Upper and Lower Probabilities of Fuzzy Events Induced by a Fuzzy Set-Valued Mapping

In this paper, we study rough set approximations under fuzzy and random environments. A fuzzy set-valued mapping defines a pair of upper and lower fuzzy rough approximations. Properties of fuzzy approximation operators are examined and the crisp representations of fuzzy approximation operators are presented. A fuzzy random variable from a universe U to a universe W carries a probability measure defined over subsets of U into a system of upper and lower probabilities over subsets of W. The connections between fuzzy approximation spaces and fuzzy belief structures are also established.

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