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 described in Pawlak (1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets.
Definitions 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 it from more recent extensions and generalizations.
Information system framework 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 I : U → V a {\displaystyle I:\mathbb {U} \rightarrow V_{a}} for every a ∈ A {\displaystyle a\in \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 P\subseteq \mathbb {A} } there is an associated equivalence relation I N D ( P ) {\displaystyle \mathrm {IND} (P)} :
I N D ( P ) = { ( x , y ) ∈ U 2 ∣ ∀ a ∈ P , a ( x ) = a ( y ) } {\displaystyle \mathrm {IND} (P)=\left\{(x,y)\in \mathbb {U} ^{2}\mid \forall a\in P,a(x)=a(y)\right\}}
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 [x]_{P}} .
Example: equivalence-class structure For example, consider the following information table:
When the full set of attributes P = { P 1 , P 2 , P 3 , P 4 , P 5 } {\displaystyle P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}} is considered, we see that we have the following seven equivalence classes:
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