Intuitionistic Fuzzy Approximations and Intuitionistic Fuzzy Sigma-Algebras
In this paper, concepts of intuitionistic fuzzy measurable spaces and intuitionistic fuzzy σ-algebras are first introduced. Relationships between intuitionistic fuzzy rough approximations and intuitionistic fuzzy measurable spaces are then discussed. It i
- PDF / 365,419 Bytes
- 8 Pages / 430 x 660 pts Page_size
- 68 Downloads / 222 Views
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan, Zhejiang 316004, P.R. China 2 Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaan’xi 710049, P.R. China [email protected], zhl [email protected]
Abstract. In this paper, concepts of intuitionistic fuzzy measurable spaces and intuitionistic fuzzy σ-algebras are first introduced. Relationships between intuitionistic fuzzy rough approximations and intuitionistic fuzzy measurable spaces are then discussed. It is proved that the family of all intuitionistic fuzzy definable sets induced from an intuitionistic fuzzy serial approximation space forms an intuitionistic fuzzy σ−algebra. Conversely, for an intuitionistic fuzzy σ−algebra generated by a crisp algebra in a finite universe, there must exist an approximation space such that the family of all intuitionistic fuzzy definable sets is the class of all measurable sets in the given intuitionistic fuzzy measurable space. Keywords: Approximation spaces, intuitionistic fuzzy rough sets, intuitionistic fuzzy sets, measurable spaces, rough sets, σ−algebras.
1
Introduction
Approximation spaces in rough set theory and measurable spaces in measure theory are two important structures to represent knowledge. An approximation space in Pawlak’s rough set theory consists of a universe of discourse and an equivalence relation imposed on it [6]. Based on the approximation space, the notions of lower and upper approximation operators are induced. A set is said to be definable if its lower and upper approximations are the same, and undefinable otherwise [6,12]. The notion of definable sets in rough set theory plays an important role. A measurable space in measure theory contains a universe of discourse and a family of measurable sets called a σ−algebra [4]. Based on the measurable space, uncertainty of knowledge can be analyzed. An equivalence relation in Pawlak’s original rough set model is a very restrictive condition which may affect the application of rough set theory. The generalization of Pawlak’s rough set model is thus one of the main directions for the study of rough set theory. Many authors have developed Pawlak’s rough set model by using nonequivalence relations in crisp and/or fuzzy environments (see e.g. literature cited in [7,8]). Other researchers have also defined rough approximations of intuitionistic fuzzy sets [2,3,5,9,10]. G. Wang et al. (Eds.): RSKT 2008, LNAI 5009, pp. 355–362, 2008. c Springer-Verlag Berlin Heidelberg 2008
356
W.-Z. Wu and L. Zhou
It seems useful to investigate connections of definability in rough set theory and measurability in measure theory. In [11], Wu and Zhang explored the connections between rough set algebras and measurable spaces in both crisp and fuzzy environments. In this paper, we will further study relationship between the two knowledge representation structures in intuitionistic fuzzy environment.
2
Concepts Related to Intuitionistic Fuzzy Sets
Let U be a nonempty set called the universe of d
Data Loading...
