Soft sets combined with fuzzy sets and rough sets: a tentative approach

Abstract

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft-rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

On Interval Soft Sets with Applications

“…Jun et al 5,6 introduced the notion of soft BCK/BCIalgebras and discussed the applications of soft sets in ideal theory of BCK/BCI-algebras. Feng et al 3,4 applied soft set theory to the study of semirings and introduced the notions of rough soft sets and soft rough sets. Qin et al 16 introduced the notion of soft equality and established lattice structures and soft quotient algebras of soft sets.…”

“…Example 5.10 Let U be the set of houses under consideration, suppose U = {h 1 , h 2 , h 3 , h 4 , h 5 , h 6 }. E 1 is the set of parameters, each parameters is a word or a sentence, suppose E 1 = {e 1 (beautiful), e 2 (wooden), e 3 (cheap), e 4 (in the green surroundings), e 5 (in good repair), e 6 (expensive)}.…”

A study on soft rough semigroups and corresponding decision making applications

“…As we all know, both rough sets and soft sets are tools for dealing with incompletion problems, and an ER can be replaced by other relations, such as binary relations in [5], therefore Feng et al [20][21][22] built rough sets based on soft sets rather than an ER, which were called SR-sets. Following that, some studies were made on SR-sets.…”

Soft Vibrational Force on Stock Market Networks

“…By the arise of the theory its algebraic [8] [9] and topological [10] [11] properties, its relation with other theories [12]- [14], and also implicational feature of the theory [15]- [18] have been studied intensively. We refer [19] to the interested readers for soft set theoretical analogues of the basic set operations.…”