Fuzzy mathematics

:"Fuzzy math" redirects here. For the controversies about mathematics education curricula that are sometimes disparaged as "fuzzy math," see Math wars."Fuzzy mathematics form a branch of mathematics related to fuzzy logic. It started in 1965 after publication by Lotfi Asker Zadeh of his 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 [0,1] . This function 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" [ Goguen, J. (1967) "L-fuzzy sets", "J. Math. Anal. Appl.", 18, 145-174. ] .

The evolution of the fuzzification of mathematical concepts can be broken into three stages [ Kerre, E.E., Mordeson, J.N. (2005) "A historical overview of fuzzy mathematis", "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" U "B" are defined as follows: ("A" ∩ "B")("x") = min("A"("x"),"B"("x")), ("A" "U" "B")("x") = max("A"("x"),"B"("x")) for all "x" $in$ "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" $in$ "X", "A"("x"*"y") ≥ min("A"("x"),"B"("x")). 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. ] , [ 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 [ 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 A. Rosenfeld in 1974 and 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. ] .

Basic types of fuzzy relations were introduced in [ 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 th'eorie des sous-ensembles flous". Paris. Masson. ] , A. Rosenfeld [A. Rosenfeld, A. (1975) "Fuzzy graphs". In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds) "Fuzzy Sets and Their Applications", Academic Press, New York, 77-95. ] and by R.T. Yeh and S.Y. Bang [ Yeh, R.T., Bang, S.Y. Fuzzy graphs, fuzzy relations and their applications to cluster analysis. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds) "Fuzzy Sets and Their Applications", Academic Press, New York, 125-149.] . Recent results can be found in [ 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 [ 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 [ 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.] .

ee also

* Fuzzy subalgebra
* t-norm
* Monoidal t-norm logic
* Possibility theory
* Fuzzy measure theory

References

- reflist

External links

* 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