Fuzzy mathematics
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Fuzzy mathematics forms a branch of mathematics including fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets.[1]
Definition
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 a characteristic function or an indicator function of a subset defined for L = {0,1}. More generally, one can use a complete lattice L in a definition of a fuzzy subset A .[2]
Fuzzification
The evolution of the fuzzification of mathematical concepts can be broken down into three stages:[3]
- 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 member 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 ,[4] 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 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)).
Fuzzy analogues
Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld .[5][6][7]
Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory,[8] fuzzy topology,[9][10] fuzzy geometry,[11][12][13][14] fuzzy orderings,[15] and fuzzy graphs.[16][17][18]
See also
References
- ^ Zadeh, L. A. (1965) "Fuzzy sets", Information and Control, 8, 338–353.
- ^ Goguen, J. (1967) "L-fuzzy sets", J. Math. Anal. Appl., 18, 145-174.
- ^ Kerre, E.E., Mordeson, J.N. (2005) "A historical overview of fuzzy mathematics", New Mathematics and Natural Computation, 1, 1-26.
- ^ Klement, E.P., Mesiar, R., Pap, E. (2000) Triangular Norms. Dordrecht, Kluwer.
- ^ 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-Verlag
- ^ Mordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005) Fuzzy Group Theory. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag.
- ^ Mordeson, J.N., Malik, D.S (1998) Fuzzy Commutative Algebra. World Scientific.
- ^ 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.
- ^ 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.
- ^ Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". Inform. Sci., 3, 177–200.
- ^ 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.
External links
- Zadeh, L.A. Fuzzy Logic - article at Scholarpedia
- Hajek, P. Fuzzy Logic - article at Stanford Encyclopedia of Philosophy
- Navara, M. Triangular Norms and Conorms - article at Scholarpedia
- Dubois, D., Prade H. Possibility Theory - article at Scholarpedia
- Center for Mathematics of Uncertainty Fuzzy Math Research - Web site hosted at Creighton University
- Seising, R. [1] 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.