Fuzziness in rough sets
In the paper the authors define a measure of fuzziness in a rough set and investigate its properties. Every approximation space \((U, R)\), where \(R\) is an equivalence relation on \(U\), and a subset \(X\) of \(U\) determine a rough set \(R(X)\). Let \(\operatorname {card}(Y)\) denote the cardinality of \(Y\). With \((U, R)\) and \(X\) we associate a fuzzy set \(F^{R}_{X}\) of \(U\): \[ F^{R}_{X}=\{(u, \mu_{F^{R}_{X}}(u)) |u\in U\wedge \mu_{F^{R}_{X}}(u)= \operatorname {card}(\{[u]_{R}\cap X\})/\operatorname {card}([u]_{R})\} \] The fuzziness in \(R(X)\) is defined by the amount of fuzziness in \(F^{R}_{X}\) and measured by a suitable (linear or quadratic) index of fuzziness.
scientific article; zbMATH DE number 1423470
scientific article