## A comparative study of fuzzy rough sets.(English)Zbl 1004.03043

A new approach to the fuzzification of well-known Pawlak’s rough sets is discussed. New sets, called $$({\mathcal I},{\mathcal T})$$-fuzzy sets, are introduced, where $${\mathcal I}$$ is an implicator, i.e., a function $${\mathcal I}:[0,1]^2 \to[0,1]$$ satisfying the conditions: $${\mathcal I}(1,0)=0$$, $${\mathcal I}(0,0)={\mathcal I}(0,1)={\mathcal I}(1,1)=1$$, and $${\mathcal T}$$ is a triangular norm, i.e., an increasing, associative and commutative mapping $${\mathcal T}:[0,1]^2\to [0,1]$$ that satisfies the boundary condition for all $$x\in[0,1]:{\mathcal T}(x,1)=x$$. The basic properties of the new sets are given.

### MSC:

 3e+72 Theory of fuzzy sets, etc.

### Keywords:

fuzzification of rough sets; implicator; triangular norm
Full Text:

### References:

 [1] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, Internat. J. general systems, 17, 2-3, 191-209, (1990) · Zbl 0715.04006 [2] Dubois, D.; Prade, H., Putting fuzzy sets and rough sets together, (), 203-232 [3] E.E. Kerre (Ed.), Introduction to the Basic Principles of Fuzzy Set Theory and Some of Its Applications, Communication & Cognition, Gent, 1993. [4] Klir, G.J.; Yuan, B., Fuzzy logic: theory and applications, (1995), Prentice-Hall Englewood Cliffs, NJ · Zbl 0915.03001 [5] T.Y. Lin (Ed.), Proceedings of the CSC’96 Workshop on Rough Sets and Database Mining, San Jose State University, 1995. [6] Lin, T.Y.; Liu, Q., Rough approximate operatorsaxiomatic rough set theory, () · Zbl 0818.03028 [7] Nachtegael, M.; Kerre, E.E., The dizzy number of fuzzy implication operators on finite chains, (), 29-35 · Zbl 0957.03033 [8] Nakamura, A., Fuzzy rough sets, Note multiple-valued logic Japan, 9, 8, 1-8, (1988) [9] E. Orłowska (Ed.), Incomplete Information: Rough Set Analysis, in Studies in Fuzziness and Soft Computing, Springer, Berlin, 1998. · Zbl 0886.68127 [10] Orłowska, E., Studying incompleteness of informationa class of information logics, (), 283-300 · Zbl 0968.03020 [11] Orłowska, E., Many-valuedness and uncertainty, proc. 27th internat. symp. on multiple-valued logics, Antigonish, Canada, may 1997; also in multiple valued logic—internat. J., 4, 207-227, (1999) [12] Pawlak, Z., Rough sets, Internat. J. comput. inform. sci., 11, 5, 341-356, (1982) · Zbl 0501.68053 [13] Pawlak, Z., Rough sets—theoretical aspects of reasoning about data, (1991), Kluwer Academic Publishers Dordrecht · Zbl 0758.68054 [14] Pawlak, Z., Hard and soft sets, (), 130-135 · Zbl 0819.04008 [15] Ruan, D.; Kerre, E.E., Fuzzy implication operators and generalized fuzzy method of cases, Fuzzy sets and systems, 54, 23-37, (1993) · Zbl 0784.68078 [16] Skowron, A.; Polkowski, L., Rough sets in knowledge discovery, vols. 1, 2, (1998), Springer Berlin · Zbl 0910.00028 [17] R. Słowiński (Ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Kluwer Academic Publishers, Boston, 1992. · Zbl 0820.68001 [18] Smets, Ph.; Magrez, P., Implication in fuzzy logic, Internat. J. approx. reasoning, 1, 327-347, (1987) · Zbl 0643.03018 [19] H. Thiele, On the definition of modal operators in fuzzy logic, Proc. ISMVL-93, Sacramento, California, 1993, pp. 62-67. [20] H. Thiele, Fuzzy rough sets versus rough fuzzy sets—an interpretation and a comparative study using concepts of modal logics, University of Dortmund, Tech. Report No CI-30/98, 1998; also in Proc. EUFIT’97, pp. 159-167. [21] H. Thiele, Generalizing the explicit concept of rough set on the basis of modal logic, in: B. Reusch (Ed.), Computational Intelligence in Theory and Practice, Advances in Soft Computing, Springer, Heidelberg, to appear. · Zbl 1002.68168 [22] Vakarelov, D., Information systems, similarity relations and modal logics, (), 492-550 [23] Ziarko (Ed.), W.P., Rough sets, fuzzy sets and knowledge discovery, workshop in computing, (1994), Springer London · Zbl 0812.00038
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