fuzzy mathematics

A fuzzy subset A of a set X is a function A: X → L, where L is the interval [0, 1]. This function is also called a membership function. A membership function is a generalization of an indicator function (also called a characteristic function) of a subset defined for L = {0, 1}. More generally, one can use any 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.

The 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 in X. Instead of {{math|min}} and {{math|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 {{math|min}} and {{math|max}} operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.

An 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 in 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 on X, i.e. R is a fuzzy subset of X × X. Then R is (fuzzy-)transitive if for all x, y, z in X, R(x, z) ≥ min(R(x, y), R(y, z)).

Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld.Rosenfeld, A. (1971) "Fuzzy groups", J. Math. Anal. Appl., 35, 512-517.Mordeson, J.N., Malik, D.S., Kuroli, N. (2003) Fuzzy Semigroups. Studies in Fuzziness and Soft Computing, vol. 131, Springer-VerlagMordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005) Fuzzy Group Theory. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag.

Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory,Mordeson, J.N., Malik, D.S (1998) Fuzzy Commutative Algebra. World Scientific. fuzzy topology,Chang, C.L. (1968) "Fuzzy topological spaces", J. Math. Anal. Appl., 24, 182—190.Liu, Y.-M., Luo, M.-K. (1997) Fuzzy Topology. Advances in Fuzzy Systems - Applications and Theory, vol. 9, World Scientific, Singapore. fuzzy geometry,Poston, Tim, "Fuzzy Geometry".Buckley, J.J., Eslami, E. (1997) "Fuzzy plane geometry I: Points and lines". Fuzzy Sets and Systems, 86, 179-187.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. fuzzy orderings,Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". Inform. Sci., 3, 177–200. and fuzzy graphs.Kaufmann, A. (1973). Introduction a la théorie des sous-ensembles flows. Paris. Masson.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.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.

• Fuzzy measure theory
• Fuzzy subalgebra
• Monoidal t-norm logic
• Possibility theory
• T-norm

{{reflist}}

• Zadeh, L.A. [http://www.scholarpedia.org/article/Fuzzy_Logic Fuzzy Logic] - article at Scholarpedia
• Hajek, P. [http://plato.stanford.edu/entries/logic-fuzzy/ Fuzzy Logic] - article at Stanford Encyclopedia of Philosophy
• Navara, M. [http://www.scholarpedia.org/article/Triangular_Norms_and_Conorms Triangular Norms and Conorms] - article at Scholarpedia
• Dubois, D., Prade H. [http://www.scholarpedia.org/article/Possibility_Theory Possibility Theory] - article at Scholarpedia
• Center [http://www.niftystats.com/ref.php?cid=C84b541099 for] Mathematics of Uncertainty [http://www.creighton.edu/ccas/fuzzymath/ Fuzzy Math Research] {{Webarchive|url=https://web.archive.org/web/20090629104833/http://www.creighton.edu/ccas/fuzzymath/ |date=2009-06-29 }} - Web site hosted at Creighton University
• Seising, R. [https://www.springer.com/engineering/computational+intelligence+and+complexity/book/978-3-540-71794-2] Book on the history of the mathematical theory of Fuzzy Sets: 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.

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Category:Fuzzy logic