Rough set

In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets. In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets. The following section contains an overview of the basic framework of rough set theory, as originally proposed by Zdzisław I. Pawlak, along with some of the key definitions. More formal properties and boundaries of rough sets can be found in Pawlak (1991) and cited references. The initial and basic theory of rough sets is sometimes referred to as 'Pawlak Rough Sets' or 'classical rough sets', as a means to distinguish from more recent extensions and generalizations. Let I = ( U , A ) {displaystyle I=(mathbb {U} ,mathbb {A} )} be an information system (attribute-value system), where U {displaystyle mathbb {U} } is a non-empty, finite set of objects (the universe) and A {displaystyle mathbb {A} } is a non-empty, finite set of attributes such that a : U → V a {displaystyle a:mathbb {U} ightarrow V_{a}} for every a ∈ A {displaystyle ain mathbb {A} } . V a {displaystyle V_{a}} is the set of values that attribute a {displaystyle a} may take. The information table assigns a value a ( x ) {displaystyle a(x)} from V a {displaystyle V_{a}} to each attribute a {displaystyle a} and object x {displaystyle x} in the universe U {displaystyle mathbb {U} } . With any P ⊆ A {displaystyle Psubseteq mathbb {A} } there is an associated equivalence relation I N D ( P ) {displaystyle mathrm {IND} (P)} : The relation I N D ( P ) {displaystyle mathrm {IND} (P)} is called a P {displaystyle P} -indiscernibility relation. The partition of U {displaystyle mathbb {U} } is a family of all equivalence classes of I N D ( P ) {displaystyle mathrm {IND} (P)} and is denoted by U / I N D ( P ) {displaystyle mathbb {U} /mathrm {IND} (P)} (or U / P {displaystyle mathbb {U} /P} ). If ( x , y ) ∈ I N D ( P ) {displaystyle (x,y)in mathrm {IND} (P)} , then x {displaystyle x} and y {displaystyle y} are indiscernible (or indistinguishable) by attributes from P {displaystyle P} . The equivalence classes of the P {displaystyle P} -indiscernibility relation are denoted [ x ] P {displaystyle _{P}} .