Fuzzy mathematics

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"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 [1] . 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 [2].

The evolution of the fuzzification of mathematical concepts can be broken into three stages [3] :

  1. straightforward fuzzification during the sixties and seventies,
  2. the explosion of the possible choices in the generalization process during the eighties,
  3. 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 [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 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)).

Contents

[edit] Some fields of mathematics using fuzzy set theory

Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld [5]. Hundreds of papers on related topics have been published. Recent results and references can be found in [6], [7].

Main results in fuzzy fields and fuzzy Galois theory are published in [8].

Fuzzy topology was introduced by C.L. Chang [9] in 1968 and further was studied in many papers [10].

Main concepts of fuzzy geometry were introduced by A. Rosenfeld in 1974 and by J.J. Buckley and E. Eslami in 1997 [11].

Basic types of fuzzy relations were introduced in [12].

The properties of fuzzy graphs have been studied by A. Kaufman [13], A. Rosenfeld [14] and by R.T. Yeh and S.Y. Bang [15]. Recent results can be found in [16].

Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in [17], [18], [19], [20], [21].

Main results and references on formal fuzzy logic can be found in [22], [23].


[edit] See also

[edit] References

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  1. ^ Zadeh, L.A. (1965) "Fuzzy sets", Information and Control, 8, 338-353.
  2. ^ Goguen, J. (1967) "L-fuzzy sets", J. Math. Anal. Appl., 18, 145-174.
  3. ^ Kerre, E.E., Mordeson, J.N. (2005) "A historical overview of fuzzy mathematis", New Mathematics and Natural Computation, 1, 1-26.
  4. ^ Klement, E.P., Mesiar, R., Pap, E. (2000) Triangular Norms. Dordrecht, Kluwer.
  5. ^ Rosenfeld, A. (1971) "Fuzzy groups", J. Math. Anal. Appl., 35, 512-517.
  6. ^ Mordeson, J.N., Malik, D.S., Kuroli, N. (2003) Fuzzy Semigroups. Studies in Fuzziness and Soft Computing, vol. 131, Springer-Verlag.
  7. ^ Mordeson, J.N., Bhutani, K.R., Rosenfeld, A. (2005) Fuzzy Group Theory. Studies in Fuzziness and Soft Computing, vol. 182. Springer-Verlag.
  8. ^ Mordeson, J.N., Malik, D.S (1998) Fuzzy Commutative Algebra. World Scientific.
  9. ^ Chang, C.L. (1968) "Fuzzy topological spaces", J. Math. Anal. Appl., 24, 182—190.
  10. ^ Liu, Y.-M., Luo, M.-K. (1997) Fuzzy Topology. Advances in Fuzzy Systems - Applications and Theory, vol. 9, World Scientific, Singapore.
  11. ^ Buckley, J.J., Eslami, E. (1997) "Fuzzy plane geometry I: Points and lines". Fuzzy Sets and Systems, 86, 179-187.
  12. ^ Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". Inform. Sci., 3, 177–200.
  13. ^ Kaufmann, A.( 1973). Introduction a la th'eorie des sous-ensembles flous. Paris. Masson.
  14. ^ 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.
  15. ^ 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.
  16. ^ Mordeson, J.N., Nair, P.S. (2000) Fuzzy Graphs and Fuzzy Hypergraphs. Studies in Fuzziness and Soft Computing, vol. 46. Springer-Verlag.
  17. ^ Zadeh, L.A. (1978) "Fuzzy sets as a basis for a theory of possibility". Fuzzy Sets and Systems, 1, 3-28.
  18. ^ Dubois, D., Prade, H. (1988) Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.
  19. ^ Wang, Z., Klir, G.J. (1992) Fuzzy Measure Theory. Plenum Press.
  20. ^ Klir, G.J. (2005) Uncertainty and Information. Foundations of Generalized Information Theory. Wiley.
  21. ^ Sugeno, M. (1974) Theory of Fuzzy Integrals and its Applications. PhD Dissertation. Tokyo, Institute of Technology.
  22. ^ Hájek, P. (1998) Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
  23. ^ 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.

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