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Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work ''Fuzzy sets''.〔Zadeh, L. A. (1965) "Fuzzy sets", ''Information and Control'', 8, 338–353.〕 A fuzzy subset ''A'' of a set ''X'' is a function ''A:X→L'', where ''L'' is the interval (). This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for ''L'' = . More generally, one can use a complete lattice ''L'' in a definition of a fuzzy subset ''A'' .〔Goguen, J. (1967) "L-fuzzy sets", ''J. Math. Anal. Appl.'', 18, 145-174.〕 The evolution of the fuzzification of mathematical concepts can be broken down into three stages:〔Kerre, E.E., Mordeson, J.N. (2005) "A historical overview of fuzzy mathematics", ''New Mathematics and Natural Computation'', 1, 1-26.〕 :# straightforward fuzzification during the sixties and seventies, :# the explosion of the possible choices in the generalization process during the eighties, :# the standardization, axiomatization and L-fuzzification in the nineties. Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let ''A'' and ''B'' be two fuzzy subsets of ''X''. Intersection ''A'' ∩ ''B'' and union ''A'' ∪ ''B'' are defined as follows: (''A'' ∩ ''B'')(''x'') = min(''A''(''x''),''B''(''x'')), (''A'' ''∪'' ''B'')(''x'') = max(''A''(''x''),''B''(''x'')) for all ''x'' ∈ ''X''. Instead of ''min'' and ''max'' one can use t-norm and t-conorm, respectively ,〔Klement, E.P., Mesiar, R., Pap, E. (2000) ''Triangular Norms''. Dordrecht, Kluwer.〕 for example, ''min(a,b)'' can be replaced by multiplication ''ab''. A straightforward fuzzification is usually based on ''min'' and ''max'' operations because in this case more properties of traditional mathematics can be extended to the fuzzy case. A very important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on ''X''. The closure property for a fuzzy subset ''A'' of ''X'' is that for all ''x,y'' ∈ ''X'', ''A''(''x'' *''y'') ≥ min(''A''(''x''),''A''(''y'')). Let (''G'', *) be a group and ''A'' a fuzzy subset of ''G''. Then ''A'' is a fuzzy subgroup of ''G'' if for all ''x,y'' in ''G'', ''A''(''x'' *''y''−1) ≥ min(''A''(''x''),''A''(''y''−1)). A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let ''R'' be a fuzzy relation in ''X'', i.e. ''R'' is a fuzzy subset of ''X×X''. Then ''R'' is transitive if for all ''x,y,z'' in ''X'', ''R''(''x'',''z'') ≥ min(''R''(''x'',''y''),''R''(''y'',''z'')). == Some fields of mathematics using fuzzy set theory == Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld .〔Rosenfeld, A. (1971) "Fuzzy groups", ''J. Math. Anal. Appl.'', 35, 512-517.〕 Hundreds of papers on related topics have been published. Recent results and references can be found in 〔Mordeson, J.N., Malik, D.S., Kuroli, N. (2003) ''Fuzzy Semigroups''. Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag〕 and.〔Mordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005) ''Fuzzy Group Theory''. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag.〕 Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.〔Mordeson, J.N., Malik, D.S (1998) ''Fuzzy Commutative Algebra''. World Scientific.〕 Fuzzy topology was introduced by C.L. Chang〔Chang, C.L. (1968) "Fuzzy topological spaces", ''J. Math. Anal. Appl.'', 24, 182—190.〕 in 1968 and further was studied in many papers.〔Liu, Y.-M., Luo, M.-K. (1997) ''Fuzzy Topology''. Advances in Fuzzy Systems - Applications and Theory, vol. 9, World Scientific, Singapore.〕 Main concepts of fuzzy geometry were introduced by Tim Poston in 1971,〔Poston, Tim, "Fuzzy Geometry".〕 A. Rosenfeld in 1974, by J.J. Buckley and E. Eslami in 1997〔Buckley, J.J., Eslami, E. (1997) "Fuzzy plane geometry I: Points and lines". ''Fuzzy Sets and Systems'', 86, 179-187.〕 and by D. Ghosh and D. Chakraborty in 2012-14 〔Ghosh, D., Chakraborty, D. (2012) "Analytical fuzzy plane geometry I". ''Fuzzy Sets and Systems'', 209, 66-83.〕 〔 Chakraborty, D. and Ghosh, D. (2014) "Analytical fuzzy plane geometry II". ''Fuzzy Sets and Systems'', 243, 84–109.〕 Basic types of fuzzy relations were introduced by Zadeh in 1971.〔Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". ''Inform. Sci.'', 3, 177–200.〕 The properties of fuzzy graphs have been studied by A. Kaufman,〔Kaufmann, A. (1973). ''Introduction a la théorie des sous-ensembles flous''. Paris. Masson.〕 A. Rosenfel,〔A. Rosenfeld, A. (1975) "Fuzzy graphs". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), ''Fuzzy Sets and their Applications to Cognitive and Decision Processes'', Academic Press, New York, ISBN 978-0-12-775260-0, pp. 77–95.〕 and by R.T. Yeh and S.Y. Bang.〔Yeh, R.T., Bang, S.Y. (1975) "Fuzzy graphs, fuzzy relations and their applications to cluster analysis". In: Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (eds.), ''Fuzzy Sets and their Applications to Cognitive and Decision Processes'', Academic Press, New York, ISBN 978-0-12-775260-0, pp. 125–149.〕 Recent results can be found in a 2000 article.〔Mordeson, J.N., Nair, P.S. (2000) ''Fuzzy Graphs and Fuzzy Hypergraphs''. Studies in Fuzziness and Soft Computing, vol. 46. Springer-Verlag.〕 Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in the cited articles and treatises.〔Zadeh, L.A. (1978) "Fuzzy sets as a basis for a theory of possibility". ''Fuzzy Sets and Systems'', 1, 3-28.〕〔Dubois, D., Prade, H. (1988) ''Possibility Theory: An Approach to Computerized Processing of Uncertainty''. Plenum Press, New York.〕〔Wang, Z., Klir, G.J. (1992) ''Fuzzy Measure Theory''. Plenum Press.〕〔Klir, G.J. (2005) ''Uncertainty and Information. Foundations of Generalized Information Theory''. Wiley.〕〔Sugeno, M. (1974) ''Theory of Fuzzy Integrals and its Applications.'' PhD Dissertation. Tokyo, Institute of Technology.〕 Main results and references on formal fuzzy logic can be found in these citations.〔Hájek, P. (1998) ''Metamathematics of Fuzzy Logic''. Dordrecht: Kluwer.〕〔Esteva, F., Godo, L. (2001) "Monoidal t-norm based logic: Towards a logic of left-continuous t-norms". ''Fuzzy Sets and Systems'', 124, 271–288.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fuzzy mathematics」の詳細全文を読む スポンサード リンク
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