Abstract
A fuzzy rough variable is defined as a rough variable on the universal set of fuzzy variables, or a rough variable taking ‘fuzzy variable’ values. In order to further discuss the mathematical properties of fuzzy rough variables, this paper extends some inequalities to the context of fuzzy rough theory based on the chance measure and the expected value operator, involving the Markov inequality, the Chebyshev inequality, the Hölder inequality, the Minkowski inequality, and the Jensen inequality. After that, linearity, monotonicity, and continuity of critical values of fuzzy rough variables are also investigated.
Highlights
Fuzzy set theory has been well developed and applied in a wide variety of real problems since it was proposed in by Zadeh [ ]
By means of a mathematical way, a fuzzy variable was defined as a function from a possibility space to the set of real numbers by Liu [ ]
In order to deal with vague description of objects, rough set theory was initialized by Pawlak [ ] in, which provides a new powerful mathematical approach to handling imperfect knowledge in the real world
Summary
Fuzzy set theory has been well developed and applied in a wide variety of real problems since it was proposed in by Zadeh [ ]. By means of a mathematical way, a fuzzy variable was defined as a function from a possibility space to the set of real numbers by Liu [ ]. A fuzzy variable is defined as a function from a possibility space ( , P( ), Pos) to the set of real numbers.
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