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

Soft Comput (2010) 14:899–911 DOI 10.1007/s00500-009-0465-6 ORIGINAL PAPER Soft sets combined with fuzzy sets and rough sets: a tentative approach Feng Feng Æ Changxing Li Æ B. Davvaz Æ M. Irfan Ali Published online: 27 June 2009
Springer-Verlag 2009 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. F. Feng (&)
C. Li Department of Applied Mathematics and Applied Physics, Xi’an Institute of Posts and Telecommunications, 710061 Xi’an, People’s Republic of China e-mail: fengnix@hotmail.com B. Davvaz Department of Mathematics, Yazd University, Yazd, Iran e-mail: davvaz@yazduni.ac.ir M. I. Ali Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan e-mail: mirfanali13@yahoo.com Keywords Soft set
Fuzzy set
Rough set
Rough fuzzy set
Approximation space
Approximation operator 1 Introduction In some sense almost all concepts we are meeting in everyday life are vague rather than precise. On the contrary, it is interesting to see that classical mathematics requires that all mathematical notions must be exact, otherwise precise reasoning would be impossible (Pawlak and Skowron 2007). This gap between the real word full of vagueness and the traditional mathematics purely concerning precise concepts becomes smaller in recent years. In fact, philosophers and recently scientists as well as engineers are showing increasing interests in vague concepts, due to the fact that many practical problems emerging within fields such as economics, ecology, engineering, environmental science, social science, and medical science require us to deal with the complexity of data containing uncertainties. The nature of the vagueness arising in these fields can be very different. Among many mathematical theories designed for modelling various types of vague concepts, fuzzy and rough sets have received much attention and been actively studied by a number of researchers worldwide. While some authors argue that one theory is more general then the other, it is accepted by majority that these two theories are closely related, but distinct in essence because they model different types of uncertainties. In general, a fuzzy set may be viewed as a class with unsharp boundaries, whereas a rough set is a coarsely described crisp set (Yao 1998). Over the years, the theories of fuzzy sets and rough sets have become much closer to each other for practical needs to use both of these two theories complementarily 123 900 for managing uncertainty that arises from inexact, noisy, or incomplete information. Hybrid models combing fuzzy set with rough sets have arisen in various guises in different settings. For instance, based on an equivalence relation, Dubois and Prade introduced the lower and upper approximations of fuzzy sets in a Pawlak approximation space to obtain an extended notion called rough fuzzy sets (Dubois and Prade 1990). Alternatively, a fuzzy similarity relation can be used to replace an equivalence relation, and the result notion is called fuzzy rough sets (Dubois and Prade 1990). In general, a rough fuzzy set is the approximation of a fuzzy set in a crisp approximation space, whereas a fuzzy rough set is the approximation of a crisp set or a fuzzy set in a fuzzy approximation space. Molodtsov (1999) initiated a novel concept called soft sets as a new mathematical tool for dealing with uncertainties. The soft set theory is free from many difficulties that have troubled the usual theoretical approaches. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktaş and Çağman 2007). Soft set theory has potential applications in many different fields including the smoothness of functions, game theory, operational research, Perron integration, probability theory, and measurement theory (Molodtsov 1999, 2004). Research works on soft sets are very active and progressing rapidly in these years. Maji et al. (2002) discussed the application of soft set theory to a decision-making problem. Maji et al. (2001) investigated the fuzzification of a soft set and obtained many useful results on fuzzy soft sets. Based on fuzzy soft sets, Roy and Maji (2007) presented a method of object recognition from an imprecise multi-observer data. Chen et al. (2005) presented a new definition of soft set parametrization reduction, and compared it with attributes reduction in rough set theory. Kong et al. (2008) introduced the notion called normal parameter reduction of soft sets, by which they investigated the problem of suboptimal choice and added parameter set in soft set parametrization reduction. Zou and Xiao (2008) discussed data analysis approaches of soft sets as well as fuzzy soft sets under incomplete information. Maji et al. (2003) defined and studied several operations on soft sets. Aktaş and Çağman (2007) related soft sets to fuzzy sets and rough sets, providing examples to clarify their differences. They also defined soft groups, derived some basic properties, and showed that soft groups extended fuzzy groups. Jun (2008) introduced and investigated soft BCK/BCI-algebras. Jun and Park (2008) discussed the applications of soft sets to study the ideal theory of BCK/BCI-algebras. Furthermore, Feng et al. (2008) applied soft set theory to the study of semirings (Feng et al. 2005, 2007; Feng and Jun 2009) and initiated the notion called soft semirings. The present paper aims at providing a framework to combine fuzzy sets, 123 F. Feng et al. rough sets, and soft sets all together, which gives rise to several interesting new concepts such as rough soft sets, soft rough sets, and soft–rough fuzzy sets. Although many results reported here are only concerned with basic properties about these new notions, one could see that this study presents a very preliminary, but potentially interesting research direction. It will be necessary to carry out further research to establish whether the notions put forth in this paper may lead to a fruitful theory. 2 Fuzzy sets and rough sets In this section, we recall some basic notions relevant to fuzzy sets and rough sets. The following notations will be used in what follows. Let U be a nonempty set, called universe. The family of all subsets of U [resp. all fuzzy sets in U] is denoted by PðUÞ [resp. FðUÞ]. The theory of fuzzy sets initiated by Zadeh (1965) provides an appropriate framework for representing and processing vague concepts by allowing partial memberships. Since established, this theory has been actively studied by both mathematicians and computer scientists. Many applications of fuzzy set theory have arisen over the years, for instance, fuzzy logic, fuzzy cellular neural networks, fuzzy automata, fuzzy control systems, and so on. A fuzzy set l in a universe U is defined by a membership function l : U ! ½0; 1: For x 2 U; the membership value l(x) essentially specifies the degree to which x 2 U belongs to the fuzzy set l. There are many different definitions for fuzzy set operations. With the min–max system proposed by Zadeh, fuzzy set intersection, union, and complement are defined componentwise as follows: • • • ðl \ mÞðxÞ ¼ lðxÞ ^ mðxÞ; ðl [ mÞðxÞ ¼ lðxÞ _ mðxÞ; lc ðxÞ ¼ 1  lðxÞ; where l; m 2 FðUÞ and x 2 U: By l  m; we mean that lðxÞ  mðxÞ for all x 2 U: Clearly l ¼ m if both l  m and m  l; i.e. lðxÞ ¼ mðxÞ for all x 2 U: A fuzzy set can be related to a family of crisp sets by means of level sets. Given a number t 2 ½0; 1; a t-level set (or t-cut) of a fuzzy set l 2 FðUÞ is a crisp subset of U defined by lt ¼ fx 2 U : lðxÞ>tg: By taking t 2 ½0; 1; a fuzzy set l determines a family of nested subsets of U; i.e., flt  U : t 2 ½0; 1g: Conversely, a fuzzy set can be reconstructed from its t-level sets by means of the folW lowing formula lðxÞ ¼ ft : x 2 lt g; where l 2 FðUÞ and x 2 U: This observation is usually summarized by a representation theorem in fuzzy set theory, which establishes a one-to-one correspondence between a fuzzy set and a family of crisp sets satisfying certain conditions. Soft sets combined with fuzzy sets and rough sets: a tentative approach One of the most important applications of fuzzy set theory is the concept of linguistic variables. The value of a linguistic variable is defined as an element of its term set, a predefined set of appropriate linguistic terms. Linguistic terms are essentially subjective categories for a linguistic variable, which do not hold exact meaning, however, and may be understood differently by different people. The boundaries of a given term are rather subjective, and may also depend on the situation. Linguistic terms therefore cannot be expressed by ordinary set theory; rather, each linguistic term is associated with a fuzzy set (Aktaş and Çağman 2007). The rough set theory proposed by Pawlak (1982) provides a systematic method for dealing with vague concepts caused by indiscernibility in situation with incomplete information or a lack of knowledge. The rough set philosophy is founded on the assumption that with every object in the universe, we associate some information (data, knowledge). From a practical point of view, it is better to define basic concepts of rough set theory in terms of data. In fact, information and knowledge are stored and represented in a data table in many data analysis applications. This data table containing rows labelled by objects and columns labelled by attributes is called an information system (also known as a knowledge representation system) which can be formulated in the following way: Definition 1 (Pawlak and Skowron 2007) An information system is a pair I ¼ ðU; AÞ, where U is a nonempty finite set of objects and A is a nonempty finite set of attributes. Every attribute a 2 A is a function a : U!Va ; where Va is the set of values of attribute a. Let R be an equivalence relation on the universe U. Then the pair (U, R) is usually called a Pawlak approximation space. Conventionally, we refer to R as an indiscernibility relation (Pawlak and Skowron 2007) for the reason that it is often obtained from an information system (see Definition 1) and gives a partition of U due to the indiscernibility of objects in U. For x; y 2 U; x and y are said to be R-indiscernible if ðx; yÞ 2 R: The family of all equivalence classes of R, i.e., the partition determined by the equivalence relation R, will be denoted by U/R. An equivalence class of R, i.e., the block of the partition U/R, containing x will be denoted by ½xR : These equivalence classes of R are referred to as R-elementary sets (or R-elementary granules). The elementary sets represent the basic building blocks (concepts) of our knowledge about reality. Using the indiscernibility relation R, we can define the following two operations: R X ¼ fx 2 U : ½xR  Xg; R X ¼ fx 2 U : ½xR \ X 6¼ ;g; 901 assigning to every subset X  U two sets R X and R X called the lower and upper approximations of X with respect to ðU; RÞ: In addition, PosR X ¼ R X; NegR X ¼ U  R X; BndR X ¼ R X  R X are called the positive, negative, and boundary regions of X, respectively. Now, we are ready to give the definition of rough sets: Definition 2 (Pawlak and Skowron 2007) Let (U, R) be a Pawlak approximation space. A subset X  U is called definable if R X ¼ R X; in the opposite case, i.e., if BndR X 6¼ ;; X is said to be rough (or inexact). Note that sometimes a pair ðA; BÞ 2 PðUÞ  PðUÞ is also called a rough set if A ¼ R X and B ¼ R X for some X  U (Radzikowska and Kerre 2002). If the set X  U is defined by a predicate P and x 2 U; then we have the following: • • • x 2 R X means that x certainly has property P, x 2 R X means that x possibly has property P, x 2 NegR X means that X definitely does not have property P: Theorem 1 Suppose that (U, R) is a Pawlak approximation space and A; B  U: Then we have 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. R ðAÞ  A  R ðAÞ; R ð;Þ ¼ ; ¼ R ð;Þ; R ðUÞ ¼ U ¼ R ðUÞ; R ðR ðAÞÞ ¼ R ðAÞ; R ðR ðAÞÞ ¼ R ðAÞ; R ðR ðAÞÞ ¼ R ðAÞ; R ðR ðAÞÞ ¼ R ðAÞ; R ðAÞ ¼ ðR ðAc ÞÞc ; R ðAÞ ¼ ðR ðAc ÞÞc ; R ðA \ BÞ ¼ R ðAÞ \ R ðBÞ; R ðA \ BÞ  R ðAÞ \ R ðBÞ; R ðA [ BÞ  R ðAÞ [ R ðBÞ; R ðA [ BÞ ¼ R ðAÞ [ R ðBÞ; A  B ) R ðAÞ  R ðBÞ; R ðAÞ  R ðBÞ: Proof See Pawlak (1991). 3 Soft sets and fuzzy soft sets Let U be an initial universe set and EU (simply denoted by E) be the set of all possible parameters with respect to U. Usually, parameters are attributes, characteristics, or properties of the objects in U. The notion of a soft set is defined as follows: 123 902 F. Feng et al. Definition 3 (Molodtsov 1999) A pair S ¼ ðF; AÞ is called a soft set over U; where A  E and F : A ! PðUÞ is a set-valued mapping. required in classical mathematics. The absence of any restrictions on the approximate description in soft set theory makes it in practice very convenient and easy to apply. In other words, a soft set over U is a parameterized family of subsets of the universe U. For  2 A; FðÞ may be considered as the set of  -approximate elements in S ¼ ðF; AÞ: For illustration, Molodtsov (1999) considered several concrete examples of soft sets. The following is one type of illuminating examples considered by many authors (Aktaş and Çağman 2007; Chen et al. 2005; Maji et al. 2002, 2003). Definition 4 Let (F, A) and (G, B) be two soft sets over U. Then (G, B) is called a soft subset of (F, A), denoted by ðF; AÞ  ðG; BÞ; if B  A and GðbÞ  FðbÞ for all b 2 B: Two soft sets ðF; AÞ and ðG; BÞ over U are said to be equal, denoted by ðF; AÞ ¼ ðG; BÞ; if ðF; AÞ  ðG; BÞ and ðG; BÞ  ðF; AÞ: Example 1 Suppose that U ¼ fh1 ; h2 ; h3 ; h4 ; h5 g is the universe consisting of five houses and the set of parameters is given by E ¼ fe1 ; e2 ; e3 ; e4 ; e5 g; where ei (i ¼ 1; 2; . . .; 5) stand for ‘‘beautiful’’, ‘‘modern’’, ‘‘cheap’’, ‘‘in green surroundings’’ and ‘‘in good repair’’, respectively. Now, let us consider a soft set ðF; EÞ which describes the ‘‘attractiveness of houses’’ that Mr. X is considering for purchase. In this case, to define the soft set means to point out beautiful houses, modern houses and so on. Consider the mapping F given by ‘‘houses(
)’’, where dot (
) is to be filled in by one of the parameters ei 2 E: For instance, Fðe1 Þ means ‘‘houses(beautiful)’’, and its functional value is the set consisting of all the beautiful houses in U. Let Fðe1 Þ ¼ fh5 g; Fðe2 Þ ¼ fh1 ; h4 g; Fðe3 Þ ¼ fh1 ; h2 ; h3 g and Fðe4 Þ ¼ fh3 ; h5 g: Then the soft set ðF; EÞ can be seen as a collection of approximations where each approximation has two parts, a predicate and an approximate value set. For instance, ðbeautiful houses; fh5 gÞ is one of the approximations. Table 1 is the tabular representation of the soft set ðF; EÞ: If hi 2 Fðej Þ; then hij ¼ 1; otherwise hij ¼ 0; where hij are the entries in the table. We shall reconsider this example in the section concerning soft–rough approximations of fuzzy sets. As is shown in Aktaş and Çağman (2007), both fuzzy sets and rough sets may be considered as soft sets. Thus, one may expect that soft set theory could provide a more general mathematical framework for dealing with uncertain data. In fact, the way of setting (or describing) any object in soft set theory differs in principle from the way used in traditional mathematics (Molodtsov 1999). In soft set theory, the initial description of the object has an approximate nature, and we do not need to introduce the notion of exact solution as Table 1 Tabular representation of the soft set ðF; EÞ h1 h2 h3 h4 h5 e1 0 0 0 0 1 e2 1 0 0 1 0 e3 1 1 1 0 0 e4 0 0 1 0 1 123 Molodtsov (1999) also proposed a general way to define binary operations over soft sets. Assume that we have a binary operation on PðUÞ; which is denoted by : Let ðF; AÞ and ðG; BÞ be soft sets over U: Then, the operation for soft sets is defined by ðF; AÞ ðG; BÞ ¼ ðH; A  BÞ; where Hða; bÞ ¼ FðaÞ GðbÞ; a 2 A; b 2 B and AB is the Cartesian product of A and B: Maji et al. (2003) introduced the following operations over soft sets, which can be seen as the implementations of Molodtsov’s idea above. Definition 5 (Maji et al. 2003) If (F, A) and (G, B) are two soft sets over a common universe U, then (F,A) AND (G,B) denoted by ðF; AÞ ^ ðG; BÞ is defined by ðF; AÞ ^ ðG; BÞ ¼ ðH; A  BÞ; where Hðx; yÞ ¼ FðxÞ \ GðyÞ for all ðx; yÞ 2 A  B: Definition 6 (Maji et al. 2003) If ðF; AÞ and ðG; BÞ are two soft sets over a common universe U; then ‘‘ðF; AÞ OR ðG; BÞ’’ denoted by ðF; AÞ _ ðG; BÞ is defined by ðF; AÞ _ ðG; BÞ ¼ ðH; A  BÞ; where Hðx; yÞ ¼ FðxÞ [ GðyÞ for all ðx; yÞ 2 A  B: It is worth noting that differing from Molodtsov’s above idea, we can define binary operations for soft sets in the following way: Definition 7 Suppose that E : PðEÞ  PðEÞ ! PðEÞ is a binary operation on PðEÞ; and U : PðUÞ  PðUÞ ! PðUÞ is a binary operation on PðUÞ: Then for any two soft sets S ¼ ðF; AÞ and T ¼ ðG; BÞ over U; S T is defined as the soft set R ¼ ðH; CÞ where C ¼ A E B and HðxÞ ¼ FðxÞ U GðxÞ for all x 2 C: Definition 8 (Ali et al. 2009) Let (F, A) and (G, B) be two soft sets over a common universe U. (1) The extended intersection of ðF; AÞ and ðG; BÞ; denoted by ðF; AÞ uE ðG; BÞ; is defined as the soft set ðH; CÞ; where C ¼ A [ B; and 8e 2 C; 8 if e 2 A  B; < FðeÞ; HðeÞ ¼ GðeÞ; if e 2 B  A; : FðeÞ \ GðeÞ; if e 2 A \ B: Soft sets combined with fuzzy sets and rough sets: a tentative approach (2) (3) The restricted intersection of ðF; AÞ and ðG; BÞ; denoted by ðF; AÞeðG; BÞ; is defined as the soft set ðH; CÞ; where C ¼ A \ B and HðcÞ ¼ FðcÞ \ GðcÞ for all c 2 C: The extended union of ðF; AÞ and ðG; BÞ; denoted by e BÞ; is defined as the soft set ðH; CÞ; where ðF; AÞ [ðG; C ¼ A [ B; and 8e 2 C; 8 if e 2 A  B; < FðeÞ; HðeÞ ¼ GðeÞ; if e 2 B  A; : FðeÞ [ GðeÞ; if e 2 A \ B: 903 S ¼ ðf ; AÞ is a soft set over U and x 2 A: Then for any y 2 U; by definition we have that y 2 fRS ðxÞ , ðx; yÞ 2 RS , y 2 f ðxÞ: That is, fRS ðxÞ ¼ f ðxÞ for all x 2 A: Hence, fRS ¼ f ; and so ðfRS ; AÞ ¼ ðf ; AÞ: That is, SRS ¼ S: Next, assume that R  A  U; x 2 A and y 2 U: Then by definition, it is clear that ðx; yÞ 2 RSR , y 2 fR ðxÞ , ðx; yÞ 2 R: The restricted union of ðF; AÞ and ðG; BÞ; denoted by ðF; AÞ [R ðG; BÞ; is defined as the soft set (H,C), where C ¼ A \ B and HðcÞ ¼ FðcÞ [ GðcÞ for all c 2 C: Therefore we conclude that RSR ¼ R as required. In what follows, RS will be called the canonical relation of the soft set S; and SR will be called the canonical soft set of the binary relation R: Note that restricted intersection was also known as bi-intersection in Feng et al. (2008), and extended union was at first introduced and called union by Maji et al. (2003). Definition 11 (Ganter and Wille 1999) A formal context is a triple (U, V, I) where U, V are two finite sets called object set and property set, and I is a binary relation between U and V. (4) Definition 9 (Ali et al. 2009) Let U be an initial universe set, E be the universe set of parameters, and A; B  E: (1) (2) f N ðU;AÞ ¼ ðN; AÞ is called a relative null soft set (with respect to the parameter set A) if NðxÞ ¼ ; for all x 2 A: f ðU;BÞ ¼ ðW; BÞ is called a relative whole soft set W (with respect to the parameter set B) if WðxÞ ¼ U for all x 2 B: The relative whole soft set with respect to the universe set of parameters E is called the absolute soft set (or whole e U instead of soft set) over U and simply denoted by A f f W ðU;EÞ : Dually N ðU;EÞ is called the null soft set over U and denoted by f N U: Definition 10 (Ali et al. 2009) The relative complement of a soft set (F, A), denoted by ðF; AÞr ; is defined as the soft set ðF r ; AÞ where F r ðxÞ ¼ U  FðxÞ for all x 2 A: The following result indicates that soft sets and binary relations are closely related. Theorem 2 Let S ¼ ðf ; AÞ be a soft set over U. Then S induces a binary relation RS  A  U; which is defined by ðx; yÞ 2 RS , y 2 f ðxÞ; where x 2 A; y 2 U: Conversely, assume that R is a binary relation from A to U. Define a set-valued mapping fR : A ! PðUÞ by fR ðxÞ ¼ fy 2 U : ðx; yÞ 2 Rg; where x 2 A: Then SR ¼ ðfR ; AÞ is a soft set over U. Moreover, we have that SRS ¼ S and RSR ¼ R: Proof One easily sees that the first part holds. Thus it suffices to show that SRS ¼ S and RSR ¼ R: Suppose that Corollary 1 Every formal context can be seen as a soft set and vice versa. Proof This follows immediately from Theorem 2. As pointed out by several researchers, information systems and soft sets are closely related (Chen et al. 2005; Zou and Xiao 2008). Given a soft set S ¼ ðF; AÞ over a universe U. If U and A are both nonempty finite sets, then S could induce an information system in a natural way. In fact, for any attribute a 2 A; one can define a function a : U ! Va ¼ f0; 1g by  1; if x 2 FðaÞ; aðxÞ ¼ 0; otherwise: Therefore, every soft set may be considered as an information system. This justifies the tabular representation of soft sets used widely in the literature. Conversely, it is worth noting that a soft set can also be applied to express an information system. Let I ¼ ðU; AÞ be an information system. Taking [ B¼ fag  Va ; a2A as the parameter set, then a soft set (F, B) can be defined by setting Fða; vÞ ¼ fx 2 U : aðxÞ ¼ vg; where a 2 A and v 2 Va : Maji et al. (2001) initiated the study on hybrid structures involving both fuzzy sets and soft sets. In Maji et al. (2001) the notion of fuzzy soft sets was introduced as a fuzzy generalization of (classical) soft sets and some basic properties are discussed in detail. 123 904 F. Feng et al. Definition 12 (Roy and Maji 2007) Let FðUÞ be the set of all fuzzy sets in U. Let E be the parameter set and A  E: A pair (F,A) is called a fuzzy soft set over U, where F is a mapping given by F : A ! FðUÞ: In the definition of a fuzzy soft set, fuzzy sets in the universe U are used as substitutes for the crisp subsets of U. Hence, every soft set may be considered as a fuzzy soft set. In addition, by analogy with soft sets, one easily sees that every fuzzy soft set can be viewed as an (fuzzy) information system and be represented by a corresponding data table with entries belonging to the unit interval [0, 1] instead of 0 and 1 only. 5 Rough soft sets Motivated by Dubois and Prade’s original idea about rough fuzzy sets, we consider the lower and upper approximations of a soft set in a Pawlak approximation space, which gives rise to the following notions in a natural way. Definition 14 Let ðU; RÞ be a Pawlak approximation space and S ¼ ðF; AÞ be a soft set over U: The lower and upper rough approximations of S ¼ ðF; AÞ with respect to ðU; RÞ are denoted by R ðSÞ ¼ ðF ; AÞ and R ðSÞ ¼ ðF  ; AÞ; which are soft sets over U with the set-valued mappings given by F ðxÞ ¼ R ðFðxÞÞ; 4 Rough fuzzy sets and F  ðxÞ ¼ R ðFðxÞÞ; Based on an equivalence relation on the universe of discourse, Dubois and Prade (1990) introduced the lower and upper approximations of fuzzy sets in a Pawlak approximation space, and obtained a new notion called rough fuzzy sets. Definition 13 (Dubois and Prade 1990) Let ðU; RÞ be a Pawlak approximation space. For a fuzzy set l 2 FðUÞ; the lower and upper rough approximations of l in ðU; RÞ are denoted by RðlÞ and RðlÞ; respectively, which are fuzzy sets in U defined by ^ RðlÞðxÞ ¼ flðyÞ : y 2 ½xR g; and RðlÞðxÞ ¼ _ flðyÞ : y 2 ½xR Þg; for all x 2 U: The operators R and R are called the lower and upper rough approximation operators on fuzzy sets. If RðlÞ ¼ RðlÞ; the fuzzy set l is said to be definable; otherwise, l is called a rough fuzzy set. By definition, it is easy to see that if l ¼ X is a crisp subset of U; then we have that RðXÞ ¼ R X ¼ fx 2 U : ½xR  Xg; where x 2 A: The operators R and R are called the lower and upper rough approximation operators on soft sets. If R ðSÞ ¼ R ðSÞ; the soft set S is said to be definable; otherwise S is called a rough soft set. One can verify the following basic properties of rough soft sets: Theorem 3 Suppose that ðU; RÞ is a Pawlak approximation space and S ¼ ðF; AÞ is a soft set over U: Then we have 1. 2. 3. 4. 5. 6. 7. 8. 9. R ðSÞ  S  R ðSÞ; R ð f N ðU;AÞ Þ ¼ f N ðU;AÞ ¼ R ð f N ðU;AÞ Þ; f ðU;AÞ Þ ¼ W f ðU;AÞ ¼ R ð W f ðU;AÞ Þ; R ð W R ðR ðSÞÞ ¼ R ðSÞ; R ðR ðSÞÞ ¼ R ðSÞ; R ðR ðSÞÞ ¼ R ðSÞ; R ðR ðSÞÞ ¼ R ðSÞ; R ðSÞ ¼ ðR ðSr ÞÞr ; R ðSÞ ¼ ðR ðSr ÞÞr : Proof This is easily obtained from Theorem 1, Definition 9, Definition 10, and Definition 14. RðXÞ ¼ R X ¼ fx 2 U : ½xR \ X 6¼ ;g: Theorem 4 Suppose that ðU; RÞ is a Pawlak approximation space and S ¼ ðF; AÞ; T ¼ ðG; BÞ are soft sets over U: Then we have In this case, it follows that X 2 PðUÞ is a rough fuzzy set if and only if X is a rough set. Thus, one can say that rough fuzzy sets are natural extensions of rough sets. In general, a rough fuzzy set can be seen as an approximation of a fuzzy set in a crisp approximation space. This hybrid model combining the concepts of both fuzzy sets and rough sets may be used to deal with knowledge acquisition in information systems with fuzzy decisions (Slowinski and Stefanowski 1999). 1. 2. 3. 4. 5. 6. 7. 8. 9. and 123 R ðSeTÞ ¼ R ðSÞ e R ðTÞ; R ðS uE TÞ ¼ R ðSÞ uE R ðTÞ; R ðSeTÞ  R ðSÞeR ðTÞ; R ðS uE TÞ  R ðSÞ uE R ðTÞ; R ðS [R TÞ  R ðSÞ [R R ðTÞ; e  R ðSÞ [e R ðTÞ; R ðS [TÞ  R ðS [R TÞ ¼ R ðSÞ [R R ðTÞ; e ¼ R ðSÞ [e R ðTÞ; R ðS [TÞ S  T ) R ðSÞ  R ðTÞ; R ðSÞ  R ðTÞ: Soft sets combined with fuzzy sets and rough sets: a tentative approach Proof Let ðU; RÞ be a Pawlak approximation space, and let S ¼ ðF; AÞ; T ¼ ðG; BÞ be soft sets over U. (1) Let S e T ¼ R ¼ ðH; CÞ: Then C ¼ A \ B and HðxÞ ¼ FðxÞ \ GðxÞ; 8x 2 C: Using Definition (Maji et al. 2002), R ðRÞ ¼ ðH ; CÞ; where H ðxÞ ¼ R ðHðxÞÞ ¼ R ðFðxÞ \ GðxÞÞ; for all x 2 C: Now by Theorem 1, R ðFðxÞ \ GðxÞÞ ¼ R ðFðxÞÞ \ R ðGðxÞÞ; and so we deduce that H ðxÞ ¼ F ðxÞ \ G ðxÞ for all x 2 C: Hence, R ðRÞ ¼ R ðSÞ e R ðTÞ: (2) Let S uE T ¼ R ¼ ðH; CÞ: Then C ¼ A [ B and for all e 2 C; 8 if e 2 A  B; < FðeÞ; HðeÞ ¼ GðeÞ; if e 2 B  A; : FðeÞ \ GðeÞ; if e 2 A \ B: Using Definition (Maji et al. 2002), R ðRÞ ¼ ðH ; CÞ; where 8 if e 2 A  B; < R ðFðeÞÞ; H ðeÞ ¼ R ðGðeÞÞ; if e 2 B  A; : R ðFðeÞ \ GðeÞÞ; if e 2 A \ B; for all e 2 C: Now by Theorem 1, R ðFðeÞ \ GðeÞÞ ¼ R ðFðeÞÞ \ R ðGðeÞÞ; and so we deduce that 8 if e 2 A  B; < F ðeÞ; H ðeÞ ¼ G ðeÞ; if e 2 B  A; : F ðeÞ \ G ðeÞ; if e 2 A \ B; (3) (4) (5) for all e 2 C: Hence, R ðRÞ ¼ R ðSÞ uE R ðTÞ: This is similar to the proof of (1). This is similar to the proof of (2). Let S [R T ¼ R ¼ ðH; CÞ: Then C ¼ A \ B and HðxÞ ¼ FðxÞ [ GðxÞ; 8x 2 C: Using Definition (Maji et al. 2002), R ðRÞ ¼ ðH ; CÞ; where H ðxÞ ¼ R ðHðxÞÞ ¼ R ðFðxÞ [ GðxÞÞ; for all x 2 C: Now by Theorem 1, R ðFðxÞ \ GðxÞÞ  R ðFðxÞÞ \ R ðGðxÞÞ; and so we deduce that H ðxÞ  F ðxÞ \ G ðxÞ for all x 2 C: Hence, R ðRÞ  R ðSÞ [R R ðTÞ: e ¼ R ¼ ðH; CÞ: Then C ¼ A [ B and for all (6) Let S [T e 2 C; 8 if e 2 A  B; < FðeÞ; HðeÞ ¼ GðeÞ; if e 2 B  A; : FðeÞ [ GðeÞ; if e 2 A \ B: Using Definition 14, R ðRÞ ¼ ðH ; CÞ; where 905 8 if e 2 A  B; < R ðFðeÞÞ; if e 2 B  A; H ðeÞ ¼ R ðGðeÞÞ; : R ðFðeÞ [ GðeÞÞ; if e 2 A \ B; for all e 2 C: Now by Theorem 1, R ðFðeÞ [ GðeÞÞ  R ðFðeÞÞ \ R ðGðeÞÞ; e  ðTÞ: and so we deduce that R ðRÞ  R ðSÞ [R (7) This is similar to the proof of (5). (8) This is similar to the proof of (6). (9) Assume that S  T: Then by Definition 4, we have A  B and FðxÞ  GðxÞ; 8x 2 A: Now it follows from Theorem 1 that R ðFðxÞÞ  R ðGðxÞÞ and R ðFðxÞÞ  R ðGðxÞÞ for all x 2 C: Hence, we conclude that R ðSÞ  R ðTÞ and R ðSÞ  R ðTÞ: This completes our proof. 6 Soft rough sets In the last section, based on an equivalence relation on the universe of discourse, we introduced rough approximations of soft sets and obtained a hybrid concept called rough soft sets. We shall discuss another way to combine soft sets with rough sets in this section. In fact, a soft set instead of an equivalence relation can be used to granulate the universe of discourse. The result is a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Definition 15 Let S ¼ ðf ; AÞ be a soft set over U: Then the pair P ¼ ðU; SÞ is called a soft approximation space. Based on P; we define the following two operations: apr P ðXÞ ¼ fu 2 U : 9a 2 A½u 2 f ðaÞ  Xg; apr P ðXÞ ¼ fu 2 U : 9a 2 A½u 2 f ðaÞ; f ðaÞ \ X 6¼ ;g; assigning to every subset X  U two sets apr P ðXÞ and apr P ðXÞ called the lower and upper soft rough approximations of X in P; respectively. Moreover, PosP ðXÞ ¼ apr P ðXÞ; NegP ðXÞ ¼ U  apr P ðXÞ; BndP ðXÞ ¼ apr P ðXÞ  apr P ðXÞ are called the soft positive, soft negative and soft boundary regions of X; respectively. If apr P ðXÞ ¼ apr P ðXÞ; X is said to be soft definable; otherwise X is called a soft rough set. By definition, we immediately have that X  U is a soft definable set if BndP ðXÞ ¼ ;: Also, it is clear that apr P ðXÞ  X and apr P ðXÞ  apr P ðXÞ for all X  U: 123 906 F. Feng et al. Table 2 Tabular representation of the soft set S u1 u2 u3 u4 u5 u6 e1 1 0 0 0 0 1 e2 0 0 1 0 0 0 e3 0 0 0 0 0 0 e4 1 1 0 0 1 0 Nevertheless, it is worth noting that X  apr P ðXÞ does not hold in general as illuminated by the following example. Example 2 Suppose that U ¼ fu1 ; u2 ; u3 ; u4 ; u5 ; u6 g; E ¼ fe1 ; e2 ; e3 ; e4 ; e5 ; e6 g and A ¼ fe1 ; e2 ; e3 ; e4 g  E: Let S ¼ ðf ; AÞ be a soft set over U given by Table 2 and the soft approximation space P ¼ ðU; SÞ: For X ¼ fu3 ; u4 ; u5 g  U; we have apr P ðXÞ ¼ fu3 g; and apr P ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g: Thus, apr P ðXÞ 6¼ apr P ðXÞ and X is a soft rough set. It is clear that X ¼ fu3 ; u4 ; u5 g * apr P ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g: Moreover, we have PosP ðXÞ ¼ fu3 g; NegP ðXÞ ¼ fu4 ; u6 g and BndP ðXÞ ¼ fu1 ; u2 ; u5 g: On the other hand, let us consider X1 ¼ fu3 ; u4 g  U: Since apr P ðX1 Þ ¼ fu3 g ¼ apr P ðX1 Þ; by definition X1 is a soft definable set. Proposition 1 Let S ¼ ðf ; AÞ be a soft set over U and P ¼ ðU; SÞ be a soft approximation space. Then, [ apr P ðXÞ ¼ ff ðaÞ : f ðaÞ  Xg they belong to U  X; using available knowledge from the soft approximation space P: If X is internally soft indefinable, this means that we are able to decide about some elements of U that they belong to U  X; but we are unable to decide for any element of U that it belongs to X; by employing P: If X is externally soft indefinable, this means that we are able to decide for some elements of U that they belong to X; but we are unable to decide, for any element of U that it belongs to U  X; by employing P: If X is totally soft indefinable, we are unable to decide for any element of U whether it belongs to X or U  X; by employing P: Theorem 5 Let S ¼ ðf ; AÞ be a soft set over U; P ¼ ðU; SÞ be a soft approximation space and X; Y  U: Then we have 1. 2. 3. 4. 5. 6. 7. 8. Proof (1) (2) a2A and apr P ðXÞ ¼ [ (3) ff ðaÞ : f ðaÞ \ X 6¼ ;g a2A for all X  U: (4) (5) Proof This is easily obtained from the definition of soft rough approximations. Let S ¼ ðf ; AÞ be a soft set over U; P ¼ ðU; SÞ be a soft approximation space and X  U: We can define four basic classes of soft rough sets: • • • • (6) (7) X is said to be roughly soft definable if apr P ðXÞ ¼ 6 ; and apr P ðXÞ 6¼ U; X is internally soft indefinable if apr P ðXÞ ¼ ; and apr P ðXÞ 6¼ U; X is externally soft indefinable if apr P ðXÞ ¼ 6 ; and apr P ðXÞ ¼ U; X is totally soft indefinable if apr P ðXÞ ¼ ; and apr P ðXÞ ¼ U: The intuitive meaning of this classification is as follows: If X is roughly soft definable, this means that we are able to decide for some elements of U that they belong to X; and meanwhile for some elements of U we are able to decide that 123 apr P ð;Þ ¼ apr P ð;Þ ¼ ;; S apr P ðUÞ ¼ apr P ðUÞ ¼ a2A f ðaÞ; X  Y ) apr P ðXÞ  apr P ðYÞ; X  Y ) apr P ðXÞ  apr P ðYÞ; apr P ðX \ YÞ  apr P ðXÞ \ apr P ðYÞ; apr P ðX [ YÞ  apr P ðXÞ [ apr P ðYÞ; apr P ðX [ YÞ ¼ apr P ðXÞ [ apr P ðYÞ; apr P ðX \ YÞ  apr P ðXÞ \ apr P ðYÞ: (8) Straightforward. This follows directly from Proposition 1 by substituting U for X: Assume that X  Y: If u 2 apr P ðXÞ; then by definition, there exists some a 2 A such that u 2 f ðaÞ  X: Hence, it follows that u 2 f ðaÞ  Y; and so u 2 apr P ðYÞ: This shows that apr P ðXÞ  apr P ðYÞ: This is similar to the proof of (3). Since X \ Y  X; we deduce from (3) that apr P ðX \ YÞ  apr P ðXÞ: Similarly we have apr P ðX \ YÞ  apr P ðYÞ; and so apr P ðX \ YÞ  apr P ðXÞ \ apr P ðYÞ: Since X  X [ Y; by (3) we have apr P ðXÞ  apr P ðX [ YÞ: Similarly apr P ðYÞ  apr P ðX [ YÞ; and so apr P ðX [ YÞ  apr P ðXÞ [ apr P ðYÞ: Let u 2 apr P ðX [ YÞ: By definition, there exists some a 2 A such that u 2 f ðaÞ and f ðaÞ \ ðX [ YÞ ¼ 6 ;: Thus we have that either f ðaÞ \ X 6¼ ; or f ðaÞ \ Y 6¼ ;; indicating that u 2 apr P ðXÞ or u 2 apr P ðYÞ: This shows that apr P ðX [ YÞ  apr P ðXÞ [ apr P ðYÞ: To prove the reverse inclusion, note that X  X [ Y and so it follows from (4) that apr P ðXÞ  apr P ðX [ YÞ . Dually we have that apr P ðYÞ  apr P ðX [ YÞ . Hence we conclude that apr P ðX [ YÞ ¼ apr P ðXÞ [ apr P ðYÞ: This is similar to the proof of (5). It should be noted that the inclusions in Theorem 5 may be strict, as shown in the following example. Soft sets combined with fuzzy sets and rough sets: a tentative approach Table 3 Tabular representation of the soft set T 907 3. 4. 5.

X P Y ) X [ ðU  YÞ P U;

X  Y; Y P ; ) X P ;;

X  Y; X P U ) Y P U; u1 u2 u3 u4 u5 u6 u7 u8 e1 0 0 0 0 0 0 1 1 e2 1 0 0 0 1 1 0 0 e3 1 1 0 1 0 0 0 0 Proof e4 0 0 0 1 0 0 1 0 (1) where X; Y; X1 ; Y1  U: Example 3 Suppose that U ¼ fu1 ; u2 ;


; u8 g; E ¼ fe1 ; e2 ;


; e6 g and B ¼ fe1 ; e2 ; e3 ; e4 g  E: Let T ¼ ðg; BÞ be a soft set over U given by Table 3 and the soft approximation space P ¼ ðU; TÞ:
Suppose that X P Y: By definition we have apr P ðXÞ ¼ apr P ðYÞ: But from Theorem 5, we know that apr P ðX [ YÞ ¼ apr P ðXÞ [ apr P ðYÞ: Hence,
apr P ðX [ YÞ ¼ apr P ðXÞ ¼ apr P ðYÞ and so X P ðX [


YÞ P Y: Conversely, if X P ðX [ YÞ P Y; then it

follows that X P Y using the transitivity of P :

Assume that X P X1 and Y P Y1 : By definition apr P ðXÞ ¼ apr P ðX1 Þ and apr P ðYÞ ¼ apr P ðY1 Þ: Also by Theorem 5, we have apr P ðX [ YÞ ¼ apr P ðXÞ [ apr P ðYÞ and apr P ðX1 [ Y1 Þ ¼ apr P ðX1 Þ [ apr P ðY1 Þ: Thus we deduce that apr P ðX [ YÞ ¼ apr P ðX1 [ Y1 Þ;
and so ðX [ YÞ P ðX1 [ Y1 Þ:
Suppose that X P Y: By definition we have apr P ðXÞ ¼ apr P ðYÞ: Also by Theorem 5, we have apr P ðX [ ðU  YÞÞ ¼ apr P ðXÞ [ apr P ðU  YÞ and apr P ðUÞ ¼ apr P ðYÞ [ apr P ðU  YÞ: It follows that apr P ðX [ ðU  YÞÞ ¼ apr P ðYÞ[ apr P ðU  YÞ ¼
apr P ðUÞ: Hence, X [ ðU  YÞ P U:
Suppose that X  Y and Y P ;: From Theorem 5, we have apr P ðXÞ  apr P ðYÞ ¼ apr P ð;Þ ¼ ;: Hence,
apr P ðXÞ ¼ ; ¼ apr P ð;Þ and so X P ;:
Assume that X  Y and X P U: By Theorem 5, we have apr P ðYÞ  apr P ðXÞ ¼ apr P ðUÞ: Also, it is clear that apr P ðYÞ  apr P ðUÞ since Y  U: Therefore,
apr P ðYÞ ¼ apr P ðUÞ; and so Y P U as required. For X ¼ fu2 ; u7 ; u8 g  U; we have apr P ðXÞ ¼ f ðe1 Þ ¼ fu7 ; u8 g; and apr P ðXÞ ¼ f ðe1 Þ [ f ðe3 Þ [ f ðe4 Þ ¼ fu1 ; u2 ; u4 ; u7 ; u8 g: For Y ¼ fu1 ; u4 ; u7 g  U; we have apr P ðYÞ ¼ f ðe4 Þ ¼ fu4 ; u7 g; and apr P ðYÞ ¼ f ðe1 Þ [ f ðe2 Þ [ f ðe3 Þ [ f ðe4 Þ ¼ fu1 ; u2 ; u4 ; u5 ; u6 ; u7 ; u8 g: Since X \ Y ¼ fu7 g and X [ Y ¼ fu1 ; u2 ; u4 ; u7 ; u8 g; we have apr P ðX \ YÞ ¼ ;; apr P ðX \ YÞ ¼ f ðe1 Þ [ f ðe4 Þ ¼ fu4 ; u7 ; u8 g; and apr P ðX [ YÞ ¼ f ðe1 Þ [ f ðe3 Þ [ f ðe4 Þ ¼ fu1 ; u2 ; u4 ; u7 ; u8 g: Now, we have apr P ðXÞ \ apr P ðYÞ ¼ fu7 g and so apr P ðX \ YÞ ¼ ; apr P ðXÞ \ apr P ðYÞ; which shows that the inclusion (5) in Theorem 5 may hold strictly. From the observation that apr P ðXÞ [ apr P ðYÞ ¼ f ðe1 Þ [ f ðe4 Þ ¼ fu4 ; u7 ; u8 g apr P ðX [ YÞ; we deduce that the inclusion (6) in Theorem 5 could be strict. Moreover, apr P ðXÞ \ apr P ðYÞ ¼ fu1 ; u2 ; u4 ; u7 ; u8 g and apr P ðX \ YÞ ¼ fu4 ; u7 ; u8 g: This shows that the inclusion (8) in Theorem 5 could be strict. (2) Definition 16 Let S ¼ ðf ; UÞ be a soft set over U and P ¼ ðU; SÞ be a soft approximation space. For all X; Y  U; we define Definition 17 Let S ¼ ðf ; AÞ be a soft set over U: If for any a1 ; a2 2 A; there exists a3 2 A such that f ða3 Þ ¼ f ða1 Þ \ f ða2 Þ whenever f ða1 Þ \ f ða2 Þ ¼ 6 ;; then S is called an intersection complete soft set. X X
P
Y , apr P ðXÞ ¼ apr P ðYÞ; , apr P ðXÞ ¼ apr P ðYÞ; PY and X P Y ,X
P Y and X
P Y: These binary relations are called the lower soft rough equal relation, the upper soft rough equal relation, and the soft rough equal relation, respectively. It is easy to verify that the relations defined above are all equivalence relations over PðUÞ: Theorem 6 Let S ¼ ðf ; AÞ be a soft set over U and P ¼ ðU; SÞ be the soft approximation space. Then, 1. 2. X X



, X P ðX [ YÞ P Y;

P X1 ; Y P Y1 ) ðX [ YÞ P ðX1 [ Y1 Þ; PY (3) (4) (5) Proposition 2 Let S ¼ ðf ; AÞ be an intersection complete soft set over U and P ¼ ðU; SÞ be a soft approximation space. Then, we have apr P ðX \ YÞ ¼ apr P ðXÞ \ apr P ðYÞ for all X; Y  U: Proof Note first that by Theorem 5, apr P ðX \ YÞ  apr P ðXÞ \ apr P ðYÞ holds for every soft set S (needless to be intersection complete). Therefore, it suffices to show the reverse inclusion apr P ðX \ YÞ  apr P ðXÞ \ apr P ðYÞ: In fact, let u 2 apr P ðXÞ \ apr P ðYÞ: Then there exist a1 ; a2 2 A such that u 2 f ða1 Þ  X and u 2 f ða2 Þ  Y: 123 908 F. Feng et al. Since by hypothesis S is an intersection complete soft set, we deduce that there exists a3 2 A such that u 2 f ða3 Þ ¼ f ða1 Þ \ f ða2 Þ  X \ Y: Hence, u 2 apr P ðX \ YÞ as required. Using the above assertion, one can verify the following result on lower soft rough equal relations. Theorem 7 Let S ¼ ðf ; AÞ be an intersection complete soft set over U and P ¼ ðU; SÞ be a soft approximation space. Then, we have 1. 2. 3. 4. 5. X Y,X ðX \ YÞ Y;
P
P
P X X1 ; Y Y1 ) ðX \ YÞ ðX1 \ Y1 Þ;
P
P
P X Y ) X \ ðU  YÞ ;;
P
P X  Y; Y ;)X ;;
P
P X  Y; X U)Y U;
P where X; Y; X1 ; Y1  U: 1. sapS ðlÞ  l  sapS ðlÞ; 2. sapS ð;Þ ¼ sapS ð;Þ ¼ ;; 3. sapS ðUÞ ¼ sapS ðUÞ ¼ U; 4. ðsapS ðlÞÞc ¼ sapS ðlc Þ; 5. ðsapS ðlÞÞc ¼ sapS ðlc Þ; 6. sapS ðl \ mÞ ¼ sapS ðlÞ \ sapS ðmÞ; 7. sapS ðl [ mÞ  sapS ðlÞ [ sapS ðmÞ; 8. sapS ðl [ mÞ ¼ sapS ðlÞ [ sapS ðmÞ; 9. sapS ðl \ mÞ  sapS ðlÞ \ sapS ðmÞ; 10. l  m ) sapS ðlÞ  sapS ðmÞ; 11. l  m ) sapS ðlÞ  sapS ðmÞ: Proof (1)
P Proof This can be obtained from Proposition 2 using similar techniques as in the proof of Theorem 6. and sapS ðlÞðxÞ ¼ 7 Soft–rough fuzzy sets In this section, we shall consider lower and upper soft rough approximations of fuzzy sets in a soft approximation space, and obtain a new hybrid model called soft–rough fuzzy sets, which can be seen as an extension of Dubois and Prade’s rough fuzzy sets. (2) (3) (4) 123 This shows that ðsapS ðlÞÞc  sapS ðlc Þ: Next, we only need to prove the reverse inclusion ðsapS ðlÞÞc  sapS ðlc Þ . To see this, note first that by virtue of similar methods as used above, we can prove that ðsapS ðmÞÞc  sapS ðmc Þ hold for all fuzzy set m 2 FðUÞ . Taking m ¼ lc , it follows that ðsapS ðlc ÞÞc  sapS ðlÞ . Consequently we have sapS ðlc Þ  ðsapS ðlÞÞc as required. _ flðyÞ : 9a 2 A½fx; yg  FðaÞg; Theorem 8 Let S ¼ ðf ; AÞ be a full soft set over U; S ¼ ðU; SÞ be a soft approximation space and l; m 2 FðUÞ: Then we have Hence, it follows that sapS ðlÞðxÞ  lðxÞ  sapS ðlÞðxÞ: This shows that sapS ðlÞ  l  sapS ðlÞ: Straightforward. Straightforward. Let l 2 FðUÞ; x 2 U and let for all y 2 NðxÞ: Thus we deduce that ^ ðsapS ðlÞÞc ðxÞ  flc ðyÞ : y 2 NðxÞg ¼ sapS ðlc ÞðxÞ: Definition 20 Let S ¼ ðf ; AÞ be a full soft set over U and S ¼ ðU; SÞ be a soft approximation space. For a fuzzy set l 2 FðUÞ; the lower and upper soft rough approximations of l with respect to S are denoted by sapS ðlÞ and sapS ðlÞ; respectively, which are fuzzy sets in U given by ^ sapS ðlÞðxÞ ¼ flðyÞ : 9a 2 A½fx; yg  FðaÞg; for all x 2 U: The operators sapS and sapS are called the lower and upper soft rough approximation operators on fuzzy sets. If sapS ðlÞ ¼ sapS ðlÞ; l is said to be soft definable; otherwise l is called a soft–rough fuzzy set. flðyÞ : 9a 2 A½fx; yg  FðaÞg: ðsapS ðlÞÞc ðxÞ ¼ 1  sapS ðlÞðxÞ  1  lðyÞ ¼ lc ðyÞ; Definition 19 A full soft set S ¼ ðf ; AÞ over U is called a covering soft set if FðaÞ ¼ 6 ;; 8a 2 A: sapS ðlÞðxÞ ¼ _ NðxÞ ¼ fy : 9a 2 A½fx; yg  FðaÞg: W Note that sapS ðlÞðxÞ ¼ flðyÞ : y 2 NðxÞg: Hence, we have that lðyÞ  sapS ðlÞðxÞ for all y 2 NðxÞ: Now it follows that Definition 18 A soft set S ¼ ðf ; AÞ over U is called a full S soft set if a2A FðaÞ ¼ U: and Let l 2 FðUÞ and x 2 U: Since S ¼ ðf ; AÞ is a full soft set over U; there exists some a0 2 A such that x 2 Fða0 Þ: By definition, we have ^ sapS ðlÞðxÞ ¼ flðyÞ : 9a 2 A½fx; yg  FðaÞg; (5) (6) This is similar to the proof of (4). Let l; m 2 FðUÞ; x 2 U and let NðxÞ ¼ fy : 9a 2 A½fx; yg  FðaÞg: At first, note that Soft sets combined with fuzzy sets and rough sets: a tentative approach sapS ðl \ mÞðxÞ ¼ ^ flðyÞ ^ mðyÞ : y 2 NðxÞg: Hence, sapS ðl \ mÞðxÞ  lðyÞ ^ mðyÞ  lðyÞ for all V y 2 NðxÞ: Since sapS ðlÞðxÞ ¼ flðyÞ : y 2 NðxÞg; it follows that sapS ðl \ mÞðxÞ  sapS ðlÞðxÞ: Similarly, we obtain sapS ðl \ mÞðxÞ  sapS ðmÞðxÞ: Therefore, If l  m; then it is easy to see that ^ sapS ðlÞðxÞ ¼ flðyÞ : y 2 NðxÞg  lðyÞ  mðyÞ; (11) for all y 2 NðxÞ: This is similar to the proof of (10). sapS ðl \ mÞðxÞ  sapS ðlÞðxÞ ^ sapS ðmÞðxÞ: To illustrate soft–rough approximations of fuzzy sets, let us consider the following example which is a continuation to Example 1. This says that sapS ðl \ mÞ  sapS ðlÞ \ sapS ðmÞ: Now it remains to show the reverse inclusion. To prove this, notefirst that Example 4 Let U; E and the soft set S ¼ ðF; EÞ over U be the same as in Example 1. Let S ¼ ðU; SÞ be a soft approximation space. Then for fuzzy set ðsapS ðlÞ \ sapS ðmÞÞðxÞ ¼ sapS ðlÞðxÞ ^ sapS ðmÞðxÞ l ¼ f0:8=h1 ; 0:5=h2 ; 0:7=h3 ; 0:2=h4 ; 0:3=h5 g;  sapS ðlÞðxÞ  lðyÞ; by definition we compute sapS ðlÞ ¼ f0:2=h1 ; 0:5=h2 ; 0:3=h3 ; 0:2=h4 ; 0:3=h5 g; for all y 2 NðxÞ . In a similar way, we have and ðsapS ðlÞ \ sapS ðmÞÞðxÞ  sapS ðmÞðxÞ  mðyÞ sapS ðlÞ ¼ f0:8=h1 ; 0:8=h2 ; 0:8=h3 ; 0:8=h4 ; 0:7=h5 g: for all y 2 NðxÞ . Thus Similarly for fuzzy set ðsapS ðlÞ \ sapS ðmÞÞðxÞ  lðyÞ ^ mðyÞ for all y 2 NðxÞ . Now it follows that ^ ðsapS ðlÞ \ sapS ðmÞÞðxÞ  flðyÞ ^ mðyÞ : y 2 NðxÞg ¼ sapS ðl \ mÞðxÞ: (7) 909 Thus, sapS ðl \ mÞ  sapS ðlÞ \ sapS ðmÞ as required. Let l; m 2 FðUÞ; x 2 U and let Then it is clear that ^ sapS ðlÞðxÞ ¼ flðyÞ : y 2 NðxÞg  lðyÞ sapS ðmÞ ¼ f0:8=h1 ; 0:6=h2 ; 0:6=h3 ; 0:8=h4 ; 0:6=h5 g: l \ m ¼ f0:1=h1 ; 0:3=h2 ; 0:6=h3 ; 0:2=h4 ; 0:3=h5 g: By computation we obtain for all y 2 NðxÞ: Thus we have ^ sapS ðl [ mÞðxÞ ¼ flðyÞ _ mðyÞ : y 2 NðxÞg sapS ðl [ mÞ ¼ f0:5=h1 ; 0:5=h2 ; 0:5=h3 ; 0:8=h4 ; 0:5=h5 g; and sapS ðlÞðxÞ: sapS ðl \ mÞ ¼ f0:6=h1 ; 0:6=h2 ; 0:6=h3 ; 0:2=h4 ; 0:6=h5 g: sapS sapS ðlÞðxÞ _ sapS ðmÞðxÞ ¼ ðsapS ðlÞ [ sapS ðmÞÞðxÞ: Thus, we conclude that sapS ðl [ mÞ  sapS ðlÞ[ sap S ðmÞ: (8) This is similar to the proof of (6). (9) This is similar to the proof of (7). (10) Let l; m 2 FðUÞ; x 2 U and let NðxÞ ¼ fy : 9a 2 A½fx; yg  FðaÞg: and and  lðyÞ _ mðyÞ; sapS ðl [ mÞðxÞ we have sapS ðmÞ ¼ f0:1=h1 ; 0:1=h2 ; 0:1=h3 ; 0:1=h4 ; 0:5=h5 g; It is easy to see that l [ m ¼ f0:8=h1 ; 0:5=h2 ; 0:7=h3 ; 0:8=h4 ; 0:5=h5 g; NðxÞ ¼ fy : 9a 2 A½fx; yg  FðaÞg: Similarly, we obtain that sapS ðl [ mÞðxÞ ðmÞðxÞ: Hence, it follows that m ¼ f0:1=h1 ; 0:3=h2 ; 0:6=h3 ; 0:8=h4 ; 0:5=h5 g; Furthermore, we have sapS ðlÞ[sapS ðmÞ ¼ f0:2=h1 ;0:5=h2 ;0:3=h3 ;0:2=h4 ;0:5=h5 g; and sapS ðlÞ\sapS ðmÞ ¼ f0:8=h1 ;0:6=h2 ;0:6=h3 ;0:8=h4 ;0:6=h5 g: This shows that the inclusions in Theorem 8 may hold strictly. Definition 21 A soft set S ¼ ðf ; AÞ over U is called a partition soft set if fFðaÞ : a 2 Ag forms a partition of U: 123 910 F. Feng et al. By definition, every partition soft set is a covering soft set. The following example shows that every quotient set may be considered a partition soft set. Example 5 Let R be an equivalence relation on U: Then the set-valued mapping fR : U ! PðUÞ in Theorem 2 coincides with the natural mapping of the equivalence relation R: That is, fR ðxÞ ¼ ½xR for all x 2 U: Then the canonical soft set SR of the equivalence relation R can be identified with the quotient set U=R: Moreover, we claim that SR ¼ ðfR ; UÞ is a partition soft set since ffR ðxÞ : x 2 Ug ¼ U=R is a partition of U: Theorem 9 Let R be an equivalence relation on U: Let SR ¼ ðfR ; UÞ be the canonical soft set of R and S ¼ ðU; SR Þ be the soft approximation space. Then, ^ sapS ðlÞðxÞ ¼ flðyÞ : y 2 ½xR g; and sapS ðlÞðxÞ ¼ _ flðyÞ : y 2 ½xR g; where l 2 FðUÞ; x 2 U: Thus in this case, l is a rough fuzzy set with respect to the approximation space ðU; RÞ if and only if l is a soft–rough fuzzy set with respect to the soft approximation space S ¼ ðU; SR Þ: Proof Let l 2 FðUÞ and x 2 U: Since SR ¼ ðfR ; UÞ is the canonical soft set of R and S ¼ ðU; SR Þ; by definition we have ^ sapS ðlÞðxÞ ¼ flðyÞ : 9a 2 U; fx; yg  fR ðaÞg ^ ¼ flðyÞ : 9a 2 U; fx; yg  ½aR g ^ ¼ flðyÞ : y 2 ½xR g: The second assertion sapS ðlÞðxÞ ¼ _ flðyÞ : y 2 ½xR g space S ¼ ðU; SÞ if and only if l is a rough fuzzy set with respect to the approximation space ðU; RÞ: Proof First, we show that the relation R induced by the partition soft set S ¼ ðf ; AÞ is an equivalence relation on U: For any x 2 U; since S ¼ ðf ; AÞ is clearly a full soft set, there exists a 2 A such that x 2 FðaÞ; and so ðx; xÞ 2 R; which shows that R is reflexive. If ðx; yÞ 2 R; then there exists a 2 A such that fx; yg ¼ fy; xg  FðaÞ: Thus, we deduce that ðy; xÞ 2 R; indicating that R is symmetric. Assume that ðx; yÞ 2 R and ðy; zÞ 2 R: Then there exist a; b 2 A such that fx; yg  FðaÞ and fy; zg  FðbÞ: Hence, FðaÞ \ FðbÞ ¼ 6 ;: But fFðaÞ : a 2 Ag is a partition of U for S ¼ ðf ; AÞ is a partition soft set over U: It follows that FðaÞ ¼ FðbÞ and so ðx; zÞ 2 R: This shows that R is transitive as required. Now let l 2 FðUÞ and x 2 U: By definition, ^ sapS ðlÞðxÞ ¼ flðyÞ : 9a 2 A; fx; yg  FðaÞg ^ ¼ flðyÞ : ðx; yÞ 2 Rg ^ ¼ flðyÞ : y 2 ½xR g: The second assertion _ sapS ðlÞðxÞ ¼ flðyÞ : y 2 ½xR g can be proved in a similar way. From the above results, one easily sees that (classical) rough fuzzy sets can be identified with soft–rough fuzzy sets when the underlying soft set in the soft approximation space is a partition soft set. Consequently, every rough fuzzy set may be considered a soft–rough fuzzy set. However, we claim that the reverse statement is generally not true, since a (full) soft set is not necessarily a partition soft set. In this sense, the notion of soft–rough fuzzy sets can be seen as a natural generalization of rough fuzzy sets by applying soft set theory. can be proved in a similar way. Theorem 10 Let S ¼ ðf ; AÞ be a partition soft set over U and S ¼ ðU; SÞ be the soft approximation space. Define a binary relation R on U by ðx; yÞ 2 R , 9a 2 A; fx; yg  FðaÞ; where x; y 2 U: Then R is an equivalence relation on U such that ^ sapS ðlÞðxÞ ¼ flðyÞ : y 2 ½xR g; and sapS ðlÞðxÞ ¼ _ flðyÞ : y 2 ½xR g; where l 2 FðUÞ; x 2 U: Thus in this case, l is a soft– rough fuzzy set with respect to the soft approximation 123 8 Conclusions We have investigated in this paper the problem of combing soft sets with fuzzy sets and rough sets. In general, three different types of hybrid models are presented, which are called rough soft sets, soft rough sets and soft–rough fuzzy sets, respectively. A rough soft set is the approximation of a soft set in a Pawlak approximation space, whereas a soft rough set is based on soft rough approximations in a soft approximation space. The approximation of a fuzzy set in a soft approximation space is also investigated to obtain soft–rough fuzzy sets which extend Dubois and Prade’s rough fuzzy sets in a natural way. In addition, we have investigated some basic properties of these new Soft sets combined with fuzzy sets and rough sets: a tentative approach hybridizations with illustrating examples. Further study will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory. Acknowledgments We are highly grateful to the anonymous referees for their helpful comments and suggestions for improving the paper. 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