Fuzzy set

Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously^[1] by Lotfi A. Zadeh^[2] and Dieter Klaua^[3] in 1965 as an extension of the classical notion of set. At the same time, Salii (1965) defined a more general kind of structures called L-relations, which were studied by him in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics (De Cock, et al, 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.^[4] 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.^[5]

Fuzzy sets can be applied, for example, to the field of genealogical research. When an individual is searching in vital records such as birth records for possible ancestors, the researcher must contend with a number of issues that could be encapsulated in a membership function.^[6] Looking for an ancestor named John Henry Pittman, who you think was born in (probably eastern) Tennessee circa 1853 (based on statements of his age in later censuses, and a marriage record in Knoxville), what is the likelihood that a particular birth record for "John Pittman" is your John Pittman? What about a record in a different part of Tennessee for "J.H. Pittman" in 1851? (It has been suggested by Thayer Watkins^[7] that Zadeh's ethnicity is an example of a fuzzy set)

Definition

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

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 $0<m(x)<1.$ ^[8] 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. 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.^[9]

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.^[10]

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."^[11]

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

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 precisely 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.^[12]

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.

Generalizations

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 and 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-Guiness, 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),

- 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),

- neutrosophic sets (Smarandache, 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)

Fuzzy sets and all their generalizations, as well as other generalizations of sets, are unified and comprised by the concept of a named set (Burgin, 1990; 2004; 2011).

References

^ Michael Winter (2007). Goguen categories: a categorical approach to L-fuzzy relations. Springer. p. ix. ISBN 978-1-4020-6163-9.
^ 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 Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1016/j.fss.2009.12.005, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1016/j.fss.2009.12.005 instead.
^ 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.
^ H. Bandemer, "Fuzzy Local Inference in Fuzzy Knowledge Bases" in V. Novak, J. Ramik, M. Mares, M. Cerny, J. Nekola (eds.), Fuzzy Approach to Reasoning and Decision-Making, 1990, Bechyne, Czechoslovakia, 47-48. [1]
^ "Fuzzy Logic: The Logic of Fuzzy Sets"
^ 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

External links

[Winter2007-1] Michael Winter (2007). Goguen categories: a categorical approach to L-fuzzy relations. Springer. p. ix. ISBN 978-1-4020-6163-9.

[2] L. A. Zadeh (1965) "Fuzzy sets". Information and Control 8 (3) 338–353.

[3] 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 Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1016/j.fss.2009.12.005, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1016/j.fss.2009.12.005 instead.

[4] D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.

[5] 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.

[6] H. Bandemer, "Fuzzy Local Inference in Fuzzy Knowledge Bases" in V. Novak, J. Ramik, M. Mares, M. Cerny, J. Nekola (eds.), Fuzzy Approach to Reasoning and Decision-Making, 1990, Bechyne, Czechoslovakia, 47-48. [1]

[7] "Fuzzy Logic: The Logic of Fuzzy Sets"

[8] AAAI

[9] Goguen, Joseph A., 196, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174

[10] Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0-86380-262-1

[11] "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.

[12] "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28

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v t e Set theory
Overview	Set (mathematics)
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