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Generalized Rough Sets in Contextual Spaces

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This paper presents a generalization of Pawlak’s conception of rough sets [6] and [7]. It is more general than Pawlak’s solution of the problem of the definability of sets, the knowledge of which is incomplete and vague. The authors’ conception is based on conception of contextual space [4], which was inspired by Ziarko’s approach [12] to rough sets. Rough sets introduced by Pawlak [6] are particular cases of contextual rough sets defined in the contextual approximation space. This space is defined axiomatically by means of so called context relations. Every contextual rough set determined by set X can be determined by the union of the lower approximation of X and a subset of the boundary of X. One of the important notions of the conception is the notion of an element of a contextual rough set which allows for formulating and proving the counterpart of the axiom of extensionality for contextual rough sets.