and Dieter Klaua^[2] in 1965 as an extension of the classical notion of set. At the same time, Salii (1965) defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, Bodenhofer & Kerre 2000) decision-making (Kuzmin 1982) and clustering (Bezdek 1978), are special cases of L-relations when L is the unit interval [0, 1].

In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.^[3] In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.^[4]

Definition

A fuzzy set is a pair $(U,m)$ where $U$ is a set and $m\colon U\rightarrow [0,1]$ a membership function.

For each $x\in U,$ the value $m(x)$ is called the grade of membership of $x$ in $(U,m).$ 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 $m(x)=0$ , $x$ is called fully included if $m(x)=1$ , and $x$ is called a fuzzy member if $0<m(x)<1$ .^[5] The set $\{x\in U\mid m(x)>0\}$ is called the support of $(U,m)$ and the set $\{x\in U\mid m(x)=1\}$ is called its kernel or core. The function $m$ is called the membership function of the fuzzy set $(U,m).$

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure $L$ of a given kind; usually it is required that $L$ be at least a poset or lattice. These are usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.^[6]

Fuzzy logic

Main article: Fuzzy logic

As an extension of the case of multi-valued logic, valuations ( $\mu :{\mathit {V}}_{o}\to {\mathit {W}}$ ) of propositional variables ( ${\mathit {V}}_{o}$ ) into a set of membership degrees ( ${\mathit {W}}$ ) can be thought of as membership functions mapping predicates into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.^[7]

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the engineering fields of automated control and knowledge engineering, and which encompasses many topics involving fuzzy sets and "approximated reasoning."^[8]

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.

Fuzzy number

Main article: Fuzzy number

A fuzzy number is a convex, normalized fuzzy set ${\tilde {\mathit {A}}}\subseteq \mathbb {R}$ whose membership function is at least segmentally continuous and has the functional value $\mu _{A}(x)=1$ at at least one element.

This can be likened to the funfair game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

Fuzzy interval

A fuzzy interval is an uncertain set ${\tilde {\mathit {A}}}\subseteq \mathbb {R}$ with a mean interval whose elements possess the membership function value $\mu _{A}(x)=1$ . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.^[9]

Fuzzy categories

The use of set membership as a key components of category theory can be generalized to fuzzy sets. This approach which initiated in 1968 shortly after the introduction of fuzzy set theory^[10] led to the development of "Goguen categories" in the 21st century.^[11] ^[12] In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.^[12]^[13]

Fuzzy relation equation

The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation, and A · R stands for the composition of A with R.

Entropy

Let A be a fuzzy variable with a continuous membership function. Then its entropy is ^[14]

H[A]=\int _{-\infty }^{\infty }S(\operatorname {Cr} \lbrace A\geq t\rbrace )\,dt

where

S(y)=-y\ln y-(1-y)\ln(1-y).

Extensions

There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965, a lot of new mathematical constructions and theories treating imprecision, inexactness, ambiguity, and uncertainty have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others try to mathematically model imprecision and uncertainty in a different way (Burgin & Chunihin 1997; Kerre 2001; Deschrijver and Kerre, 2003).

The diversity of such constructions and corresponding theories includes:

interval sets (Moore, 1966),
L-fuzzy sets (Goguen, 1967),
flou sets (Gentilhomme, 1968),
Boolean-valued fuzzy sets (Brown, 1971),
type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),
set-valued sets (Chapin, 1974; 1975),
interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),
functions as generalizations of fuzzy sets and multisets (Lake, 1976),
level fuzzy sets (Radecki, 1977)
underdetermined sets (Narinyani, 1980),
rough sets (Pawlak, 1982),
intuitionistic fuzzy sets (Atanassov, 1983),
fuzzy multisets (Yager, 1986),
intuitionistic L-fuzzy sets (Atanassov, 1986),
rough multisets (Grzymala-Busse, 1987),
fuzzy rough sets (Nakamura, 1988),
real-valued fuzzy sets (Blizard, 1989),
named sets (Burgin, 1990),
vague sets (Wen-Lung Gau and Buehrer, 1993),
Q-sets (Gylys, 1994)
shadowed sets (Pedrycz, 1998),
α-level sets (Yao, 1997),
genuine sets (Demirci, 1999),
soft sets (Molodtsov, 1999),
intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)
blurry sets (Smith, 2004)
L-fuzzy rough sets (Radzikowska and Kerre, 2004),
generalized rough fuzzy sets (Feng, 2010)
rough intuitionistic fuzzy sets (Thomas and Nair, 2011),
soft rough fuzzy sets (Meng, Zhang and Qin, 2011)
soft fuzzy rough sets (Meng, Zhang and Qin, 2011)
soft multisets (Alkhazaleh, Salleh and Hassan, 2011)
fuzzy soft multisets (Alkhazaleh and Salleh, 2012)
bipolar fuzzy sets (Wen-Ran Zhang, 1998)

While most of the above can be generally categorized as truth-based extensions to fuzzy sets, bipolar fuzzy set theory presents a philosophically and logically different, equilibrium-based generalization of fuzzy sets.^[15]^[16]^[17]

Notes

↑ L. A. Zadeh (1965) "Fuzzy sets". Information and Control 8 (3) 338–353.
↑ Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided by Gottwald, S. (2010). "An early approach toward graded identity and graded membership in set theory". Fuzzy Sets and Systems. 161 (18): 2369–2379. doi:10.1016/j.fss.2009.12.005.
↑ D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
↑ Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, "FM-test: A Fuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis", BMC Bioinformatics, 7 (Suppl 4): S7. 2006.
↑ AAAI
↑ Goguen, Joseph A., 196, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
↑ Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0-86380-262-1
↑ "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.
↑ "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28
↑ J. A. Goguen "Categories of fuzzy sets : applications of non-Cantorian set theory" PhD Thesis University of California, Berkeley, 1968
↑ Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations" 2007 Springer ISBN 9781402061639
1 2 Michael Winter "Representation theory of Goguen categories" Fuzzy Sets and Systems Volume 138, Issue 1, 16 August 2003, Pages 85–126
↑ Goguen, J.A., "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18(1):145–174, 1967
↑ Xuecheng, Liu. "Entropy, distance measure and similarity measure of fuzzy sets and their relations." Fuzzy sets and systems 52.3 (1992): 305–318.
↑ Zhang, W. -R. (1998). (Yin)(Yang) Bipolar Fuzzy Sets. Proc. of IEEE World Congress on Computational Intelligence – Fuzz-IEEE, Anchorage, AK, May 1998, 835-840.
↑ Zhang, W. -R. & Zhang, L. (2004). YinYang Bipolar Logic and Bipolar Fuzzy Logic. Information Sciences. Vol. 165, No. 3-4, 2004, 265–287.
↑ Zhang, W.-R. (2011), YinYang Bipolar Relativity: A Unifying Theory of Nature, Agents and Causality with Applications in Quantum Computing, Cognitive Informatics and Life Sciences. IGI Global, Hershey and New York, 2011.

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Non-classical logic

Modal	Alethic Axiologic Deontic Doxastic Epistemic Temporal

Intuitionistic	Intuitionistic logic Constructive analysis Heyting arithmetic Intuitionistic type theory Constructive set theory

Fuzzy	Degree of truth Fuzzy rule Fuzzy set Fuzzy finite element Fuzzy set operations

Substructural	Structural rule Relevance logic Linear logic

Paraconsistent	Dialetheism

Description	Ontology Ontology language

Many-valued	Three-valued Four-valued Łukasiewicz

Set theory

Axioms	Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification

Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union

Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram

Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal

Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck

Paradoxes Problems	Russell's paradox Suslin's problem

Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

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