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Rough Sets: Foundations and Perspectives

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Abbreviations

Approximation:

The replacement of mathematical objects by others that resemble them in certain respects [64].

Approximation space :

An approximation space is denoted by \( { \left(\mathcal{O}, \mathcal{F}, \sim_B\right) } \), where \( { \mathcal{O} } \) is a set of perceived objects, \( { \mathcal{F} } \) is a set of probe functions representing object features, and \( { \sim_{B} } \) is an indiscernibility relation defined relative to \( { B\subseteq\mathcal{F} } \).This approximation space is considered fundamental because it provided a framework for the original rough set theory [37,40]. Severalgeneralizations of this definition of approximation space have been proposed(see, e.?g., [40,44,54,55,56,58,59,69]).

Attribute :

A quality regarded as characteristic or inherent inan object [29]. In rough set theory, an attribute a of an object x is represented by a partialfunction \( { f_a(x) = v }\), where v is a value in the range off a . In rough set theory,the function f a is oftencalled an attribute [38,45].

Boundary region :

The B-boundary region of an approximation of a set X is denoted by \( { \mathrm{Bnd}_B X } \) and is defined relative to a set of functions B representing features of objects in X as well as the lower approximation \( { B_{\ast}X } \) and theupper approximation \( { B^{\ast}X } \), where

$$\mathrm{Bnd}_B X = B^{\ast} X \setminus B_{\ast}X =\{x \mid x \in B^{\ast}X\text{ and } x \notin B_{\ast} X\} \:.$$
Elementary set:

A B-class in the quotient set \( { X/\sim_B } \).

Equivalence class:

Given a relation ~, anequivalence class is a set denoted by \( {\left[x\right] } \) or \( { \left[x\right]_{\sim} } \) [10] in the quotient set \( { X/\sim } \) (See Glossary item “Quotientset”), where

$$\left[x\right] = \{x^\prime \in X \mid x \sim x^\prime\}\:.$$
Equivalence relation:

A reflexive, symmetric and transitive relation \( { \sim\subseteq X\times X } \). An equivalence relation ~ on a set X defines a partition of X into classes.

Feature :

Make, form, fashion, shape (of an object) [29]. A characteristic of an object perceived by the senses or knowable by the mind [41,52]. In rough set theory, a feature f of an object x is represented by a function \( { \phi_{f}(x) = v } \), where v is a value in the range of ? f (e.?g., ??(x) as a measure of the tonality ? feature of a Chopin Mazurka x) [41,52]. The function ? f is sometimes also termed an attribute [38,45].

Indiscernibility relation :

An equivalence relation

$$\sim_B = \{(x,x^\prime) \in X \times X | f(x)=f(x^\prime)\text{ for any } f \in B\} \:,$$

where X denotes a set of objects, B denotesa set of functions, and \( { f\in B } \) is a function representing a feature of anobject \( { x \in X } \). The notation used to denote an equivalencerelation in rough set theory has varied widely over time. Forexample, \( { \widetilde B } \) was originally introduced byZdzislaw Pawlak in 1981 [37]. Later,\( { \mathrm{Ind}(B) } \) [18,30,38,66] or\( { \mathrm{IND}_B } \) [45] or \( { \mathrm{Ind}_B } \) [14] or IND [66]or I(B) [40] or \( { =_B } \) [16] has also been used todenote an equivalence relation on a set X defined relative toattributes of objects. In rough set theory, the equivalence relation\( { \sim_B } \) was introduced by Zdzislaw Pawlak [37].

Information granule :

Information granules are obtained in theprocess of granulation. Granulation can be viewed as a human wayof achieving data compression and it plays a key role inimplementing the divide-and-conquer strategy in humanproblem-solving. An information granule represents a set of objects that have descriptions matching the granule [52], e.?g. elementary set\( { \left[x\right]_B } \), lower approximation \( { B_{\ast}X } \), quotient set \( { X/\sim_B } \).

Lower approximation :

The B-lower approximation of a set X is denoted by \( { B_\ast X } \) and is defined relative to a set of functions B representing features of objects in X and the quotient set \( { X/\sim_B } \), where

$$B_{\ast}X = \bigcup_{x: [x]_B\subseteq X} [x]_B \:.$$
Object:

Something perceptible to the senses or knowable by the mind [29].

Information:

Whatever is conveyed or represented by a particular sequence of symbols [29]. In rough set theory, information is derivable either from the patterns in a particular information table or from what can be observed in a particular approximation space [37,40].

Information system :

A system to represent knowledge [25,36,40]. Syntactic representation of knowledge in table form [25,45].

Partition of a non-empty set X :

A family of non-empty, pairwise disjoint subsets of X(called classes) such that the union of this family is equal to X.

Quotient set:

Set of all classes in a partition defined by an equivalence relation ~ on a set X (denoted by \( { X/\sim } \)).

Rough set :

A set X is considered a rough set if, and only if it has a non-empty B-boundary \( { \mathrm{Bnd}_B X } \), i.?e., the B-approximation of X has a non-empty boundary.

Upper approximation :

The B-upper approximation of X is denoted by \( { B^\ast X } \) and is defined relative to a set of functions B representing features of objects in X and the quotient set \( { X/\sim_B } \), where

$$B^{\ast}X = \bigcup_{x: [x]_B\cap X \neq \emptyset}[x]_B \:.$$

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Acknowledgments

The research has been supported by the grant fromMinistry of Scientific Research and Information Technology of theRepublic of Poland and by grant 185986 from the Natural Sciencesand Engineering Research Council of Canada (NSERC).

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Authors and Affiliations

  1. Computational Intelligence Laboratory, University of Manitoba, Winnipeg, Canada

    James F. Peters

  2. Institute of Mathematics, Warsaw University, Warsaw, Poland

    Andrzej Skowron

  3. Computer Science, Bialystok University of Technology, Bialystok, Poland

    Jaroslaw Stepaniuk

Authors
  1. James F. Peters
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  2. Andrzej Skowron
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  3. Jaroslaw Stepaniuk
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Peters, J.F., Skowron, A., Stepaniuk, J. (2009). Rough Sets: Foundations and Perspectives. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_461

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