Fuzzy mathematics

For other uses, see Fuzzy math (disambiguation).

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. 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 .

The evolution of the fuzzification of mathematical concepts can be broken down into three stages:

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 , 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 . Hundreds of papers on related topics have been published. Recent results and references can be found in and.

Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.

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

Main concepts of fuzzy geometry were introduced by Tim Poston in 1971, A. Rosenfeld in 1974, by J.J. Buckley and E. Eslami in 1997 and by D. Ghosh and D. Chakraborty in 2012-14

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

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

Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in the cited articles and treatises.

Main results and references on formal fuzzy logic can be found in these citations.

See also

• Fuzzy measure theory
• Fuzzy subalgebra
• Monoidal t-norm logic
• Possibility theory
• T-norm

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.

• 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 flous. 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.
• Mordeson, J.N., Nair, P.S. (2000) Fuzzy Graphs and Fuzzy Hypergraphs. Studies in Fuzziness and Soft Computing, vol. 46. Springer-Verlag.
• 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.
• 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.