Editing Fuzzy set (section)

{{Short description|Sets whose elements have degrees of membership}}

In [[mathematics]], '''fuzzy sets''' (also known as '''uncertain sets''') are [[Set (mathematics)|sets]] whose [[Element (mathematics)|elements]] have degrees of membership. Fuzzy sets were introduced independently by [[Lotfi Asker Zadeh|Lotfi A. Zadeh]] in 1965 as an extension of the classical notion of set.<ref>L. A. Zadeh (1965) [http://www.cs.berkeley.edu/~zadeh/papers/Fuzzy%20Sets-Information%20and%20Control-1965.pdf "Fuzzy sets"] {{Webarchive|url=https://web.archive.org/web/20150813153834/http://www.cs.berkeley.edu/~zadeh/papers/Fuzzy%20Sets-Information%20and%20Control-1965.pdf |date=2015-08-13 }}. ''Information and Control'' 8 (3) 338–353.</ref><ref>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 {{Cite journal | last1 = Gottwald | first1 = S. | title = An early approach toward graded identity and graded membership in set theory | doi = 10.1016/j.fss.2009.12.005 | journal = Fuzzy Sets and Systems | volume = 161 | issue = 18 | pages = 2369–2379 | year = 2010 }}</ref>
At the same time, {{harvtxt|Salii|1965}} defined a more general kind of structure called an "[[L-relation|''L''-relation]]", which he studied in an [[abstract algebra]]ic context; 
fuzzy relations are special cases of ''L''-relations when ''L'' is the [[unit interval]] [0,&thinsp;1].
They are now used throughout [[fuzzy mathematics]], having applications in areas such as [[linguistics]] {{harv|De Cock|Bodenhofer|Kerre|2000}}, [[Decision making|decision-making]] {{harv|Kuzmin|1982}}, and [[Cluster analysis|clustering]] {{harv|Bezdek|1978}}.

In classical [[set theory]], the membership of elements in a set is assessed in binary terms according to a [[Principle of bivalence|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 (mathematics)|membership function]] valued in the [[Real number|real]] unit interval [0,&thinsp;1]. Fuzzy sets generalize classical sets, since the [[indicator function]]s (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1.<ref name=":0">D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.</ref> 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]].<ref>{{Cite journal | doi=10.1186/1471-2105-7-S4-S7| pmid=17217525| pmc=1780132| title=FM-test: A fuzzy-set-theory-based approach to differential gene expression data analysis| journal=BMC Bioinformatics| volume=7| pages=S7| year=2006| last1=Liang| first1=Lily R.| last2=Lu| first2=Shiyong| last3=Wang| first3=Xuena| last4=Lu| first4=Yi| last5=Mandal| first5=Vinay| last6=Patacsil| first6=Dorrelyn| last7=Kumar| first7=Deepak| issue=Suppl 4| doi-access=free}}</ref>