Fuzzy mathematics

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]

  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 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]

gollark: I can imagine, for limited values of "imagine" (I can't picture them usefully, or anything like that, but I can't do that anyway), impossible things like "colourless green", say.
gollark: Well, they're related to things people know about, sure. I don't think you clarified what you meant by "spiritual understanding".
gollark: And nonanimals.
gollark: These are just slight variations on existing animals.
gollark: I don't think this is true, except in a very broadly defined sense.

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 flows. 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.