Fuzzy mathematics

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.[1] 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]

The evolution of the fuzzification of mathematical concepts can be broken down 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 ∪ 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 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 .[5] Hundreds of papers on related topics have been published. Recent results and references can be found in [6] and.[7]

Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.[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 Tim Poston in 1971,[11] A. Rosenfeld in 1974, by J.J. Buckley and E. Eslami in 1997[12] and by D. Ghosh and D. Chakraborty in 2012-14 [13] [14]

Basic types of fuzzy relations were introduced by Zadeh in 1971.[15]

The properties of fuzzy graphs have been studied by A. Kaufman,[16] A. Rosenfel,[17] and by R.T. Yeh and S.Y. Bang.[18] Recent results can be found in a 2000 article.[19]

Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in the cited articles and treatises.[20][21][22][23][24]

Main results and references on formal fuzzy logic can be found in these citations.[25][26]

See also

References

  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 mathematics", 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. ↑ Poston, Tim, "Fuzzy Geometry".
  12. ↑ Buckley, J.J., Eslami, E. (1997) "Fuzzy plane geometry I: Points and lines". Fuzzy Sets and Systems, 86, 179-187.
  13. ↑ Ghosh, D., Chakraborty, D. (2012) "Analytical fuzzy plane geometry I". Fuzzy Sets and Systems, 209, 66-83.
  14. ↑ Chakraborty, D. and Ghosh, D. (2014) "Analytical fuzzy plane geometry II". Fuzzy Sets and Systems, 243, 84–109.
  15. ↑ Zadeh L.A. (1971) "Similarity relations and fuzzy orderings". Inform. Sci., 3, 177–200.
  16. ↑ Kaufmann, A. (1973). Introduction a la théorie des sous-ensembles flous. Paris. Masson.
  17. ↑ 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.
  18. ↑ 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.
  19. ↑ Mordeson, J.N., Nair, P.S. (2000) Fuzzy Graphs and Fuzzy Hypergraphs. Studies in Fuzziness and Soft Computing, vol. 46. Springer-Verlag.
  20. ↑ Zadeh, L.A. (1978) "Fuzzy sets as a basis for a theory of possibility". Fuzzy Sets and Systems, 1, 3-28.
  21. ↑ Dubois, D., Prade, H. (1988) Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.
  22. ↑ Wang, Z., Klir, G.J. (1992) Fuzzy Measure Theory. Plenum Press.
  23. ↑ Klir, G.J. (2005) Uncertainty and Information. Foundations of Generalized Information Theory. Wiley.
  24. ↑ Sugeno, M. (1974) Theory of Fuzzy Integrals and its Applications. PhD Dissertation. Tokyo, Institute of Technology.
  25. ↑ Hájek, P. (1998) Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
  26. ↑ 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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