Generalized Rough Sets in Contextual Spaces

KEY WORDS Contextual spaces, context relation, contextual rough sets, elements of a contextual rough set, the counterpart of the axiom of extensionality. ABSTRACT This paper originates from the conceptions of rough sets presented by Pawlak (1982, 1992). It also refers to Ziarko's conception (1993) and the conceptions of Blizard's multisets (1989a, 1989b), Zadeh's fuzzy sets (1965), and the authors' conceptions of generalized rough sets and an approximation space called the contextual space (1995). The authors' conception of contextual space is inspired by Ziarko's approach (1993) to rough sets. Rough sets introduced by Pawlak (1982) are particular cases of contextual rough sets deened in the contextual approximation space. This space is deened axiomatically by means of so called context relations. Every contextual rough set determined by set X can be determined by the sum 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.