Abstract
The essence of granular computing (GrC) is to replace the concept of points in classical mathematics by that of granules. Usual fuzzy number systems are obtained by using type I fuzzy sets as granules. These fuzzy number systems have a common weakness - lack of existence theorem. Let R be the real number system, the trapezoidal membership functions at r ∈ R is a base of fuzzified topological neighborhood system FNS(r). By taking FNS(r) as the granule, a new (but not type I) fuzzy number system \(\mathcal{F}\) is formed. Surprisingly, we have found that such a new \(\mathcal{F}\) is abstractly isomorphic to the classical real number system.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
References
Birkhof, G., MacLane, S.: A Survey of Modern Algebra. MacMillan Publishers (1970)
Delillo, N.J.: Adanced Calculus with Applications. Macmillan Publishing Co. (1982)
Dobois, D., Prade, H.: Fuzzy Sets and System: Theory and Applications. Academic (1980)
Robinson, A.: Non-standard analysis. North-Holland Pub. Co. (1966)
Kelley, J.: General topology. Springer, New York (1975) ISBN 0387901256
Lin, T.Y.: Chinese Wall security policy -an aggressive model. In: Proceedings of the Fifth Aerospace Computer Security Application Conference, December 4-8 (1989)
Lin, T.Y.: Neighborhood Systems and Approximation in Database and Knowledge Base Systems. In: Proceedings of the Fourth International Symposium on Methodologies of Intelligent Systems, Poster Session, October 12-15 (1989)
Lin, T.Y.: Qualitative Fuzzy Sets: A comparison of three approaches. In: Proceeding of Joint 9th IFSA World Congress and 20th NAFIPS International Conference, Vancouver, Canada, July 25-28, pp. 2359–2363 (2001)
Lin, T.Y., Syau, Y.-R.: Granular Mathematics foundation and current state. GrC (2011)
Zadeh, L.: Fuzzy sets and Information Granularity. In: Gupta, M., Ragade, R., Yager, R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 3–18. North-Holland, Amsterdam (1979)
Zadeh, L.: The Key Roles of Information Granulation and Fuzzy logic in Human Reasoning. In: 1996 IEEE International Conference on Fuzzy Systems, September 8-11, p. 1 (1996)
Zadeh, L.: Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information/intelligent systems. Soft Comput. 2(1), 23–25 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lin, T.Y. (2013). A Mathematical Theory of Fuzzy Numbers. In: Ciucci, D., Inuiguchi, M., Yao, Y., Ślęzak, D., Wang, G. (eds) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. RSFDGrC 2013. Lecture Notes in Computer Science(), vol 8170. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41218-9_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-41218-9_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41217-2
Online ISBN: 978-3-642-41218-9
eBook Packages: Computer ScienceComputer Science (R0)