## 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:

 3e+72 Theory of fuzzy sets, etc.
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### 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 [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
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