Generalized Rough Sets

In our real-life problems, there are situations with the uncertain data that may not be successfully modelled by the classical mathematics. There are some mathematical tools for dealing with uncertainties—they are fuzzy set theory introduced by Zadeh [10], rough set theory introduced by Pawlak [7], and soft set theory initiated by Molodtsov [5]. In this chapter, we recall some basic notions relevant to our Chaps. 2–10, such as fuzzy sets, intuitionistic fuzzy sets, interval fuzzy sets, soft set, fuzzy soft sets, rough sets, fuzzy rough sets, fuzzy rough soft set, and others.

Molodtsov initiated the concept of fuzzy soft set theory in 1999. Maji et al. introduced the notion of fuzzy soft sets. By introducing the concept of intuitionistic fuzzy sets into the theory of soft sets, Maji et al. proposed the concept of intuitionistic fuzzy soft set theory. The notion of the interval-valued intuitionistic fuzzy sets was first introduced by Atanassov and Gargov. It is characterised by an interval-valued membership degree and an interval-valued non-membership degree. In 2010, Y. Jiang et al. introduced the concept of interval-valued intuitionistic fuzzy soft sets. In this chapter, the concept of generalised interval-valued intuitionistic fuzzy soft sets is introduced. The basic properties of these sets are presented. Also, an application of generalised interval-valued intuitionistic fuzzy soft sets in decision-making with respect to interval of degree of preference is investigated.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

In this chapter, the concept of interval-valued intuitionistic fuzzy soft rough sets is introduced. Also interval-valued intuitionistic fuzzy soft rough set-based multi-criteria group decision-making scheme is presented, which refines the primary evaluation of the whole expert group and enables us to select the optimal object in a most reliable manner. The proposed scheme is illustrated by an example regarding the candidate selection problem.

In this chapter, the concept of interval-valued intuitionistic fuzzy soft topological space (IVIFS topological space) together with intuitionistic fuzzy soft open sets (IVIFS open sets) and intuitionistic fuzzy soft closed sets (IVIFS closed sets) are introduced. We define neighbourhood of an IVIFS set, interior IVIFS set, interior of an IVIFS set, exterior IVIFS set, exterior of an IVIFS set, closure of a IVIFS set, IVIFS basis, and IVIFS subspace. Some examples and theorems regarding these concepts are presented.

In this chapter, we introduce the concept of interval-valued intuitionistic fuzzy soft multi-sets and study its properties and operations. Then, the concept of interval-valued intuitionistic fuzzy soft multi-set relations (IVIFSMS-relations) is proposed. The basic properties of the IVIFSMS-relations are also discussed. Finally, various types of IVIFSMS-relations are presented.

In this chapter, the concepts of interval-valued neutrosophic sets (IVNS in short), interval-valued neutrosophic soft sets (IVNSS in short) and IVNSS relations (IVNSS-relations in short) are proposed. The basic properties of IVNS-, IVNSS-, and IVNSS-relations are also presented and discussed. Also various types of IVNSS-relations are presented. Finally, a solution to a decision-making problem using IVNSS-relation is presented.

The purpose of this chapter was to study the concept of topological structure formed by soft multi-sets. The notion of relative complement of soft multi-set, soft multi-point, soft multi-open set, soft multi-closed set, soft multi-basis, soft multi-sub-basis, neighbourhoods and neighbourhood system, interior and closure of a soft multi-set, etc., is to be introduced, and their basic properties are also to be investigated. It is seen that a soft multi-topological space gives a parameterised family of topological spaces. Lastly, the concept of soft multi-compact space is also introduced.

The vagueness or the representation of imperfect knowledge has been a problem for a long time for the mathematicians. There are many mathematical tools for dealing with uncertainties; some of them are fuzzy set theory, rough set theory, and soft set theory. In this chapter, the concept of soft interval-valued intuitionistic fuzzy rough sets is introduced. Also some properties based on soft interval-valued intuitionistic fuzzy rough sets are presented. Also a soft interval-valued intuitionistic fuzzy rough set-based multi-criteria group decision-making scheme is presented. The proposed scheme is illustrated by an example regarding the car selection problem.

In this chapter, we introduce the concept of intuitionistic fuzzy parameterised intuitionistic fuzzy soft (ifpifs) sets and their operations with examples. We also define the approximate functions of ifpifs-set from the intuitionistic fuzzy parameterised set to the intuitionistic fuzzy subsets [1] of universal set. Lastly, we construct an ifpifs-set decision-making problem and try to solve the problem.