Fusion Research, Based On The Theory Of Soft Sets Uncertainty

The real world is full of various type of uncertainties, including randomness, fuzziness, vagueness, imprecision and roughness. Actually, most of the concepts we are meeting in everyday life are vague rather than absolutely precise. In recent years, researchers from a wide range of areas such as mathematics, quantum physics, com-puter science, information science, decision analysis and artificial intelligence have been increasingly interested in modelling and manipulating uncertainty. But classical mathematics, with its foundation based on crisp sets and Boolean logic, is in many cases no longer suitable for dealing with uncertainty. If we simply make use of tradi-tional methods by ignoring the uncertainty inherent in concepts and collected data, the right answers would not always been there; sometimes what we have finally obtained would be quite ridiculous and unreasonable.To solve these difficulties, American cybernetician Zadeh proposed the theory of fuzzy sets as a formal method dedicated to the analysis and manipulation of uncer-tainty in1965. Fuzzy sets are very useful extensions of crisp sets, which have been extensively used in almost every branches of modern mathematics. In fuzzy set theory, uncertainty is characterized in terms of membership functions, and this is philosoph-ically justified by the idea of gradualness. In1982, Polish mathematician Pawlak initiated rough set theory, in which uncertainty is expressed by the lower and upper approximations constructed from equivalence classes of indiscernibility relations. The underlying philosophy for rough sets can be understood using the idea of granularity. In order to cope with uncertainty in a more general scheme, Molodtsov at the Russian Academy of Science initiated the theory of soft sets in1999. A soft set is a pair con-sisting of a parameter set and a set-valued mapping from the parameter set into the power set of the universe of discourse. Each parameter corresponds to a crisp set called approximate set, which gives an approximate description of the concepts with uncer-tainty. All of the approximate sets can be naturally viewed as a granular structure of the universe. The soft set philosophy is founded on the idea of parametrization, which suggests that complicated concepts with uncertainty should be characterized from many different aspects and all related facets should be organized as a whole. As a newly-emerging area of interdisciplinary research, soft set theory have re-ceived much attention from a number of researchers all over the world. Evidence of this can be found in the increasing number of high-quality works on soft sets and re-lated topics that have been published over past few years. However, we should never forget Molodtsov's original idea for introducing soft sets, with the hope of establishing a more general scheme and obtaining more powerful tools for dealing with uncertainty. In view of these facts, in the present study we concentrate on the fundamental theory of soft sets, with particular emphasis on the fusion of soft sets and other uncertain theories like fuzzy sets and rough sets. Specifically, the main contributions of this dissertation can be summarized as follows:(1) We analyze various types of soft subsets and introduce the notion of soft L-subsets, which is precisely in the midway between the existing concepts of soft M-subsets and soft J-subsets. Some useful characterizations of these soft subsets are obtained and the relationships among several types of soft subsets and the induced soft equalities are ascertained.(2) Various types of soft set operations and their related properties are investi-gated in a systematic way. We divide operations of soft sets into two types, namely Molodtsov product operations and set-theoretic operations. Concerning the study of Molodtsov product operations, we introduce some new notions such as (?)-product,(?)'-product,↑-product and↓-product. Some non-classical interesting algebraic prop-erties are obtained, which are not always similar to the cases of crisp sets or fuzzy sets. For instance, we point out that the Λ-product and V-product do not satisfy the classical commutative and associative laws, but these algebraic properties indeed hold in the sense of soft L-equality. But even in the weakest sense of soft J-equality, the two operations still do not satisfy the distributive laws. With regard to set-theoretic operations of soft sets, we explore some algebraic systems and ordered algebraic struc-tures of soft sets with respect to operations like extended union, extended intersection, restricted union and restricted intersection. The obtain structures include monoids, semirings, anti-semirings, almost semirings, distributive lattices, ordered semigroups, semilattice-ordered semigroups and ordered semirings.(3) We carry out a thorough analysis on the fusion of soft sets and fuzzy sets. The notion of canonical level soft sets is introduced, which can serve as a vital nexus between soft sets and fuzzy sets. It is shown that classical fuzzy sets can be identified with their canonical level soft sets, and all the basic operations of fuzzy sets can be directly implemented by corresponding operations of nested soft sets. Actually, we show that fuzzy lattices are isomorphic to the quotient algebra of nested soft sets. By ignoring the operations, soft sets can formally be regarded as L-fuzzy sets. On the other hand, we find that most of the soft set operations cannot be simply implemented by L-fuzzy set operations. In addition, some decomposition theorems of fuzzy soft sets are obtained by using the notion of scalar product of a threshold fuzzy set and a fuzzy soft set. We also incorporate various types of soft set models into a uniform framework called L-fuzzy soft sets.(4) We explore the combination of soft sets and rough sets in detail. We indi-cate that soft sets, information systems and rough sets are closely related. Soft sets naturally induce Boolean information systems, whence we can discuss the correspond-ing rough approximations and Pawlak rough sets. Pawlak approximate operators and rough sets both can be seen as soft sets. Based on Pawlak approximation spaces, we introduce the notion of rough soft sets, and prove that Dubois and Prade's rough fuzzy sets can be seen as a special case of rough soft sets. We propose the concept of soft approximation spaces, in which a soft set is used to provide the granular structure of the universe. Based on soft approximation spaces, soft rough approximations and soft rough sets are introduced. It is shown that soft rough approximations degenerate into Pawlak approximations when the soft set contained in the soft approximation space is a partition soft set. Finally, we further apply the idea of gradualness and obtain some hybrid soft computing models of great generality, such as rough fuzzy soft sets and soft rough fuzzy sets.