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.

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 ${\displaystyle I=(\mathbb {U} ,\mathbb {A} )}$ be an information system (attribute-value system), where ${\displaystyle \mathbb {U} }$ is a non-empty, finite set of objects (the universe) and ${\displaystyle \mathbb {A} }$ is a non-empty, finite set of attributes such that ${\displaystyle a:\mathbb {U} \rightarrow V_{a}}$ for every ${\displaystyle a\in \mathbb {A} }$. ${\displaystyle V_{a}}$ is the set of values that attribute ${\displaystyle a}$ may take. The information table assigns a value ${\displaystyle a(x)}$ from ${\displaystyle V_{a}}$ to each attribute ${\displaystyle a}$ and object ${\displaystyle x}$ in the universe ${\displaystyle \mathbb {U} }$.

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