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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:

03E72 Theory of fuzzy sets, etc.
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