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 be an information system (attribute–value system), where is a non-empty, finite set of objects (the universe) and is a non-empty, finite set of attributes such that for every . is the set of values that attribute may take. The information table assigns a value from to each attribute and object in the universe .
With any there is an associated equivalence relation :
The relation is called a -indiscernibility relation. The partition of is a family of all equivalence classes of and is denoted by (or ).
If , then and are indiscernible (or indistinguishable) by attributes from .
The equivalence classes of the -indiscernibility relation are denoted .
For example, consider the following information table:
{| class="wikitable" style="text-align:center; width:30%" border="1"
|+ Sample Information System
! Object !! !! !! !! !!
|-
!
| 1 || 2 || 0 || 1 || 1
|-
!
| 1 || 2 || 0 || 1 || 1
|-
!
| 2 || 0 || 0 || 1 || 0
|-
!
| 0 || 0 || 1 || 2 || 1
|-
!
| 2 || 1 || 0 || 2 || 1
|-
!
| 0 || 0 || 1 || 2 || 2
|-
!
| 2 || 0 || 0 || 1 || 0
|-
!
| 0 || 1 || 2 || 2 || 1
|-
!
| 2 || 1 || 0 || 2 || 2
|-
!
| 2 || 0 || 0 || 1 || 0
|}
When the full set of attributes is considered, we see that we have the following seven equivalence classes:
Thus, the two objects within the first equivalence class, , cannot be distinguished from each other based on the available attributes, and the three objects within the second equivalence class, , cannot be distinguished from one another based on the available attributes.
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Granular computing is an emerging computing paradigm of information processing that concerns the processing of complex information entities called "information granules", which arise in the process of data abstraction and derivation of knowledge from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like.
Type-2 fuzzy sets and systems generalize standard Type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Lotfi A.
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
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