×

Fuzzy sets. (English) Zbl 0139.24606

A fuzzy set is a “set” of elements with a continuum of “grades of membership”. The rigorous definition is: let \(X\) be a set of objects (elements); a fuzzy set \(A\) in \(X\) is defined by a “membership (characteristic) function” \(f_A\), which associates with each element \(x\in X\) a real number \(f_A(x)\in [0,1]\). The value \(f_A(x)\) of \(f_A\) at \(x\) represents the grade of membership of \(x\) in \(A\). If \(A\) is an “ordinary” set, its membership function \(f_A\) can take on only the values 0 and 1: \(x\in A\Leftrightarrow f_A(x) = 1\) and \(x\neq A \Leftrightarrow f_A(x)=0\). Various usual notions are extended to such sets (for instance, the notions of union, intersection and convexity).

MSC:

03E72 Theory of fuzzy sets, etc.
Full Text: DOI

References:

[1] Zadeh L A. Fuzzy sets. Inf Control, 1965, 3: 338-353 · Zbl 0139.24606
[2] Pawlak Z. Rough sets. Int J Comput Inf Sci, 1982, 11: 341-356 · Zbl 0501.68053
[3] Zhang L, Zhang B. Theory and Applications of Problem Solving (in Chinese). Beijing: Tsinghua University Press, 1990
[4] Liang P, Song F. What does a probabilistic interpretation of fuzzy sets mean? IEEE Trans Fuzzy Syst, 1996, 4: 200-205
[5] Lucero Y C, Nava P A. A method for membership function generation from training samples. http://www.ece.utep.edu/research/webfuzzy/docs/electro99.doc
[6] Mitsuishi T, Endou N, Shidama Y. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. J Formal Math, 2000, 12: 1-6
[7] Lin T Y. Neighborhood systems and approximation in relational databases and knowledge bases. In: Proceedings of the 4th International Symposium on Methodologies of Intelligent Systems, Turin, 1988
[8] Lin T Y. Topological and fuzzy rough Sets. In: Slowinski R, ed. Decision Support by Experience-Application of the Rough Sets Theory. Kluwer Academic Publishers, 1992. 287-304 · Zbl 0820.68001
[9] Lin T Y. Neighborhood systems-application to qualitative fuzzy and rough sets, in: Wang P P, ed. Advances in Machine Intelligence and Soft-Computing. Department of Electrical Engineering, Duke University, Durham, North Carolina, USA, 1997. 132-155
[10] Lin T Y. Context free fuzzy sets and information tables. In: European Congress on Intelligent Techniques and Soft Computing, 1998. 76-80
[11] Skowron A, Stepaniuk J. Tolerance approximation spaces. Fund Inf, 1996, (27): 245-253 · Zbl 0868.68103
[12] Doherty P, Lukaszewicz W, Szalas A. Tolerance spaces and approximative representational structures. In: Proceeding of 26 th German Conference on Artificial Intelligence, Hamburg, Germany, 2003. 475-489 · Zbl 1274.68482
[13] Cattaneo G. Abstract Approximation Spaces for Rough Theories, Ruogh Sets in Knowledge Discovery 1: Methodology and Applications. Heidelberg: Physica-Verlag, 1998. 59-98 · Zbl 0927.68087
[14] Cattaneo G, Ciucci D. Algebraic structures for rough sets. In: Lecture Notes in Computer Science 3135. Berlin: Springer, 2004. 208-252 · Zbl 1109.68115
[15] Slowinski R, Vanderpooten D. A generalized definition of rough approximations based on similarity. IEEE Trans Knowl Data Eng, 2000, 12: 331-336
[16] Shi Z Z. The model and application of tolerance granular spaces. In: Miao D Q, ed. Granular Computing: Past, Present and Future (Chinese), Scientific Publishing Company. 2007. 42-82
[17] Zhu F, Wang F Y. Covering based granular computing. In: Miao D Q. ed. Granular Computing: Past, Present and Future (Chinese). Scientific Publishing Compony, 2007. 83-111
[18] Yao Y Y. The art of granular computing. In: Miao D Q, ed. Granular Computing: Past, Present and Future (in Chinese). Scientific Publishing Compony, 2007. 1-20
[19] Zhang L, Zhang B. Fuzzy quotient spaces (fuzzy granular computing approaches). Chin J Soft, 2003, 14: 770-776 · Zbl 1025.68040
[20] Zhang L, Zhang B. The theory and application of tolerance relations. Int J Granular Compu, Rough Sets Intell Syst, (accepted)
[21] Zhang L, Zhang B. The structural analysis of fuzzy sets. Int J Approx Reason, 2005, 40: 92-108 · Zbl 1074.03028
[22] Boixader D, Jacas J, Recasens J. Upper and lower approximations of fuzzy sets. Int J General Sys, 2000, 29: 555-568 · Zbl 0955.03056
[23] Radzikowska A M, Kerre E E. A comparative study of fuzzy rough sets. Fuzzy Sets Syst, 2002, 126: 137-155 · Zbl 1004.03043
[24] Wu W Z, Zhang W X. Generalized fuzzy rough sets. Inf Sci, 2003, 151: 263-282 · Zbl 1019.03037
[25] Mi J S, Zhang W X. An axiomatic characterization of fuzzy generalization of rough sets. Int J General Syst, 2004, 160: 235-249 · Zbl 1041.03038
[26] Yeung D S, Chen D G, Tsang E C, et al. On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst, 2005, 13: 343-361
[27] Mi J S, Leung Y, Wu W Z. An uncertainty in partition-based fuzzy rough sets. Int J General Syst, 2005, 34: 77-90 · Zbl 1125.03309
[28] Zadeh L A. Similarity relations and fuzzy orderings. Inf Sci, · Zbl 0218.02058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.