In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh and Dieter Klaua 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. 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.
A fuzzy set is a pair where is a set and . (membership function)
For each the value is called the grade of membership of in For a finite set the fuzzy set is often denoted by
Let Then is called not included in the fuzzy set if , is called fully included if , and is called a fuzzy member if . The set is called the support of and the set is called its kernel or core. The function is called the membership function of the fuzzy set
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 of a given kind; usually it is required that 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.
As an extension of the case of multi-valued logic, valuations () of propositional variables () into a set of membership degrees () 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.
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."
Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.
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).
A fuzzy interval is an uncertain set with a mean interval whose elements possess the membership function value . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.
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 led to the development of "Goguen categories" in the 21st century.  In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.
Fuzzy relation equation
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Let A be a fuzzy variable with a continuous membership function. Then its entropy is 
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.
- 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.
- 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
- 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.
- Alkhazaleh, S. and Salleh, A.R. Fuzzy Soft Multiset Theory, Abstract and Applied Analysis, 2012, article ID 350600, 20 p.
- Alkhazaleh, S., Salleh, A.R. and Hassan, N. Soft Multisets Theory, Applied Mathematical Sciences, v. 5, No. 72, 2011, pp. 3561–3573
- Atanassov, K. T. (1983) Intuitionistic fuzzy sets, VII ITKR's Session, Sofia (deposited in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)
- Atanasov, K. (1986) Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, v. 20, No. 1, pp. 87–96
- Bezdek, J.C. (1978). "Fuzzy partitions and relations and axiomatic basis for clustering". Fuzzy Sets and Systems 1. pp. 111–127.
- Blizard, W.D. (1989) Real-valued Multisets and Fuzzy Sets, Fuzzy Sets and Systems, v. 33, pp. 77–97
- Brown, J.G. (1971) A Note on Fuzzy Sets, Information and Control, v. 18, pp. 32–39
- Burgin, M. Theory of Named Sets as a Foundational Basis for Mathematics, in Structures in Mathematical Theories, San Sebastian, 1990, pp. 417-420
- Burgin M. and Chunihin, A. (1997) Named Sets in the Analysis of Uncertainty, in Methodological and Theoretical Problems of Mathematics and Information Sciences, Kiev, pp. 72-85
- Chapin, E.W. (1974) Set-valued Set Theory, I, Notre Dame J. Formal Logic, v. 15, pp. 619–634
- Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp. 255–267
- Chris Cornelis, Martine De Cock and Etienne E. Kerre, Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge, Expert Systems, v. 20, issue 5, pp. 260–270, 2003
- Cornelis, C., Deschrijver, C., and Kerre, E. E. (2004) Implication in intuitionistic and interval-valued fuzzy set theory: construction, classification, application, International Journal of Approximate Reasoning, v. 35, pp. 55–95
- De Cock, Martine; Bodenhofer, Ulrich; Kerre, Etienne E. (1–4 October 2000). Modelling Linguistic Expressions Using Fuzzy Relations. Proceedings of the 6th International Conference on Soft Computing. Iizuka, Japan. pp. 353–360.
- Demirci, M. (1999) Genuine Sets, Fuzzy Sets and Systems, v. 105, pp. 377–384
- Deschrijver, G.; Kerre, E.E. (2003). "On the relationship between some extensions of fuzzy set theory". Fuzzy Sets and Systems 133 (2): 227–235.
- Didier Dubois, Henri M. Prade, ed. (2000). Fundamentals of fuzzy sets. The Handbooks of Fuzzy Sets Series 7. Springer. ISBN 978-0-7923-7732-0.
- Feng F. Generalized Rough Fuzzy Sets Based on Soft Sets, Soft Computing, July 2010, Volume 14, Issue 9, pp 899–911
- Gentilhomme, Y. (1968) Les ensembles flous en linguistique, Cahiers Linguistique Theoretique Appliqee, 5, pp. 47–63
- Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp. 145–174
- Gottwald, S. (2006). "Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches". Studia Logica 82 (2): 211–244. doi:10.1007/s11225-006-7197-8.. Gottwald, S. (2006). "Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches". Studia Logica 84: 23–50. doi:10.1007/s11225-006-9001-1. preprint..
- Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik. Grundladen Math. 22, pp. 149–160.
- Grzymala-Busse, J. Learning from examples based on rough multisets, in Proceedings of the 2nd International Symposium on Methodologies for Intelligent Systems, Charlotte, NC, USA, 1987, pp. 325–332
- Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 9–31.
- Ulrich Höhle, Stephen Ernest Rodabaugh, ed. (1999). Mathematics of fuzzy sets: logic, topology, and measure theory. The Handbooks of Fuzzy Sets Series 3. Springer. ISBN 978-0-7923-8388-8.
- Jahn, K. U. (1975) Intervall-wertige Mengen, Math.Nach. 68, pp. 115–132
- Kerre, E.E. (2001). B. Reusch; K-H. Temme, eds. "A first view on the alternatives of fuzzy set theory". Computational Intelligence in Theory and Practice (Heidelberg: Physica-Verlag): 55–72. ISBN 3-7908-1357-5.
- George J. Klir; Bo Yuan (1995). Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall. ISBN 978-0-13-101171-7.
- Kuzmin, V.B. (1982). "Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations" (in Russian). Nauka, Moscow.
- Lake, J. (1976) Sets, fuzzy sets, multisets and functions, J. London Math. Soc., II Ser., v. 12, pp. 323–326
- Meng, D., Zhang, X. and Qin, K. Soft rough fuzzy sets and soft fuzzy rough sets, 'Computers & Mathematics with Applications', v. 62, issue 12, 2011, pp. 4635–4645
- Miyamoto, S. Fuzzy Multisets and their Generalizations, in 'Multiset Processing', LNCS 2235, pp. 225–235, 2001
- Molodtsov, O. (1999) Soft set theory – first results, Computers & Mathematics with Applications, v. 37, No. 4/5, pp. 19–31
- Moore, R.E. Interval Analysis, New York, Prentice-Hall, 1966
- Nakamura, A. (1988) Fuzzy rough sets, 'Notes on Multiple-valued Logic in Japan', v. 9, pp. 1–8
- Narinyani, A.S. Underdetermined Sets – A new datatype for knowledge representation, Preprint 232, Project VOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences, 1980
- Pedrycz, W. Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on System, Man, and Cybernetics, Part B, 28, 103–109, 1998.
- Radecki, T. Level Fuzzy Sets, 'Journal of Cybernetics', Volume 7, Issue 3-4, 1977
- Radzikowska, A.M. and Etienne E. Kerre, E.E. On L-Fuzzy Rough Sets, Artificial Intelligence and Soft Computing – ICAISC 2004, 7th International Conference, Zakopane, Poland, June 7–11, 2004, Proceedings; 01/2004
- Salii, V.N. (1965). "Binary L-relations". Izv. Vysh. Uchebn. Zaved. Matematika (in Russian) 44 (1): 133–145.
- Sambuc, R. Fonctions φ-floues: Application a l'aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis Univ. Marseille, France, 1975.
- Seising, Rudolf: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications—Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]: Springer 2007.
- Smith, N.J.J. (2004) Vagueness and blurry sets, 'J. of Phil. Logic', 33, pp. 165–235
- Thomas, K.V. and L. S. Nair, Rough intuitionistic fuzzy sets in a lattice, 'International Mathematical Forum', Vol. 6, 2011, no. 27, 1327–1335
- Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp. 23–37
- Yao, Y.Y., Combination of rough and fuzzy sets based on α-level sets, in: Rough Sets and Data Mining: Analysis for Imprecise Data, Lin, T.Y. and Cercone, N. (Eds.), Kluwer Academic Publishers, Boston, pp. 301–321, 1997.
- Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1-4, 1998, pp. 227 – 242
- Zadeh, L. (1975) The concept of a linguistic variable and its application to approximate reasoning–I, Inform. Sci., v. 8, pp. 199–249
- Hans-Jürgen Zimmermann (2001). Fuzzy set theory—and its applications (4th ed.). Kluwer. ISBN 978-0-7923-7435-0.
- Gianpiero Cattaneo and Davide Ciucci, "Heyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets" in J.J. Alpigini et al. (Eds.): RSCTC 2002, LNAI 2475, pp. 77–84, 2002. doi:10.1007/3-540-45813-1_10