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Soft sets and soft rough sets
Information Sciences 181 (2011) 1125–1137
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Soft sets and soft rough sets Feng Feng a,b,⇑, Xiaoyan Liu b, Violeta Leoreanu-Fotea c, Young Bae Jun d a
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China Department of Applied Mathematics, Faculty of Science, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China Faculty of Mathematics, ‘‘Al.I.Cuza’’ University, 6600 Iasßi, Romania d Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea b c
a r t i c l e
i n f o
Article history: Received 28 June 2008 Received in revised form 12 March 2010 Accepted 8 November 2010
Keywords: Soft sets Rough sets Soft rough sets Soft approximation spaces Soft rough approximations Granular computing
a b s t r a c t In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak’s rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak’s rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The volume and complexity of the collected data in our modern society is growing rapidly. There often exist various types of uncertainties in those data related to complex problems in biology, economics, ecology, engineering, environmental science, medical science, social science, and many other fields. In order to describe and extract the useful information hidden in uncertain data, researchers in mathematics, computer science and related areas have proposed a number of theories such as probability theory, fuzzy set theory [35], intuitionistic fuzzy set theory [2,6], rough set theory [21–25], vague set theory [8] and interval mathematics [9]. While all these theories are well-known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out by Molodtsov [19]. In 1999, Molodtsov [19] proposed soft set theory as a new mathematical tool for dealing with uncertainties which is free from the difficulties affecting existing methods. As reported in [19,20], a wide range of applications of soft sets have been developed in many different fields, including the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory. There has been a rapid growth of interest in soft set theory and its applications in recent years. Maji et al. [17] discussed the application of soft set theory in a decision making problem. The same authors also extended classical soft sets to fuzzy soft sets [18]. By combining the interval-valued fuzzy set and soft set models, Yang et al. [31] further introduced interval-valued fuzzy soft sets. Based on fuzzy soft sets, Roy and Maji
⇑ Corresponding author at: Department of Applied Mathematics, Faculty of Science, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China. Tel.: +86 29 88166086. E-mail addresses: [email protected] (F. Feng), [email protected] (X. Liu), [email protected] (V. Leoreanu-Fotea), [email protected] (Y.B. Jun). 0020-0255/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2010.11.004
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[26] presented a method of object recognition from an imprecise multi-observer data and applied it to decision making problems. Chen et al. [4] proposed a new definition of soft set parametrization reduction, and compared it with the related concept of attributes reduction in rough set theory. Maji et al. [16] introduced several operations in soft set theory and carried out a detailed theoretical study on soft sets. Since soft set theory emerged as a novel approach for modelling vagueness and uncertainty, it soon invoked an interesting question concerning a possible connection between soft sets and algebraic systems. As the first results in this research direction, Aktasß and Çag˘man [1] initiated soft groups and showed that fuzzy groups can be viewed as a special case of the soft groups. Jun [11] applied soft sets to the theory of BCK/BCI-algebras, and introduced the concept of soft BCK/BCI-algebras. Jun and Park [12] reported applications of soft sets in ideal theory of BCK/BCI-algebras. Soft semirings and several related notions are defined in [7] in order to establish a connection between soft sets and semirings. Rough set theory was initiated by Pawlak [21] for dealing with vagueness and granularity in information systems. This theory deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [22–25]. It is well-known that Pawlak’s rough set model is based on equivalence relations. In the real world, the equivalence relation is, however, too restrictive for many practical applications. To address this issue, many interesting and meaningful extensions of Pawlak’s rough sets have been presented in the literature. Equivalence relations can be replaced by tolerance relations [28], similarity relations [27], dominance relations [10], general binary relations [15,32,33,36], coverings of the universe of discourse [5,13,30,38] or neighborhood systems [29,34]. All these proposals share the common feature that they deal with approximations of concepts in terms of granules. Actually, different types of generalized rough sets are based on different interpretations of granules. In Pawlak’s rough set model, each equivalence class may be viewed as a granule consisting of indistinguishable elements. The granulation structure induced by an equivalence relation is a partition of the universe. By weakening the requirement of equivalence relations, we get more general granulation structures based on general binary relations or coverings of the universe. In this study we initiate the notion of soft rough sets, which can be seen as a generalized rough set model based on soft sets. The standard soft set model is used to form the granulation structure of the universe, namely the soft approximation space. Based on this granulation structure, we then define soft rough approximations, soft rough sets and some related notions. As the hybrid model combining rough sets with soft sets, soft rough sets could be exploited to extend many practical applications based on rough sets or soft sets. The remainder of this paper is organized as follows. Section 2 presents some fundamental concepts in rough set and soft set theories and also some interesting connections between them. Section 3 is devoted to a detailed discussion on the rudiments of soft rough sets. We define soft approximation spaces, soft rough approximations and soft rough sets. Basic properties of soft rough approximations are reported and supported by examples. Meanwhile full soft sets and intersection complete soft sets are introduced. We also define soft rough equal relations in terms of soft rough approximations and explore some related properties. In Section 4 we define partition soft sets and point out a close connection between soft sets and binary relations. We show that the standard rough set model may be considered a special case of the soft rough sets. Also, through an example we present a comparative analysis between rough sets and soft rough sets. Finally, conclusions are presented in the last section. Although this study is a preliminary proposal concerning the soft rough set model, we hope it will give rise to a potentially interesting research direction. 2. Overview of rough sets and soft sets In this section, we first recall some fundamental facts about Pawlak’s rough sets [21] and Molodtsov’s soft sets [19]. Then we point out some interesting connections between these two different models. The rough set philosophy is founded on the assumption that with every object of interest, we associate some information (data, knowledge). For example, if objects are patients suffering from a certain disease, symptoms of the disease form information about patients. Objects characterized by the same information are indiscernible in view of the available information about them. In many data analysis applications, information is stored and represented in a data table, where a set of objects are described by a set of attributes. Definition 2.1. [23] An information system (or a knowledge representation system) is a pair I ¼ ðU; AÞ of non-empty finite sets U and A, where U is a set of objects and A is a set of attributes; each attribute a 2 A is a function a:U ? Va, where Va is the set of values (called domain) of attribute a. Let U be a non-empty finite universe and R be an equivalence relation on U. The pair (U, R) is called a Pawlak approximation space. The equivalence relation R is often called an indiscernibility relation and related to an information system. Specifically, if I ¼ ðU; AÞ is an information system and B # A, then an indiscernibility relation R = I(B) can be defined by
ðx; yÞ 2 IðBÞ () aðxÞ ¼ aðyÞ;
8a 2 B;
where x, y 2 U, and a(x) denotes the value of attribute a for object x. Using the indiscernibility relation R, one can define the following two operations
R X ¼ fx 2 U : ½xR # Xg;
R X ¼ fx 2 U : ½xR \ X – ;g
F. Feng et al. / Information Sciences 181 (2011) 1125–1137
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assigning to every subset X # U two sets R*X and R*X called the R-lower and the R-upper approximation of X, respectively. Moreover, the sets
PosR X ¼ R X;
Neg R X ¼ U R X;
BndR X ¼ R X R X
are referred to as the R-positive, the R-negative and the R-boundary region of X, respectively. If the R-boundary region of X is empty, i.e., R*X = R*X, then X is crisp (or exact) with respect to R; in the opposite case, i.e., BndRX – ;, X is said to be rough (or inexact) with respect to R [23]. Note that sometimes the pair (R*X, R*X) is also referred to as the rough set of X with respect to R [14,15]. If X # U is defined by a predicate P and x 2 U, we have the following interpretation: x 2 PosRX = R*X means that x certainly has property P; x 2 R*X means that x possibly has property P; x 2 NegRX means that x definitely does not have property P. Let us recall now the soft set notion, which is a newly-emerging mathematical approach to vagueness. Let U be an initial universe of objects and EU (simply denoted by E) the set of certain parameters in relation to the objects in U. Parameters are often attributes, characteristics, or properties of the objects in U. Let PðUÞ denote the power set of U. Following the definition in [1,16], the concept of soft sets is defined as follows. Definition 2.2. [1,16]A pair S = (F, A) is called a soft set over U, where A # E and F : A ! PðUÞ is a set-valued mapping. In other words, a soft set over U is a parameterized family of subsets of the universe U. For 2 A, F() may be considered as the set of -approximate elements in the soft set S = (F, A). It is worth noting that F() may be arbitrary: some of them may be empty, and some may have nonempty intersection [19]. The absence of any restrictions on the approximate description in soft set theory makes this theory very convenient and easily applicable in practice. Actually, we can use any suitable parametrization—with the help of words and sentences, real numbers, functions, mappings, and so on. Molodtsov [19] 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 (F, A) and (G, B) be soft sets over U. Then, the operation for soft sets is defined by (F, A) (G, B) = (H, A B), where H(a, b) = F(a) G(b), a 2 A, b 2 B and A B is the Cartesian product of A and B. Maji et al. [16] carried out Molodtsov’s seminal idea by introducing several operations in soft set theory, such as AND, OR, intersection and union. Definition 2.3. 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 (G, B) # (F, A), if B # A and G(b) # F(b) for all b 2 B. Although rough sets and soft sets are two different mathematical tools for modelling vagueness, there are some interesting connections between them. At first, we note that information systems and soft sets are closely related. Let S = (F, A) be a soft set over U. If U and A are both non-empty finite sets, then S could induce an information system I ¼ ðU; AÞ in a natural way. In fact, for any attribute a 2 A, one can define a function a: U ? Va = {0, 1} by
aðxÞ ¼
1; if x 2 FðaÞ; 0; otherwise:
Therefore every soft set may be considered as an information system. This justifies the tabular representation of soft sets widely used in the literature (e.g., see the soft set tables used in what follows). Conversely, it is worth noting that soft sets S can also be applied to represent information systems. Let I ¼ ðU; AÞ be an information system. Taking B = a2A{a} Va, as the parameter set, then a soft set (F, B) can be defined by setting
Fða; v Þ ¼ fx 2 U : aðxÞ ¼ v g; where a 2 A and v 2 Va. By the above discussion, it is easy to see that once given a soft set S = (F, A) over U, we obtain an information system I ¼ ðU; AÞ corresponding to the soft set S, whence we shall be able to construct rough approximations and discuss Pawlak’s rough sets based on the Pawlak approximation space (U, R) induced by the information system I. On the other hand, it is interesting to find that Pawlak’s rough set model may be considered as a special case of Molodtsov’s soft sets [1]. To see this, suppose that (U, R) is a Pawlak approximation space and X # U. Let R(X) = (R*X, R*X) be the rough set of X with respect to R. Consider two predicates p1(x), p2(x), which mean ‘‘[x]R # X’’ and ‘‘[x]R \ X – ;’’, respectively. The predicates p1(x), p2(x) may be treated as elements of a parameter set; that is, E = {p1(x), p2(x)}. Then we can define a setvalued mapping
F : E ! PðUÞ;
pi ðxÞ # Fðpi ðxÞÞ ¼ fx 2 U : pi ðxÞ is trueg;
where i = 1, 2. It follows that the rough set R(X) may be considered a soft set (F, E) with the following representation
ðF; EÞ ¼ fðp1 ðxÞ; R XÞ; ðp2 ðxÞ; R XÞg:
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Maji et al. [18] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced in [18] the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of (classical) soft sets. Definition 2.4. [26] Let FðUÞ be the set of all fuzzy sets in the universe U. Let E be a set of parameters and A # E. A pair ð e F ; AÞ is called a fuzzy soft set over U, where e F is a mapping given by e F : A ! FðUÞ. 3. Soft rough approximations and soft rough sets In this section we introduce soft rough approximations and soft rough sets. If U is the initial universe, then the complement of X in U is denoted by X. Definition 3.1. Let S = (F, A) be a soft set over U. Then the pair P = (U, S) is called a soft approximation space. Based on the soft approximation space P, we define the following two operations
aprP ðXÞ ¼ fu 2 U : 9a 2 A; ½u 2 FðaÞ # X g;
ð1Þ
aprP ðXÞ ¼ fu 2 U : 9a 2 A; ½u 2 FðaÞ; FðaÞ \ X – ;g
ð2Þ
assigning to every subset X # U two sets aprP(X) and apr P ðXÞ, which are called the soft P-lower approximation and the soft Pupper approximation of X, respectively. In general, we refer to aprP(X) and apr P ðXÞ as soft rough approximations of X with respect to P. Moreover, the sets
PosP ðXÞ ¼ aprP ðXÞ;
ð3Þ
Neg P ðXÞ ¼ aprP ðXÞ;
ð4Þ
BndP ðXÞ ¼ aprP ðXÞ aprP ðXÞ
ð5Þ
are called the soft P-positive region, the soft P-negative region and the soft P-boundary region of X, respectively. If aprP ðXÞ ¼ aprP ðXÞ, X is said to be soft P-definable; otherwise X is called a soft P-rough set. Remark 3.2. Although soft rough approximations seem to be similar to covering rough set approximations, it is worth noting that in most cases they are closely related but distinct in essence. More specifically, this can be illustrated by an example as follows. One can find various definitions and results about generalized rough sets induced by coverings, c.f. [3,5,13,14,30,37,38]. Here, we follow the definitions and notations used in [14]. Let U = {u1, u2, u3, u4} be a universe and S = (F, A) a soft set over U, where A = {e1, e2}, F(e1) = {u1, u2} and F(e2) = {u1, u3, u4}. Then P = (U, S) is a soft approximation space. It is clear that C = {F(e1), F(e2)} is a covering of the universe U, whence hU, Ci forms a covering approximation space. For x 2 U, the minimal description of x is defined as
MdðxÞ ¼ fK 2 C : x 2 K ^ ð8S 2 C ^ x 2 S # K ) K ¼ SÞg: By definition, one easily sees that Md(u1) = {F(e1), F(e2)}, Md(u2) = {F(e1)} and Md(u3) = {F(e2)} = Md(u4). Zhu [37] proposed the following lower and upper approximations of X # U in a covering approximation space hU, Ci:
X þ ¼ [fK 2 C : K # Xg;
ð6Þ
X þ ¼ X þ [ fNðxÞ : x 2 X X þ g;
ð7Þ
where N(x) = \{K 2 C: x 2 K} = \Md(x) is called the neighborhood of x 2 U. By direct computation, we have N(u1) = {u1}, N(u2) = F(e1), and N(u3) = F(e2) = N(u4). Zhu [37] proved that X+ = [{N(x): x 2 X}, which can be viewed as an alternative representation of the upper approximation. On the other hand, Xu and Zhang [30] introduced another type of covering-based approximations as follows:
C X ¼ fx 2 U : NðxÞ # Xg;
ð8Þ
C X ¼ fx 2 U : NðxÞ \ X – ;g:
ð9Þ
For X1 = {u1}, by the definition of soft rough approximations, we have aprP(X1) = ; and apr P ðX 1 Þ ¼ U. The covering-based approximations of X1 are (X1)+ = ; and (X1)+ = {u1}, according to the definition in [37]. In contrast to these approximations, the covering lower and upper approximations of X1, according to the definition in [30], are C*X1 = {u1} and C*X1 = U, respectively. Consequently, these three types of approximations are different. By Definition 3.1, we immediately have that X # U is a soft P-definable set if the soft P-boundary region BndP(X) of X is empty. Also, it is clear that aprP(X) # X and apr P ðXÞ # apr P ðXÞ for all X # U. Nevertheless, it is worth noticing that X # aprP ðXÞ does not hold in general. Example 3.3. Let U = {u1, u2, u3, u4, u5, u6}, E = {e1, e2, e3, e4, e5, e6} and A = {e1, e2, e3, e4} # E. Let S = (F, A) be a soft set over U given by Table 1 and the soft approximation space P = (U, S).
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F. Feng et al. / Information Sciences 181 (2011) 1125–1137 Table 1 Tabular representation of the soft set S.
e1 e2 e3 e4
u1
u2
u3
u4
u5
u6
1 0 0 1
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 1
1 0 0 0
For X = {u3, u4, u5} # U, we have aprP(X) = {u3}, and aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g. Thus apr P ðXÞ – apr P ðXÞ and X is a soft Prough set. Note that X ¼ fu3 ; u4 ; u5 g aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g in this case. Moreover, it is easy to see that PosP(X) = {u3}, NegP(X) = {u4, u6} and BndP(X) = {u1, u2, u5}. On the other hand, one can consider X1 = {u3, u4} # U. Since apr P ðX 1 Þ ¼ fu3 g ¼ apr P ðX 1 Þ, by definition, X1 is a soft P-definable set. The following result is easily obtained from the definition of soft rough approximations. Proposition 3.4. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then we have
aprP ðXÞ ¼
[
fFðaÞ : FðaÞ # Xg
a2A
and
aprP ðXÞ ¼
[
fFðaÞ : FðaÞ \ X – ;g
a2A
for all X # U. Suppose that S = (F, A) is a soft set over U and P = (U, S) is the corresponding soft approximation space. One can verify that soft rough approximations satisfy the following properties:
aprP ð;Þ ¼ aprP ð;Þ ¼ ;; [ aprP ðUÞ ¼ aprP ðUÞ ¼ f ðaÞ; a2A
aprP ðX \ YÞ # aprP ðXÞ \ aprP ðYÞ; aprP ðX [ YÞ aprP ðXÞ [ aprP ðYÞ; aprP ðX [ YÞ ¼ aprP ðXÞ [ aprP ðYÞ; aprP ðX \ YÞ # aprP ðXÞ \ aprP ðYÞ; X # Y ) aprP ðXÞ # aprP ðYÞ; X # Y ) aprP ðXÞ # aprP ðYÞ:
Proposition 3.5. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then for any X # U, X is soft Pdefinable if and only if apr P ðXÞ # X.
Proof. Note first that if X is soft P-definable, then apr P ðXÞ ¼ apr P ðXÞ, and so apr P ðXÞ ¼ apr P ðXÞ # X. Conversely, suppose that apr P ðXÞ # X for X # U. To show that X is soft P-definable, we only need to prove that aprP ðXÞ # apr P ðXÞ since the reverse inequality is trivial. Let u 2 apr P ðXÞ. Then u 2 F(a) and F(a) \ X – ; for some a 2 A. It follows that u 2 FðaÞ # apr P ðXÞ # X. Hence u 2 aprP(X), and so aprP ðXÞ # aprP ðXÞ as required. h Remark 3.6. To illustrate the above result, we revisit Example 3.3. For X = {u3, u4, u5} # U, we already know that aprP(X) = {u3}, and aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g. Thus apr P ðXÞ X and X is a soft P-rough set. On the other hand, for X 1 ¼ fu3 ; u4 g # U; apr P ðX 1 Þ ¼ fu3 g ¼ apr P ðX 1 Þ. Hence aprP ðX 1 Þ # X 1 and X1 is a soft P-definable set. Theorem 3.7. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then we have (1) (2) (3) (4)
aprP ðapr P ðXÞÞ ¼ aprP ðXÞ; aprP ðapr P ðXÞÞ aprP ðXÞ; aprP(aprP(X)) = aprP(X); aprP ðapr P ðXÞÞ aprP ðXÞ,
for all X # U.
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Proof. (1) Let Y ¼ aprP ðXÞ and u 2 Y. Then u 2 F(a) and F(a) \ X – ; for some a 2 A. By Proposition 3.4, S Y ¼ apr P ðXÞ ¼ a2A fFðaÞ : FðaÞ \ X – ;g. There exists a 2 A such that u 2 F(a) # Y. Hence u 2 aprP(Y), and so Y # aprP(Y). On the other hand, we know that aprP(Y) # Y holds for any Y # U. Thus it follows that Y = aprP(Y) as required. S (2) Let Y = aprP(X) and u 2 Y. Then u 2 F(a) # X for some a 2 A. Since Y = aprP(X) = a2A{F(a): F(a) # X} by Proposition 3.4, we deduce that u 2 F(a) and F(a) \ Y = F(a) – ;. Hence u 2 apr P ðYÞ, and so Y # apr P ðYÞ. S (3) Let Y = aprP(X) and u 2 Y. Then u 2 F(a) # X for some a 2 A. But Y = aprP(X) = a2A{F(a): F(a) # X}, we deduce that u 2 F(a) # Y for a 2 A. Thus u 2 aprP(Y), and so Y # aprP(Y). Since aprP(Y) # Y for any Y # U, we conclude that Y = aprP(Y). S (4) Let Y ¼ aprP ðXÞ and u 2 Y. Then u 2 F(a) and F(a) \ X – ; for some a 2 A. But Y ¼ apr P ðXÞ ¼ a2A fFðaÞ : FðaÞ \ X – ;g, it follows that u 2 F(a) and F(a) \ Y = F(a) – ;. Hence u 2 aprP ðYÞ, and so Y # aprP ðYÞ. h Note that the inclusion relations in Proposition 3.7 may be strict, as shown in the following example. Example 3.8. Let U = {u1, u2, . . . , u8} and E = {e1, e2, . . . , e6}. Let A = {e1, e2, e3, e4} # E, S1 = (F, A) a soft set over U given by Table 2, and the soft approximation space P = (U, S1). For X = {u4, u7, u8} # U, we have aprP(X) = F(e1) [ F(e4) = X, and aprP ðXÞ ¼ Fðe1 Þ [ Fðe3 Þ [ Fðe4 Þ ¼ fu2 ; u3 ; u4 ; u5 ; u7 ; u8 g. Let Y ¼ apr P ðXÞ. Then we have
aprP ðaprP ðXÞÞ ¼ aprP ðYÞ ¼ Fðe1 Þ [ Fðe3 Þ [ Fðe4 Þ ¼ Y: Also, we have aprP ðapr P ðXÞÞ ¼ aprP ðXÞ ¼ Y%X ¼ apr P ðXÞ, which suggests that the inclusion (2) in Proposition 3.7 may hold strictly. Moreover, it is easy to see that aprP(aprP(X)) = aprP(X), and
aprP ðaprP ðXÞÞ ¼ aprP ðYÞ ¼ Fðe1 Þ [ Fðe2 Þ [ Fðe3 Þ [ Fðe4 Þ ¼ U%Y; which indicates that the inclusion (4) in Proposition 3.7 may be strict. Definition 3.9. Let S = (F, A) be a soft set over U. If
S
a2AF(a)
= U, then S is said to be a full soft set.
Theorem 3.10. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then the following conditions are equivalent: (1) (2) (3) (4) (5)
S is a full soft set. aprP(U) = U. aprP ðUÞ ¼ U. X # apr P ðXÞ for all X # U. aprP ðfugÞ – ; for all u 2 U.
Proof. Note first that
aprP ðUÞ ¼
[
fFðaÞ : FðaÞ # Ug ¼
a2A
[
FðaÞ:
a2A
Hence by definition, S = (F, A) is a full soft set if and only if aprP(U) = U. That is, conditions (1) and (2) are equivalent. Similarly, we can show that (1) and (3) are equivalent conditions. On the other hand, we can prove that conditions (4), (5) and (1) are also equivalent. Now assume that condition (4) holds. We prove that condition (5) is true. Given u 2 U, by condition (4), we have fug # aprP ðfugÞ. Thus apr P ðfugÞ – ; since u 2 apr P ðfugÞ. This shows that the condition (4) implies the condition (5). Next, we prove that condition (5) implies condition (1) as well. So suppose that the condition (5) holds. For any u 2 U, we have that apr P ðfugÞ – ;. Let v be an element in aprP ðfugÞ. Then there exists some a 2 A such that v 2 F(a) and F(a) \ {u} – ;. It follows that u = v 2 F(a), and so we have that
Table 2 Tabular representation of the soft set S1.
e1 e2 e3 e4
u1
u2
u3
u4
u5
u6
u7
u8
0 1 0 0
0 0 1 0
0 0 1 0
0 0 1 1
0 1 1 0
0 1 0 0
1 0 0 1
1 0 0 0
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u 2 [a2A{F(a)}. Hence S = (F, A) is a full soft set. This shows that the condition (5) implies the condition (1). To complete the proof, it remains to show that (1) implies (4). Assume that S = (F, A) is a full soft set and X # U. For any x 2 X, since S is full, there exists some a 2 A such that x 2 F(a). Also, it is clear that X \ F(a) – ; since x 2 X \ F(a). Hence we have that x 2 aprP ðXÞ and so X # aprP ðXÞ as required. h The following result is a direct consequence of Propositions 3.5 and 3.10. Corollary 3.11. Let S = (F, A) be a full soft set over U and P = (U, S) a soft approximation space. Then for any X # U, X is soft Pdefinable if and only if apr P ðXÞ ¼ X. As one might suspect from the analogy with Pawlak’s rough sets, if S = (F, A) is a full soft set over U, we have the following interpretations of the concepts introduced in Definition 3.1. Intuitively, the soft lower approximation (equivalently, the soft positive region) of a set X consists of all elements that surely belong to X, whereas the soft negative region constitutes of all elements that surely do not belong to X; meanwhile, the soft upper approximation of the set X constitutes of all elements that possibly belong to X, and the soft boundary region consists of all elements that cannot be classified uniquely to X or its complement X, by employing available knowledge from the soft approximation space. Let S = (F, A) be a full soft set over U, P = (U, S) a soft approximation space and X # U. We define the following four basic types of soft rough sets:
X X X X
is is is is
roughly soft P-definable if aprP(X) – ; and apr P ðXÞ – U; internally soft P-indefinable if aprP(X) = ; and apr P ðXÞ – U; externally soft P-indefinable if aprP(X) – ; and apr P ðXÞ ¼ U; totally soft P-indefinable if aprP(X) = ; and apr P ðXÞ ¼ U.
The intuitive meaning of this classification is as follows: If X is roughly soft P-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 they belong to X, by using available knowledge from the soft approximation space P. If X is internally soft P-indefinable, this means that we are able to decide about some elements of U that they belong to 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 P-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 X, by employing P. If X is totally soft P-indefinable, we are unable to decide for any element of U whether it belongs to X or X, by employing P. Proposition 3.12. Let S = (F, A) be a full soft set over U and P = (U, S) a soft approximation space. Then for any X # U, we have the following: (1) aprP ðXÞ # apr P ðXÞ; (2) Neg P ðXÞ ¼ apr P ðXÞ # aprP ðXÞ.
Proof (1) If the set aprP(X) is empty, then clearly we have that the inclusion apr P ðXÞ # aprP ðXÞ holds. So suppose that aprP(X) – ;. Let u be any element of aprP(X). Since S = (F, A) is a full soft set, there exists some a0 2 A such that u 2 F(a0). Note also that
aprP ðXÞ ¼ fu 2 U : 8a 2 A; u 2 FðaÞ ) FðaÞ \ ðXÞ – ;g: Thus it follows that F(a0) \ (X) – ; since u 2 F(a0). Hence we have that u 2 aprP ðXÞ as required. (2) It is clear that the inclusion Neg P ðXÞ ¼ aprP ðXÞ # apr P ðXÞ holds when the set aprP ðXÞ is empty. So suppose that aprP ðXÞ – ;. Let u 2 aprP ðXÞ. Since S = (F, A) is a full soft set, there exists some a0 2 A such that u 2 F(a0). But we have that
Neg P ðXÞ ¼ aprP ðXÞ ¼ fu 2 U : 8a 2 A; u 2 FðaÞ ) FðaÞ # ðXÞg: Thus we deduce that F(a0) # (X) since u 2 F(a0). Hence we have that u 2 aprP(X) as required. h
Remark 3.13. As illustration of the above result, we revisit Example 3.8. It is clear that the soft set S1 given by Table 2 is a full soft set. For X = {u4, u7, u8} # U, we already know that aprP(X) = X, and aprP ðXÞ ¼ Y ¼ fu2 ; u3 ; u4 ; u5 ; u7 ; u8 g. Hence aprP(X) = X = {u1, u2, u3, u5, u6} and Neg P ðXÞ ¼ aprP ðXÞ ¼ Y ¼ fu1 ; u6 g. In addition, we have that aprP(X) = F(e2) = {u1, u5, u6} and
aprP ðXÞ ¼ Fðe2 Þ [ Fðe3 Þ ¼ fu1 ; u2 ; u3 ; u4 ; u5 ; u6 g:
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It is easy to see that apr P ðXÞ$aprP ðXÞ and Neg P ðXÞ ¼ aprP ðXÞ$aprP ðXÞ. This shows that the inclusions in Proposition 3.12 may be strict if the soft set is full. On the other hand, it is worth noting that these inclusions may not hold when the soft set is not full. To see this, let us reconsider Example 3.3. Note first that the soft set S given by Table 1 is not a full soft set. For X1 = {u3, u4} # U, we already know that aprP ðX 1 Þ ¼ aprP ðX 1 Þ ¼ fu3 g. Hence aprP ðX 1 Þ ¼ aprP ðX 1 Þ ¼ fu1 ; u2 ; u4 ; u5 ; u6 g. Moreover, we have that
aprP ðX 1 Þ ¼ aprP ðX 1 Þ ¼ Fðe1 Þ [ Fðe4 Þ ¼ fu1 ; u2 ; u5 ; u6 g: Thus apr P ðX 1 Þ apr P ðX 1 Þ and Neg P ðX 1 Þ ¼ apr P ðX 1 Þ aprP ðX 1 Þ. Definition 3.14. Let S = (F, U) be a soft set over U and P = (U, S) a soft approximation space. For any X, Y # U, we define
X P Y () aprP ðXÞ ¼ aprP ðYÞ;
X _ P Y () aprP ðXÞ ¼ aprP ðYÞ; 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. Proposition 3.15. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then for any X, Y, X1, Y1 # U, we have (1) (2) (3) (4) (5)
X _ P Y () X _ P ðX [ YÞ _ P Y; X _ P X 1 ; Y _ P Y 1 ) ðX [ YÞ _ P ðX 1 [ Y 1 Þ; X _ P Y ) X [ ðYÞ _ P U; X # Y; Y _ P ; ) X _ P ;; X # Y; X _ P U ) Y _ P U.
Proof. (1) If X _ P Y, then aprP ðXÞ ¼ aprP ðYÞ. Since apr P ðX [ YÞ ¼ aprP ðXÞ [ apr P ðYÞ, we deduce
aprP ðX [ YÞ ¼ aprP ðXÞ ¼ aprP ðYÞ and so X _ P ðX [ YÞ _ P Y. Conversely, if X _ P ðX [ YÞ _ P Y, then we immediately have that X _ P Y by using the transitivity of the relation _ P . (2) Assume that X _ P X 1 and Y _ P Y 1 . Then by definition, we know that aprP ðXÞ ¼ aprP ðX 1 Þ and apr P ðYÞ ¼ aprP ðY 1 Þ. Since aprP ðX [ YÞ ¼ aprP ðXÞ [ aprP ðYÞ and aprP ðX 1 [ Y 1 Þ ¼ aprP ðX 1 Þ [ apr P ðY 1 Þ, we deduce that apr P ðX [ YÞ ¼ apr P ðX 1 [ Y 1 Þ, whence ðX [ YÞ _ P ðX 1 [ Y 1 Þ. (3) Suppose that X _ P Y. Then by definition, apr P ðXÞ ¼ aprP ðYÞ. Since aprP ðX [ ðYÞÞ ¼ aprP ðXÞ [ apr P ðYÞ and aprP ðUÞ ¼ aprP ðYÞ [ apr P ðYÞ, it follows that apr P ðX [ ðYÞÞ ¼ aprP ðUÞ; hence X [ ðYÞ _ P U as required. (4) Let X # Y and Y _ P ;. Then we deduce
aprP ðXÞ # aprP ðYÞ ¼ aprP ð;Þ ¼ ;: Hence aprP ðXÞ ¼ ; ¼ aprP ð;Þ, and so we have that X _ P ;. (5) Suppose that X # Y and X _ P U. Then we deduce
aprP ðYÞ aprP ðXÞ ¼ aprP ðUÞ: But it is easy to see 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. h
Definition 3.16. Let S = (F, A) be a soft set over U. If for any a1, a2 2 A, there exists a3 2 A such that F(a3) = F(a1) \ F(a2) whenever F(a1) \ F(a2) – ;, then S is said to be an intersection complete soft set. Lemma 3.17. Let S = (F, A) be an intersection complete soft set over U and P = (U, S) a soft approximation space. Then we have
aprP ðX \ YÞ ¼ aprP ðXÞ \ aprP ðYÞ for all X, Y # U.
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Proof. Note first that for any soft set S (need not be intersection complete),
aprP ðX \ YÞ # aprP ðXÞ \ aprP ðYÞ: Therefore, it suffices to show that the reverse inclusion
aprP ðX \ YÞ aprP ðXÞ \ aprP ðYÞ also holds. Now, let u 2 aprP(X) \ aprP(Y). Then there exist a1, a2 2 A such that u 2 F(a1) # X and u 2 F(a2) # Y. Since by assumption, S is an intersection complete soft set, we deduce that there exists a3 2 A such that
u 2 Fða3 Þ ¼ Fða1 Þ \ Fða2 Þ # X \ Y: Hence we conclude that u 2 aprP(X \ Y) as required. h Remark 3.18. Notice that the requirement of an intersection complete soft set is not a necessary condition for the equality in Lemma 3.17. To illustrate this, we consider an example as follows. Let U = {u1, u2, u3, u4} be a universe and S = (F, A) a soft set over U, where A = {e1, e2, e3, e4}, and F is a set-valued mapping such that F(e1) = {u1, u2, u3}, F(e2) = {u2, u3, u4}, F(e3) = {u2} and F(e4) = {u3}. Let P = (U, S) be a soft approximation space. Then, one can verify that
aprP ðX \ YÞ ¼ aprP ðXÞ \ aprP ðYÞ for all X, Y # U. On the other hand, it is easy to see that F(e1) \ F(e2) = {u2, u3}, but there does not exist any ei 2 A such that F(ei) = {u2, u3}, whence S = (F, A) is not an intersection complete soft set. By using the above lemma, we can prove the following result in relation to the lower soft rough equal relation. Proposition 3.19. Let S = (F, A) be an intersection complete soft set over U and P = (U, S) a soft approximation space. Then for any X, Y, X1, Y1 # U, we have (1) (2) (3) (4) (5)
X P Y () X P ðX \ YÞ P Y;
X P X 1 ; Y P Y 1 ) ðX \ YÞ P ðX 1 \ Y 1 Þ;
X P Y ) X \ ðYÞ P ;;
X # Y; Y P ; ) X P ;;
X # Y; X P U ) Y P U.
Theorem 3.20. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then we have
aprP ðXÞ ¼
\ fY # U : X P Yg
for all X # U. Moreover, if S is a full soft set, then we have
aprP ðXÞ
[ fY # U : X _ P Yg;
for all X # U. Proof. Let u 2 aprP(X). If X P Y, then by definition aprP(X) = aprP(Y). But aprP(Y) # Y for any Y # U. It follows that
u 2 aprP ðXÞ ¼ aprP ðYÞ # Y: T T Hence u 2 fY # U : X P Yg, and so aprP ðXÞ # fY # U : X P Yg. Next, we show that the reverse inclusion T T
fY # U : X P Yg # apr P ðXÞ also holds. Let u 2 fY # U : X P Yg. Then by Proposition 3.7, we have aprP(aprP(X)) = aprP(X). T
Thus X P apr P ðXÞ, and it follows that u 2 aprP(X). Consequently, we conclude that aprP ðXÞ ¼ fY # U : X P Yg. S
Now assume that S is a full soft set and u 2 fY # U : X _ P Yg. Then there exists some Y0 # U such that u 2 Y0 and X _ P Y 0 . Since S is full, we have Y 0 # aprP ðY 0 Þ by Proposition 3.10. It follows that u 2 Y 0 # apr P ðY 0 Þ ¼ aprP ðXÞ. Hence we conclude that S apr P ðXÞ fY # U : X _ P Yg as required. h Note that for a full soft set S, the inclusion in Theorem 3.20 may be strict, i.e.,
aprP ðXÞ%
[ fY # U : X _ P Yg:
Moreover, if S is not a full soft set, this inclusion may not hold. For illustration, we consider the following example. Example 3.21. Let U = {u1, u2, . . ., u8} and E = {e1, e2, . . ., e6}. Now, we reconsider the full soft set S1 = (F, A) given in Example 3.8, and take the soft approximation space P = (U, S1). For X = {u4, u7, u8} # U, we already know that aprP(X) = X, and apr P ðXÞ ¼ Y ¼ fu2 ; u3 ; u4 ; u5 ; u7 ; u8 g. Then it is easy to see that
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\ fY # U : X P Yg ¼ X ¼ aprP ðXÞ
and
[ fY # U : X _ P Yg ¼ fu2 ; u3 ; u4 ; u7 ; u8 g$Y: Next, let B = {e1, e3, e5} and consider the soft set S2 = (G, B) over U given by Table 3. Note that S2 is not a full soft set. Let Q = (U, S2) be a new soft approximation space. For the same set X = {u4, u7, u8} # U, we have aprQ(X) = ;, and
aprQ ðXÞ ¼ Gðe1 Þ [ Gðe3 Þ ¼ fu1 ; u2 ; u4 ; u8 g ¼ W: Then it is clear that
\ fY # U : X Q Yg ¼ ; ¼ aprQ ðXÞ
and
[ fY # U : X _ Q Yg ¼ fu1 ; u2 ; u4 ; u5 ; u6 ; u7 ; u8 g W:
4. Relationships between soft rough sets and Pawlak’s rough sets In this section, we shall explore the relationships between soft rough sets and Pawlak’s rough sets. We start with the following basic notion. Definition 4.1. A soft set S = (F, A) over U is call a partition soft set if {F(a): a 2 A} forms a partition of U. By definition, we immediately have that every partition soft set is a full soft set. The following result indicates that soft sets and binary relations are closely related. Theorem 4.2. Let S = (F, A) be a soft set over U. Then S induces a binary relation qS # A U, which is defined by
ðx; yÞ 2 qS () y 2 FðxÞ for all x 2 A and y 2 U. Conversely, let q be a binary relation from A to U. Define a set-valued mapping F q : A ! PðUÞ by
F q ðxÞ ¼ fy 2 U : ðx; yÞ 2 qg for all x 2 A. Then Sq = (Fq, A) is a soft set over U. Moreover, we have that SqS ¼ S and qSq ¼ q. Proof. It is easy to see that the first part of the statement holds. Thus we only need to show that SqS ¼ S and qSq ¼ q. Suppose that S = (F, A) is a soft set over U and x 2 A. Then for any y 2 U, by definition we have that
y 2 F qS ðxÞ () ðx; yÞ 2 qS () y 2 FðxÞ: That is, F qS ðxÞ ¼ FðxÞ for all x 2 A. Thus F qS ¼ F, whence SqS ¼ S. Next, assume that q # A U, x 2 A and y 2 U. Then by definition, it is clear that
ðx; yÞ 2 qSq () y 2 F q ðxÞ () ðx; yÞ 2 q: Therefore we conclude that qSq ¼ q as required. h In what follows, qS is called the canonical relation of the soft set S, and Sq is referred to as the canonical soft set of the binary relation q. The following example shows that every quotient set can be viewed as a partition soft set. Example 4.3. Let q be an equivalence relation on U. Then the set-valued mapping F q : U ! PðUÞ in Theorem 4.2 coincides with the nature mapping of the equivalence relation q. That is, Fq(x) = [x]q for all x 2 U. Thus the canonical soft set of the equivalence relation q can be identified with the quotient set U/q. Moreover, it is easy to see that Sq is a partition soft set since {Fq(x): x 2 U} = U/q is a partition of U.
Table 3 Tabular representation of the soft set S2.
e1 e3 e5
u1
u2
u3
u4
u5
u6
u7
u8
1 0 0
0 1 0
0 0 1
1 1 0
0 0 0
0 0 0
0 0 0
0 1 0
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Theorem 4.4. Let R be an equivalence relation on U, SR = (FR, U) the canonical soft set of R and P = (U, SR) a soft approximation space. Then for all X # U,
R X ¼ aprP ðXÞ and R X ¼ aprP ðXÞ: Thus in this case, X # U is a (Pawlak) rough set if and only if X is a soft P-rough set. Proof. Let X # U and u 2 U. First, we show that R*X = aprP(X). If u 2 R*X, then [u]R # X. Thus there exists u 2 U such that u 2 [u]R = FR(u) # X. Hence u 2 aprP(X), and so R*X # aprP(X). Conversely, assume that u 2 aprP(X). By definition, there exists v 2 U such that u 2 FR(v) = [v]R # X. It follows that [u]R = [v]R # X, whence u 2 R*X. Therefore, we conclude that R*X = aprP(X). Now it remains to show that R X ¼ apr P ðXÞ. Suppose that u 2 R*X. Then we have that [u]R \ X – ;. Hence there exists u 2 U such that u 2 [u]R = FR(u) and FR(u) \ X – ;. Thus u 2 aprP ðXÞ, and so R X # apr P ðXÞ. Conversely, assume that u 2 aprP ðXÞ. Then there exists v 2 U such that u 2 FR(v) = [v]R and FR(v) \ X – ;. It follows that [u]R = [v]R and [u]R \ X – ;. Hence u 2 R*X, and so we conclude that R X ¼ apr P ðXÞ. h Theorem 4.5. Let S = (F, A) be a partition soft set over U and P = (U, S) a soft approximation space. Define an equivalence relation R on U by
ðx; yÞ 2 R () 9a 2 A;
fx; yg # FðaÞ
for all x, y 2 U. Then, for all X # U,
R X ¼ aprP ðXÞ and R X ¼ aprP ðXÞ: Proof. Let X # U and u 2 U. Note first that there exists a unique element au 2 A such that u 2 F(au) since S = (F, A) is a partition soft set over U. We claim that F(au) = [u]R, where R is the equivalence relation on U given as above. In fact, this is true since for all x 2 U
x 2 ½uR () ðx; uÞ 2 R () 9a 2 A;
fx; ug # FðaÞ () x 2 Fðau Þ:
Next, we show that R*X = aprP(X). If u 2 R*X, then [u]R # X. Thus there exists au 2 A such that u 2 F(au) = [u]R # X. Hence u 2 aprP(X), and so R*X # aprP(X). Conversely, assume that u 2 aprP(X). Then there exists a 2 A such that u 2 F(a) # X. But by uniqueness of the element au 2 A, we have that a = au. It follows that [u]R = F(au) = F(a) # X, and so u 2 R*X. Hence we conclude that R*X = aprP(X). To complete the proof, we only need to show that R X ¼ apr P ðXÞ. Assume that u 2 R*X. Then we have that [u]R \ X – ;. Hence there exists au 2 A such that u 2 F(au) = [u]R and F(au) \ X – ;. Thus u 2 apr P ðXÞ, and so R X # apr P ðXÞ. Conversely, suppose that u 2 apr P ðXÞ. Then there exists a 2 A such that u 2 F(a) and F(a) \ X – ;. By uniqueness of the element au 2 A, we have that a = au. Thus it follows that [u]R = F(au) = F(a) and [u]R \ X – ;. Hence u 2 R*X, and so we conclude that R X ¼ apr P ðXÞ. h The above results reveal that Pawlak’s rough set model can be viewed as a special instance of soft rough sets. Since one can consider F() as a basic granule of the universe, it follows that a soft set can be used to granulate the universe. The granulation structure induced by a soft set is referred to as a soft approximation space. Moreover, it is worth noting that in contrast to equivalence classes, there is no restriction on the subsets F(), whence the granulation structure induced by a soft set may not form a partition or even a covering of the universe. Soft rough sets are also associated with generalized rough sets based on covering or binary relation. Next, we consider the relationship between soft rough sets and generalized rough sets based on binary relation [10,15,27,28,32,33,36]. Let R # U U be a binary relation on U without any additional constraints. For x, y 2 U, if (x, y) 2 R, y is said to be R-related to x, and denoted by xRy. In this case, we say that x is a predecessor of y, and y is a successor of x. Note that a binary relation R can be equivalently defined by a mapping from U to the power set PðUÞ given by Rs(x) = {y 2 U: xRy} for all x 2 U. By Theorem 4.2, (Rs, U) is the canonical soft set of the binary relation R on U, and so we can discuss soft rough sets in the corresponding soft approximation space. According to [32], the set Rs(x) of R-related elements of x may be interpreted as a successor neighborhood of x, and Rs is called the successor operator. In general, by substituting [x]R with Rs(x), one may approximate a subset X # U by a pair of subsets of U with respect to the binary relation R. These facts indicate that rough sets constructed from arbitrary binary relations by using neighborhood operators can also be related to soft rough sets. Finally, we present a comparative analysis between rough sets and soft rough sets, by giving an interesting example. Example 4.6. Let us consider the following soft set S = (F, E) which describes ‘‘life expectancy’’. Suppose that the universe U = {u1, u2, u3, u4, u5, u6} consists of six persons and E = {e1, e2, e3, e4} is a set of decision parameters. The ei (i = 1, 2, 3, 4) stands for ‘‘under stress’’, ‘‘young’’, ‘‘drug addict’’ and ‘‘healthy’’. Set F(e1) = {u5}, F(e2) = {u1, u2}, F(e3) = ; and F(e4) = {u1, u2, u3, u6}. The soft set (F, E) can be viewed as the following collection of approximations:
ðF; EÞ ¼ fðunder stress; fu5 gÞ; ðyoung;fu1 ; u2 gÞ; ðdrug addict; ;Þ; ðhealthy;fu1 ; u2 ; u3 ; u6 gÞg:
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Table 4 An information table.
Sex Age category Living area Habits
u1
u2
u3
u4
u5
u6
Woman Young City NSND
Woman Young City NSND
Man Mature age City Smoke
Man Old Village SD
Man Mature age City Smoke
Man Baby Village NSND
On the other hand, ‘‘life expectancy’’ topic can also be described using rough sets as follows: The evaluation will be done in terms of attributes: ‘‘sex’’, ‘‘age category’’, ‘‘living area’’, ‘‘habits’’, characterized by the value sets ‘‘{man, woman}’’, ‘‘{baby, young, mature age, old}’’, ‘‘{village, city}’’, ‘‘{smoke, drinking, smoke and drinking, no smoke and no drinking}’’. We denote ‘‘smoke and drinking’’ by SD and ‘‘no smoke and no drinking’’ by NSND. The information will be given by Table 4, where the rows are labeled by attributes and the table entries are the attribute values for each person. From here we obtain the following equivalence classes, induced by the above mentioned attributes:
½u1 R ¼ ½u2 R ¼ fu1 ; u2 g;
½u3 R ¼ ½u5 R ¼ fu3 ; u5 g;
½u4 R ¼ fu4 g;
½u6 R ¼ fu6 g:
Let X be a target subset of U, that we wish to represent using the above equivalence classes. Hence we analyze the upper and lower approximations of X, in some particular cases: 1. Set X = {u1, u2, u3, u6}. It follows that
R ðXÞ ¼ fu1 ; u2 ; u6 g;
R ðXÞ ¼ fu1 ; u2 ; u3 ; u5 ; u6 g:
Let us calculate now the soft P-lower and P-upper approximations of X, where P = (U, S). By Proposition 3.4. we obtain
aprP ðXÞ ¼
[ fFðeÞ : FðeÞ # Xg ¼ fu1 ; u2 ; u3 ; u6 g ¼ X;
aprP ðXÞ ¼
[ fFðeÞ : FðeÞ \ X – ;g ¼ fu1 ; u2 ; u3 ; u6 g ¼ X;
e2E
e2E
hence X is soft P-definable. 2. Set X = {u5}. It follows that R*(X) = ;, R*(X) = {u3, u5}. On the other hand, apr P ðXÞ ¼ apr P ðXÞ ¼ X, hence X is soft P-definable. 3. Set X = {u4, u5}. It follows that R*(X) = {u4}, R*(X) = {u3, u4, u5}. On the other hand, apr P ðXÞ ¼ apr P ðXÞ ¼ fu5 g. In this case, X is not a subset of apr P ðXÞ. 4. Set X = {u2, u3}. Then R*(X) = aprP(X) = ;, while R*(X) = {u1, u2, u3, u5} and aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u6 g. The above results show that soft rough set approximation is a worth considering alternative to the rough set approximation. Soft rough sets could provide a better approximation than rough sets do, depending on the structure of the equivalence classes and of the subsets F(e), where e 2 E. 5. Conclusion In this study, we have proposed the new concept of soft rough sets, which can be viewed as a soft set based generalization of the standard rough set model. We presented important properties of soft rough approximations based on soft approximation spaces, giving interesting examples. Several new types of soft sets were proposed, and soft rough equal relations were discussed as well. Furthermore, we examined the relationship between soft rough sets and Pawlak’s rough sets, and compared these two different models. As future work, connections between soft rough sets and various types of generalized rough set models could be explored. Acknowledgements The authors are highly grateful to the anonymous referees and Professor Witold Pedrycz, Editor-in-Chief, for their valuable comments and suggestions which greatly improve the quality of this paper. We are also indebted to Professor Yongming Li for helpful discussion and valuable suggestions. This work was supported by the National Natural Science Foundation of China (Nos. 10926031 and 10571112), the Natural Science Foundation of Education Department of Shaanxi Province of China (No. 2010JK831), and by the Shaanxi Provincial Research and Development Plan of Science and Technology (No. 2008K01-33). References [1] H. Aktasß, N. Çag˘man, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735. [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.
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[3] Z. Bonikowski, E. Bryniariski, V.W. Skardowska, Extension and intensions in the rough set theory, Inform. Sci. 107 (1998) 149–167. [4] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The parametrization reduction of soft sets and its applications, Comput. Math. Appl. 49 (2005) 757–763. [5] D. Chen, C.Z. Wang, Q.H. Hu, A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets, Inform. Sci. 177 (2007) 3500–3518. [6] B. Davvaz, W.A. Dudek, Y.B. Jun, Intuitionistic fuzzy Hv-submodules, Inform. Sci. 176 (2006) 285–300. [7] F. Feng, Y.B. Jun, X.Z. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621–2628. [8] W.L. Gau, D.J. Buehrer, Vague sets, IEEE Trans. Syst. Man Cybernet. 23 (2) (1993) 610–614. [9] M.B. Gorzalzany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets Syst. 21 (1987) 1–17. [10] S. Greco, B. Matarazzo, R. Slowinski, Rough approximation by dominance relations, Int. J. Intell. Syst. 17 (2002) 153–171. [11] Y.B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413. [12] Y.B. Jun, C.H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178 (2008) 2466–2475. [13] T.J. Li, Y. Leung, W.X. Zhang, Generalized fuzzy rough approximation operators based on fuzzy coverings, Int. J. Approx. Reason. 48 (2008) 836–856. [14] G. Liu, Y. Sai, A comparison of two types of rough sets induced by coverings, Int. J. Approx. Reason. 50 (2009) 521–528. [15] G. Liu, W. Zhu, The algebraic structures of generalized rough set theory, Inform. Sci. 178 (2008) 4105–4113. [16] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555–562. [17] P.K. Maji, A.R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083. [18] P.K. Maji, R. Biswas, A.R. Roy, Fuzzy soft sets, J. Fuzzy Math. 9 (3) (2001) 589–602. [19] D. Molodtsov, Soft set theory – First results, Comput. Math. Appl. 37 (1999) 19–31. [20] D. Molodtsov, The Theory of Soft Sets (in Russian), URSS Publishers, Moscow, 2004. [21] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (1982) 341–356. [22] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers., Dordrecht, 1991. [23] Z. Pawlak, A. Skowron, Rudiments of rough sets, Inform. Sci. 177 (2007) 3–27. [24] Z. Pawlak, A. Skowron, Rough sets: some extensions, Inform. Sci. 177 (2007) 28–40. [25] Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Inform. Sci. 177 (2007) 41–73. [26] A.R. Roy, P.K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math. 203 (2007) 412–418. [27] R. Słowin´ski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Trans. Knowledge Data Eng. 12 (2000) 331– 336. [28] A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fundam. Inform. 27 (1996) 245–253. [29] W.Z. Wu, W.X. Zhang, Neighborhood operator systems and approximations, Inform. Sci. 144 (2002) 201–217. [30] W.X. Xu, W.X. Zhang, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets Syst. 158 (2007) 2443–2455. [31] X.B. Yang, T.Y. Lin, J.Y. Yang, Y. Li, D.J. Yu, Combination of interval-valued fuzzy set and soft set, Comput. Math. Appl. 58 (2009) 521–527. [32] Y.Y. Yao, T.Y. Lin, Generalization of rough sets using modal logic, Intell. Autom. Soft Comput. – Int. J. 2 (1996) 103–120. [33] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inform. Sci. 109 (1998) 21–47. [34] Y.Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci. 111 (1998) 239–259. [35] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [36] W. Zhu, Generalized rough sets based on relations, Inform. Sci. 177 (22) (2007) 4997–5011. [37] W. Zhu, Topological approaches to covering rough sets, Inform. Sci. 177 (6) (2007) 1499–1508. [38] W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Inform. Sci. 179 (2009) 210–225.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Soft sets and soft rough sets Feng Feng a,b,⇑, Xiaoyan Liu b, Violeta Leoreanu-Fotea c, Young Bae Jun d a
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China Department of Applied Mathematics, Faculty of Science, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China Faculty of Mathematics, ‘‘Al.I.Cuza’’ University, 6600 Iasßi, Romania d Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea b c
a r t i c l e
i n f o
Article history: Received 28 June 2008 Received in revised form 12 March 2010 Accepted 8 November 2010
Keywords: Soft sets Rough sets Soft rough sets Soft approximation spaces Soft rough approximations Granular computing
a b s t r a c t In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak’s rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak’s rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction The volume and complexity of the collected data in our modern society is growing rapidly. There often exist various types of uncertainties in those data related to complex problems in biology, economics, ecology, engineering, environmental science, medical science, social science, and many other fields. In order to describe and extract the useful information hidden in uncertain data, researchers in mathematics, computer science and related areas have proposed a number of theories such as probability theory, fuzzy set theory [35], intuitionistic fuzzy set theory [2,6], rough set theory [21–25], vague set theory [8] and interval mathematics [9]. While all these theories are well-known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out by Molodtsov [19]. In 1999, Molodtsov [19] proposed soft set theory as a new mathematical tool for dealing with uncertainties which is free from the difficulties affecting existing methods. As reported in [19,20], a wide range of applications of soft sets have been developed in many different fields, including the smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability theory and measurement theory. There has been a rapid growth of interest in soft set theory and its applications in recent years. Maji et al. [17] discussed the application of soft set theory in a decision making problem. The same authors also extended classical soft sets to fuzzy soft sets [18]. By combining the interval-valued fuzzy set and soft set models, Yang et al. [31] further introduced interval-valued fuzzy soft sets. Based on fuzzy soft sets, Roy and Maji
⇑ Corresponding author at: Department of Applied Mathematics, Faculty of Science, Xi’an Institute of Posts and Telecommunications, Xi’an 710061, PR China. Tel.: +86 29 88166086. E-mail addresses: [email protected] (F. Feng), [email protected] (X. Liu), [email protected] (V. Leoreanu-Fotea), [email protected] (Y.B. Jun). 0020-0255/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2010.11.004
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[26] presented a method of object recognition from an imprecise multi-observer data and applied it to decision making problems. Chen et al. [4] proposed a new definition of soft set parametrization reduction, and compared it with the related concept of attributes reduction in rough set theory. Maji et al. [16] introduced several operations in soft set theory and carried out a detailed theoretical study on soft sets. Since soft set theory emerged as a novel approach for modelling vagueness and uncertainty, it soon invoked an interesting question concerning a possible connection between soft sets and algebraic systems. As the first results in this research direction, Aktasß and Çag˘man [1] initiated soft groups and showed that fuzzy groups can be viewed as a special case of the soft groups. Jun [11] applied soft sets to the theory of BCK/BCI-algebras, and introduced the concept of soft BCK/BCI-algebras. Jun and Park [12] reported applications of soft sets in ideal theory of BCK/BCI-algebras. Soft semirings and several related notions are defined in [7] in order to establish a connection between soft sets and semirings. Rough set theory was initiated by Pawlak [21] for dealing with vagueness and granularity in information systems. This theory deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [22–25]. It is well-known that Pawlak’s rough set model is based on equivalence relations. In the real world, the equivalence relation is, however, too restrictive for many practical applications. To address this issue, many interesting and meaningful extensions of Pawlak’s rough sets have been presented in the literature. Equivalence relations can be replaced by tolerance relations [28], similarity relations [27], dominance relations [10], general binary relations [15,32,33,36], coverings of the universe of discourse [5,13,30,38] or neighborhood systems [29,34]. All these proposals share the common feature that they deal with approximations of concepts in terms of granules. Actually, different types of generalized rough sets are based on different interpretations of granules. In Pawlak’s rough set model, each equivalence class may be viewed as a granule consisting of indistinguishable elements. The granulation structure induced by an equivalence relation is a partition of the universe. By weakening the requirement of equivalence relations, we get more general granulation structures based on general binary relations or coverings of the universe. In this study we initiate the notion of soft rough sets, which can be seen as a generalized rough set model based on soft sets. The standard soft set model is used to form the granulation structure of the universe, namely the soft approximation space. Based on this granulation structure, we then define soft rough approximations, soft rough sets and some related notions. As the hybrid model combining rough sets with soft sets, soft rough sets could be exploited to extend many practical applications based on rough sets or soft sets. The remainder of this paper is organized as follows. Section 2 presents some fundamental concepts in rough set and soft set theories and also some interesting connections between them. Section 3 is devoted to a detailed discussion on the rudiments of soft rough sets. We define soft approximation spaces, soft rough approximations and soft rough sets. Basic properties of soft rough approximations are reported and supported by examples. Meanwhile full soft sets and intersection complete soft sets are introduced. We also define soft rough equal relations in terms of soft rough approximations and explore some related properties. In Section 4 we define partition soft sets and point out a close connection between soft sets and binary relations. We show that the standard rough set model may be considered a special case of the soft rough sets. Also, through an example we present a comparative analysis between rough sets and soft rough sets. Finally, conclusions are presented in the last section. Although this study is a preliminary proposal concerning the soft rough set model, we hope it will give rise to a potentially interesting research direction. 2. Overview of rough sets and soft sets In this section, we first recall some fundamental facts about Pawlak’s rough sets [21] and Molodtsov’s soft sets [19]. Then we point out some interesting connections between these two different models. The rough set philosophy is founded on the assumption that with every object of interest, we associate some information (data, knowledge). For example, if objects are patients suffering from a certain disease, symptoms of the disease form information about patients. Objects characterized by the same information are indiscernible in view of the available information about them. In many data analysis applications, information is stored and represented in a data table, where a set of objects are described by a set of attributes. Definition 2.1. [23] An information system (or a knowledge representation system) is a pair I ¼ ðU; AÞ of non-empty finite sets U and A, where U is a set of objects and A is a set of attributes; each attribute a 2 A is a function a:U ? Va, where Va is the set of values (called domain) of attribute a. Let U be a non-empty finite universe and R be an equivalence relation on U. The pair (U, R) is called a Pawlak approximation space. The equivalence relation R is often called an indiscernibility relation and related to an information system. Specifically, if I ¼ ðU; AÞ is an information system and B # A, then an indiscernibility relation R = I(B) can be defined by
ðx; yÞ 2 IðBÞ () aðxÞ ¼ aðyÞ;
8a 2 B;
where x, y 2 U, and a(x) denotes the value of attribute a for object x. Using the indiscernibility relation R, one can define the following two operations
R X ¼ fx 2 U : ½xR # Xg;
R X ¼ fx 2 U : ½xR \ X – ;g
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assigning to every subset X # U two sets R*X and R*X called the R-lower and the R-upper approximation of X, respectively. Moreover, the sets
PosR X ¼ R X;
Neg R X ¼ U R X;
BndR X ¼ R X R X
are referred to as the R-positive, the R-negative and the R-boundary region of X, respectively. If the R-boundary region of X is empty, i.e., R*X = R*X, then X is crisp (or exact) with respect to R; in the opposite case, i.e., BndRX – ;, X is said to be rough (or inexact) with respect to R [23]. Note that sometimes the pair (R*X, R*X) is also referred to as the rough set of X with respect to R [14,15]. If X # U is defined by a predicate P and x 2 U, we have the following interpretation: x 2 PosRX = R*X means that x certainly has property P; x 2 R*X means that x possibly has property P; x 2 NegRX means that x definitely does not have property P. Let us recall now the soft set notion, which is a newly-emerging mathematical approach to vagueness. Let U be an initial universe of objects and EU (simply denoted by E) the set of certain parameters in relation to the objects in U. Parameters are often attributes, characteristics, or properties of the objects in U. Let PðUÞ denote the power set of U. Following the definition in [1,16], the concept of soft sets is defined as follows. Definition 2.2. [1,16]A pair S = (F, A) is called a soft set over U, where A # E and F : A ! PðUÞ is a set-valued mapping. In other words, a soft set over U is a parameterized family of subsets of the universe U. For 2 A, F() may be considered as the set of -approximate elements in the soft set S = (F, A). It is worth noting that F() may be arbitrary: some of them may be empty, and some may have nonempty intersection [19]. The absence of any restrictions on the approximate description in soft set theory makes this theory very convenient and easily applicable in practice. Actually, we can use any suitable parametrization—with the help of words and sentences, real numbers, functions, mappings, and so on. Molodtsov [19] 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 (F, A) and (G, B) be soft sets over U. Then, the operation for soft sets is defined by (F, A) (G, B) = (H, A B), where H(a, b) = F(a) G(b), a 2 A, b 2 B and A B is the Cartesian product of A and B. Maji et al. [16] carried out Molodtsov’s seminal idea by introducing several operations in soft set theory, such as AND, OR, intersection and union. Definition 2.3. 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 (G, B) # (F, A), if B # A and G(b) # F(b) for all b 2 B. Although rough sets and soft sets are two different mathematical tools for modelling vagueness, there are some interesting connections between them. At first, we note that information systems and soft sets are closely related. Let S = (F, A) be a soft set over U. If U and A are both non-empty finite sets, then S could induce an information system I ¼ ðU; AÞ in a natural way. In fact, for any attribute a 2 A, one can define a function a: U ? Va = {0, 1} by
aðxÞ ¼
1; if x 2 FðaÞ; 0; otherwise:
Therefore every soft set may be considered as an information system. This justifies the tabular representation of soft sets widely used in the literature (e.g., see the soft set tables used in what follows). Conversely, it is worth noting that soft sets S can also be applied to represent information systems. Let I ¼ ðU; AÞ be an information system. Taking B = a2A{a} Va, as the parameter set, then a soft set (F, B) can be defined by setting
Fða; v Þ ¼ fx 2 U : aðxÞ ¼ v g; where a 2 A and v 2 Va. By the above discussion, it is easy to see that once given a soft set S = (F, A) over U, we obtain an information system I ¼ ðU; AÞ corresponding to the soft set S, whence we shall be able to construct rough approximations and discuss Pawlak’s rough sets based on the Pawlak approximation space (U, R) induced by the information system I. On the other hand, it is interesting to find that Pawlak’s rough set model may be considered as a special case of Molodtsov’s soft sets [1]. To see this, suppose that (U, R) is a Pawlak approximation space and X # U. Let R(X) = (R*X, R*X) be the rough set of X with respect to R. Consider two predicates p1(x), p2(x), which mean ‘‘[x]R # X’’ and ‘‘[x]R \ X – ;’’, respectively. The predicates p1(x), p2(x) may be treated as elements of a parameter set; that is, E = {p1(x), p2(x)}. Then we can define a setvalued mapping
F : E ! PðUÞ;
pi ðxÞ # Fðpi ðxÞÞ ¼ fx 2 U : pi ðxÞ is trueg;
where i = 1, 2. It follows that the rough set R(X) may be considered a soft set (F, E) with the following representation
ðF; EÞ ¼ fðp1 ðxÞ; R XÞ; ðp2 ðxÞ; R XÞg:
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Maji et al. [18] initiated the study on hybrid structures involving both fuzzy sets and soft sets. They introduced in [18] the notion of fuzzy soft sets, which can be seen as a fuzzy generalization of (classical) soft sets. Definition 2.4. [26] Let FðUÞ be the set of all fuzzy sets in the universe U. Let E be a set of parameters and A # E. A pair ð e F ; AÞ is called a fuzzy soft set over U, where e F is a mapping given by e F : A ! FðUÞ. 3. Soft rough approximations and soft rough sets In this section we introduce soft rough approximations and soft rough sets. If U is the initial universe, then the complement of X in U is denoted by X. Definition 3.1. Let S = (F, A) be a soft set over U. Then the pair P = (U, S) is called a soft approximation space. Based on the soft approximation space P, we define the following two operations
aprP ðXÞ ¼ fu 2 U : 9a 2 A; ½u 2 FðaÞ # X g;
ð1Þ
aprP ðXÞ ¼ fu 2 U : 9a 2 A; ½u 2 FðaÞ; FðaÞ \ X – ;g
ð2Þ
assigning to every subset X # U two sets aprP(X) and apr P ðXÞ, which are called the soft P-lower approximation and the soft Pupper approximation of X, respectively. In general, we refer to aprP(X) and apr P ðXÞ as soft rough approximations of X with respect to P. Moreover, the sets
PosP ðXÞ ¼ aprP ðXÞ;
ð3Þ
Neg P ðXÞ ¼ aprP ðXÞ;
ð4Þ
BndP ðXÞ ¼ aprP ðXÞ aprP ðXÞ
ð5Þ
are called the soft P-positive region, the soft P-negative region and the soft P-boundary region of X, respectively. If aprP ðXÞ ¼ aprP ðXÞ, X is said to be soft P-definable; otherwise X is called a soft P-rough set. Remark 3.2. Although soft rough approximations seem to be similar to covering rough set approximations, it is worth noting that in most cases they are closely related but distinct in essence. More specifically, this can be illustrated by an example as follows. One can find various definitions and results about generalized rough sets induced by coverings, c.f. [3,5,13,14,30,37,38]. Here, we follow the definitions and notations used in [14]. Let U = {u1, u2, u3, u4} be a universe and S = (F, A) a soft set over U, where A = {e1, e2}, F(e1) = {u1, u2} and F(e2) = {u1, u3, u4}. Then P = (U, S) is a soft approximation space. It is clear that C = {F(e1), F(e2)} is a covering of the universe U, whence hU, Ci forms a covering approximation space. For x 2 U, the minimal description of x is defined as
MdðxÞ ¼ fK 2 C : x 2 K ^ ð8S 2 C ^ x 2 S # K ) K ¼ SÞg: By definition, one easily sees that Md(u1) = {F(e1), F(e2)}, Md(u2) = {F(e1)} and Md(u3) = {F(e2)} = Md(u4). Zhu [37] proposed the following lower and upper approximations of X # U in a covering approximation space hU, Ci:
X þ ¼ [fK 2 C : K # Xg;
ð6Þ
X þ ¼ X þ [ fNðxÞ : x 2 X X þ g;
ð7Þ
where N(x) = \{K 2 C: x 2 K} = \Md(x) is called the neighborhood of x 2 U. By direct computation, we have N(u1) = {u1}, N(u2) = F(e1), and N(u3) = F(e2) = N(u4). Zhu [37] proved that X+ = [{N(x): x 2 X}, which can be viewed as an alternative representation of the upper approximation. On the other hand, Xu and Zhang [30] introduced another type of covering-based approximations as follows:
C X ¼ fx 2 U : NðxÞ # Xg;
ð8Þ
C X ¼ fx 2 U : NðxÞ \ X – ;g:
ð9Þ
For X1 = {u1}, by the definition of soft rough approximations, we have aprP(X1) = ; and apr P ðX 1 Þ ¼ U. The covering-based approximations of X1 are (X1)+ = ; and (X1)+ = {u1}, according to the definition in [37]. In contrast to these approximations, the covering lower and upper approximations of X1, according to the definition in [30], are C*X1 = {u1} and C*X1 = U, respectively. Consequently, these three types of approximations are different. By Definition 3.1, we immediately have that X # U is a soft P-definable set if the soft P-boundary region BndP(X) of X is empty. Also, it is clear that aprP(X) # X and apr P ðXÞ # apr P ðXÞ for all X # U. Nevertheless, it is worth noticing that X # aprP ðXÞ does not hold in general. Example 3.3. Let U = {u1, u2, u3, u4, u5, u6}, E = {e1, e2, e3, e4, e5, e6} and A = {e1, e2, e3, e4} # E. Let S = (F, A) be a soft set over U given by Table 1 and the soft approximation space P = (U, S).
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F. Feng et al. / Information Sciences 181 (2011) 1125–1137 Table 1 Tabular representation of the soft set S.
e1 e2 e3 e4
u1
u2
u3
u4
u5
u6
1 0 0 1
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 1
1 0 0 0
For X = {u3, u4, u5} # U, we have aprP(X) = {u3}, and aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g. Thus apr P ðXÞ – apr P ðXÞ and X is a soft Prough set. Note that X ¼ fu3 ; u4 ; u5 g aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g in this case. Moreover, it is easy to see that PosP(X) = {u3}, NegP(X) = {u4, u6} and BndP(X) = {u1, u2, u5}. On the other hand, one can consider X1 = {u3, u4} # U. Since apr P ðX 1 Þ ¼ fu3 g ¼ apr P ðX 1 Þ, by definition, X1 is a soft P-definable set. The following result is easily obtained from the definition of soft rough approximations. Proposition 3.4. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then we have
aprP ðXÞ ¼
[
fFðaÞ : FðaÞ # Xg
a2A
and
aprP ðXÞ ¼
[
fFðaÞ : FðaÞ \ X – ;g
a2A
for all X # U. Suppose that S = (F, A) is a soft set over U and P = (U, S) is the corresponding soft approximation space. One can verify that soft rough approximations satisfy the following properties:
aprP ð;Þ ¼ aprP ð;Þ ¼ ;; [ aprP ðUÞ ¼ aprP ðUÞ ¼ f ðaÞ; a2A
aprP ðX \ YÞ # aprP ðXÞ \ aprP ðYÞ; aprP ðX [ YÞ aprP ðXÞ [ aprP ðYÞ; aprP ðX [ YÞ ¼ aprP ðXÞ [ aprP ðYÞ; aprP ðX \ YÞ # aprP ðXÞ \ aprP ðYÞ; X # Y ) aprP ðXÞ # aprP ðYÞ; X # Y ) aprP ðXÞ # aprP ðYÞ:
Proposition 3.5. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then for any X # U, X is soft Pdefinable if and only if apr P ðXÞ # X.
Proof. Note first that if X is soft P-definable, then apr P ðXÞ ¼ apr P ðXÞ, and so apr P ðXÞ ¼ apr P ðXÞ # X. Conversely, suppose that apr P ðXÞ # X for X # U. To show that X is soft P-definable, we only need to prove that aprP ðXÞ # apr P ðXÞ since the reverse inequality is trivial. Let u 2 apr P ðXÞ. Then u 2 F(a) and F(a) \ X – ; for some a 2 A. It follows that u 2 FðaÞ # apr P ðXÞ # X. Hence u 2 aprP(X), and so aprP ðXÞ # aprP ðXÞ as required. h Remark 3.6. To illustrate the above result, we revisit Example 3.3. For X = {u3, u4, u5} # U, we already know that aprP(X) = {u3}, and aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u5 g. Thus apr P ðXÞ X and X is a soft P-rough set. On the other hand, for X 1 ¼ fu3 ; u4 g # U; apr P ðX 1 Þ ¼ fu3 g ¼ apr P ðX 1 Þ. Hence aprP ðX 1 Þ # X 1 and X1 is a soft P-definable set. Theorem 3.7. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then we have (1) (2) (3) (4)
aprP ðapr P ðXÞÞ ¼ aprP ðXÞ; aprP ðapr P ðXÞÞ aprP ðXÞ; aprP(aprP(X)) = aprP(X); aprP ðapr P ðXÞÞ aprP ðXÞ,
for all X # U.
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Proof. (1) Let Y ¼ aprP ðXÞ and u 2 Y. Then u 2 F(a) and F(a) \ X – ; for some a 2 A. By Proposition 3.4, S Y ¼ apr P ðXÞ ¼ a2A fFðaÞ : FðaÞ \ X – ;g. There exists a 2 A such that u 2 F(a) # Y. Hence u 2 aprP(Y), and so Y # aprP(Y). On the other hand, we know that aprP(Y) # Y holds for any Y # U. Thus it follows that Y = aprP(Y) as required. S (2) Let Y = aprP(X) and u 2 Y. Then u 2 F(a) # X for some a 2 A. Since Y = aprP(X) = a2A{F(a): F(a) # X} by Proposition 3.4, we deduce that u 2 F(a) and F(a) \ Y = F(a) – ;. Hence u 2 apr P ðYÞ, and so Y # apr P ðYÞ. S (3) Let Y = aprP(X) and u 2 Y. Then u 2 F(a) # X for some a 2 A. But Y = aprP(X) = a2A{F(a): F(a) # X}, we deduce that u 2 F(a) # Y for a 2 A. Thus u 2 aprP(Y), and so Y # aprP(Y). Since aprP(Y) # Y for any Y # U, we conclude that Y = aprP(Y). S (4) Let Y ¼ aprP ðXÞ and u 2 Y. Then u 2 F(a) and F(a) \ X – ; for some a 2 A. But Y ¼ apr P ðXÞ ¼ a2A fFðaÞ : FðaÞ \ X – ;g, it follows that u 2 F(a) and F(a) \ Y = F(a) – ;. Hence u 2 aprP ðYÞ, and so Y # aprP ðYÞ. h Note that the inclusion relations in Proposition 3.7 may be strict, as shown in the following example. Example 3.8. Let U = {u1, u2, . . . , u8} and E = {e1, e2, . . . , e6}. Let A = {e1, e2, e3, e4} # E, S1 = (F, A) a soft set over U given by Table 2, and the soft approximation space P = (U, S1). For X = {u4, u7, u8} # U, we have aprP(X) = F(e1) [ F(e4) = X, and aprP ðXÞ ¼ Fðe1 Þ [ Fðe3 Þ [ Fðe4 Þ ¼ fu2 ; u3 ; u4 ; u5 ; u7 ; u8 g. Let Y ¼ apr P ðXÞ. Then we have
aprP ðaprP ðXÞÞ ¼ aprP ðYÞ ¼ Fðe1 Þ [ Fðe3 Þ [ Fðe4 Þ ¼ Y: Also, we have aprP ðapr P ðXÞÞ ¼ aprP ðXÞ ¼ Y%X ¼ apr P ðXÞ, which suggests that the inclusion (2) in Proposition 3.7 may hold strictly. Moreover, it is easy to see that aprP(aprP(X)) = aprP(X), and
aprP ðaprP ðXÞÞ ¼ aprP ðYÞ ¼ Fðe1 Þ [ Fðe2 Þ [ Fðe3 Þ [ Fðe4 Þ ¼ U%Y; which indicates that the inclusion (4) in Proposition 3.7 may be strict. Definition 3.9. Let S = (F, A) be a soft set over U. If
S
a2AF(a)
= U, then S is said to be a full soft set.
Theorem 3.10. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then the following conditions are equivalent: (1) (2) (3) (4) (5)
S is a full soft set. aprP(U) = U. aprP ðUÞ ¼ U. X # apr P ðXÞ for all X # U. aprP ðfugÞ – ; for all u 2 U.
Proof. Note first that
aprP ðUÞ ¼
[
fFðaÞ : FðaÞ # Ug ¼
a2A
[
FðaÞ:
a2A
Hence by definition, S = (F, A) is a full soft set if and only if aprP(U) = U. That is, conditions (1) and (2) are equivalent. Similarly, we can show that (1) and (3) are equivalent conditions. On the other hand, we can prove that conditions (4), (5) and (1) are also equivalent. Now assume that condition (4) holds. We prove that condition (5) is true. Given u 2 U, by condition (4), we have fug # aprP ðfugÞ. Thus apr P ðfugÞ – ; since u 2 apr P ðfugÞ. This shows that the condition (4) implies the condition (5). Next, we prove that condition (5) implies condition (1) as well. So suppose that the condition (5) holds. For any u 2 U, we have that apr P ðfugÞ – ;. Let v be an element in aprP ðfugÞ. Then there exists some a 2 A such that v 2 F(a) and F(a) \ {u} – ;. It follows that u = v 2 F(a), and so we have that
Table 2 Tabular representation of the soft set S1.
e1 e2 e3 e4
u1
u2
u3
u4
u5
u6
u7
u8
0 1 0 0
0 0 1 0
0 0 1 0
0 0 1 1
0 1 1 0
0 1 0 0
1 0 0 1
1 0 0 0
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u 2 [a2A{F(a)}. Hence S = (F, A) is a full soft set. This shows that the condition (5) implies the condition (1). To complete the proof, it remains to show that (1) implies (4). Assume that S = (F, A) is a full soft set and X # U. For any x 2 X, since S is full, there exists some a 2 A such that x 2 F(a). Also, it is clear that X \ F(a) – ; since x 2 X \ F(a). Hence we have that x 2 aprP ðXÞ and so X # aprP ðXÞ as required. h The following result is a direct consequence of Propositions 3.5 and 3.10. Corollary 3.11. Let S = (F, A) be a full soft set over U and P = (U, S) a soft approximation space. Then for any X # U, X is soft Pdefinable if and only if apr P ðXÞ ¼ X. As one might suspect from the analogy with Pawlak’s rough sets, if S = (F, A) is a full soft set over U, we have the following interpretations of the concepts introduced in Definition 3.1. Intuitively, the soft lower approximation (equivalently, the soft positive region) of a set X consists of all elements that surely belong to X, whereas the soft negative region constitutes of all elements that surely do not belong to X; meanwhile, the soft upper approximation of the set X constitutes of all elements that possibly belong to X, and the soft boundary region consists of all elements that cannot be classified uniquely to X or its complement X, by employing available knowledge from the soft approximation space. Let S = (F, A) be a full soft set over U, P = (U, S) a soft approximation space and X # U. We define the following four basic types of soft rough sets:
X X X X
is is is is
roughly soft P-definable if aprP(X) – ; and apr P ðXÞ – U; internally soft P-indefinable if aprP(X) = ; and apr P ðXÞ – U; externally soft P-indefinable if aprP(X) – ; and apr P ðXÞ ¼ U; totally soft P-indefinable if aprP(X) = ; and apr P ðXÞ ¼ U.
The intuitive meaning of this classification is as follows: If X is roughly soft P-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 they belong to X, by using available knowledge from the soft approximation space P. If X is internally soft P-indefinable, this means that we are able to decide about some elements of U that they belong to 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 P-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 X, by employing P. If X is totally soft P-indefinable, we are unable to decide for any element of U whether it belongs to X or X, by employing P. Proposition 3.12. Let S = (F, A) be a full soft set over U and P = (U, S) a soft approximation space. Then for any X # U, we have the following: (1) aprP ðXÞ # apr P ðXÞ; (2) Neg P ðXÞ ¼ apr P ðXÞ # aprP ðXÞ.
Proof (1) If the set aprP(X) is empty, then clearly we have that the inclusion apr P ðXÞ # aprP ðXÞ holds. So suppose that aprP(X) – ;. Let u be any element of aprP(X). Since S = (F, A) is a full soft set, there exists some a0 2 A such that u 2 F(a0). Note also that
aprP ðXÞ ¼ fu 2 U : 8a 2 A; u 2 FðaÞ ) FðaÞ \ ðXÞ – ;g: Thus it follows that F(a0) \ (X) – ; since u 2 F(a0). Hence we have that u 2 aprP ðXÞ as required. (2) It is clear that the inclusion Neg P ðXÞ ¼ aprP ðXÞ # apr P ðXÞ holds when the set aprP ðXÞ is empty. So suppose that aprP ðXÞ – ;. Let u 2 aprP ðXÞ. Since S = (F, A) is a full soft set, there exists some a0 2 A such that u 2 F(a0). But we have that
Neg P ðXÞ ¼ aprP ðXÞ ¼ fu 2 U : 8a 2 A; u 2 FðaÞ ) FðaÞ # ðXÞg: Thus we deduce that F(a0) # (X) since u 2 F(a0). Hence we have that u 2 aprP(X) as required. h
Remark 3.13. As illustration of the above result, we revisit Example 3.8. It is clear that the soft set S1 given by Table 2 is a full soft set. For X = {u4, u7, u8} # U, we already know that aprP(X) = X, and aprP ðXÞ ¼ Y ¼ fu2 ; u3 ; u4 ; u5 ; u7 ; u8 g. Hence aprP(X) = X = {u1, u2, u3, u5, u6} and Neg P ðXÞ ¼ aprP ðXÞ ¼ Y ¼ fu1 ; u6 g. In addition, we have that aprP(X) = F(e2) = {u1, u5, u6} and
aprP ðXÞ ¼ Fðe2 Þ [ Fðe3 Þ ¼ fu1 ; u2 ; u3 ; u4 ; u5 ; u6 g:
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It is easy to see that apr P ðXÞ$aprP ðXÞ and Neg P ðXÞ ¼ aprP ðXÞ$aprP ðXÞ. This shows that the inclusions in Proposition 3.12 may be strict if the soft set is full. On the other hand, it is worth noting that these inclusions may not hold when the soft set is not full. To see this, let us reconsider Example 3.3. Note first that the soft set S given by Table 1 is not a full soft set. For X1 = {u3, u4} # U, we already know that aprP ðX 1 Þ ¼ aprP ðX 1 Þ ¼ fu3 g. Hence aprP ðX 1 Þ ¼ aprP ðX 1 Þ ¼ fu1 ; u2 ; u4 ; u5 ; u6 g. Moreover, we have that
aprP ðX 1 Þ ¼ aprP ðX 1 Þ ¼ Fðe1 Þ [ Fðe4 Þ ¼ fu1 ; u2 ; u5 ; u6 g: Thus apr P ðX 1 Þ apr P ðX 1 Þ and Neg P ðX 1 Þ ¼ apr P ðX 1 Þ aprP ðX 1 Þ. Definition 3.14. Let S = (F, U) be a soft set over U and P = (U, S) a soft approximation space. For any X, Y # U, we define
X P Y () aprP ðXÞ ¼ aprP ðYÞ;
X _ P Y () aprP ðXÞ ¼ aprP ðYÞ; 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. Proposition 3.15. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then for any X, Y, X1, Y1 # U, we have (1) (2) (3) (4) (5)
X _ P Y () X _ P ðX [ YÞ _ P Y; X _ P X 1 ; Y _ P Y 1 ) ðX [ YÞ _ P ðX 1 [ Y 1 Þ; X _ P Y ) X [ ðYÞ _ P U; X # Y; Y _ P ; ) X _ P ;; X # Y; X _ P U ) Y _ P U.
Proof. (1) If X _ P Y, then aprP ðXÞ ¼ aprP ðYÞ. Since apr P ðX [ YÞ ¼ aprP ðXÞ [ apr P ðYÞ, we deduce
aprP ðX [ YÞ ¼ aprP ðXÞ ¼ aprP ðYÞ and so X _ P ðX [ YÞ _ P Y. Conversely, if X _ P ðX [ YÞ _ P Y, then we immediately have that X _ P Y by using the transitivity of the relation _ P . (2) Assume that X _ P X 1 and Y _ P Y 1 . Then by definition, we know that aprP ðXÞ ¼ aprP ðX 1 Þ and apr P ðYÞ ¼ aprP ðY 1 Þ. Since aprP ðX [ YÞ ¼ aprP ðXÞ [ aprP ðYÞ and aprP ðX 1 [ Y 1 Þ ¼ aprP ðX 1 Þ [ apr P ðY 1 Þ, we deduce that apr P ðX [ YÞ ¼ apr P ðX 1 [ Y 1 Þ, whence ðX [ YÞ _ P ðX 1 [ Y 1 Þ. (3) Suppose that X _ P Y. Then by definition, apr P ðXÞ ¼ aprP ðYÞ. Since aprP ðX [ ðYÞÞ ¼ aprP ðXÞ [ apr P ðYÞ and aprP ðUÞ ¼ aprP ðYÞ [ apr P ðYÞ, it follows that apr P ðX [ ðYÞÞ ¼ aprP ðUÞ; hence X [ ðYÞ _ P U as required. (4) Let X # Y and Y _ P ;. Then we deduce
aprP ðXÞ # aprP ðYÞ ¼ aprP ð;Þ ¼ ;: Hence aprP ðXÞ ¼ ; ¼ aprP ð;Þ, and so we have that X _ P ;. (5) Suppose that X # Y and X _ P U. Then we deduce
aprP ðYÞ aprP ðXÞ ¼ aprP ðUÞ: But it is easy to see 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. h
Definition 3.16. Let S = (F, A) be a soft set over U. If for any a1, a2 2 A, there exists a3 2 A such that F(a3) = F(a1) \ F(a2) whenever F(a1) \ F(a2) – ;, then S is said to be an intersection complete soft set. Lemma 3.17. Let S = (F, A) be an intersection complete soft set over U and P = (U, S) a soft approximation space. Then we have
aprP ðX \ YÞ ¼ aprP ðXÞ \ aprP ðYÞ for all X, Y # U.
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Proof. Note first that for any soft set S (need not be intersection complete),
aprP ðX \ YÞ # aprP ðXÞ \ aprP ðYÞ: Therefore, it suffices to show that the reverse inclusion
aprP ðX \ YÞ aprP ðXÞ \ aprP ðYÞ also holds. Now, let u 2 aprP(X) \ aprP(Y). Then there exist a1, a2 2 A such that u 2 F(a1) # X and u 2 F(a2) # Y. Since by assumption, S is an intersection complete soft set, we deduce that there exists a3 2 A such that
u 2 Fða3 Þ ¼ Fða1 Þ \ Fða2 Þ # X \ Y: Hence we conclude that u 2 aprP(X \ Y) as required. h Remark 3.18. Notice that the requirement of an intersection complete soft set is not a necessary condition for the equality in Lemma 3.17. To illustrate this, we consider an example as follows. Let U = {u1, u2, u3, u4} be a universe and S = (F, A) a soft set over U, where A = {e1, e2, e3, e4}, and F is a set-valued mapping such that F(e1) = {u1, u2, u3}, F(e2) = {u2, u3, u4}, F(e3) = {u2} and F(e4) = {u3}. Let P = (U, S) be a soft approximation space. Then, one can verify that
aprP ðX \ YÞ ¼ aprP ðXÞ \ aprP ðYÞ for all X, Y # U. On the other hand, it is easy to see that F(e1) \ F(e2) = {u2, u3}, but there does not exist any ei 2 A such that F(ei) = {u2, u3}, whence S = (F, A) is not an intersection complete soft set. By using the above lemma, we can prove the following result in relation to the lower soft rough equal relation. Proposition 3.19. Let S = (F, A) be an intersection complete soft set over U and P = (U, S) a soft approximation space. Then for any X, Y, X1, Y1 # U, we have (1) (2) (3) (4) (5)
X P Y () X P ðX \ YÞ P Y;
X P X 1 ; Y P Y 1 ) ðX \ YÞ P ðX 1 \ Y 1 Þ;
X P Y ) X \ ðYÞ P ;;
X # Y; Y P ; ) X P ;;
X # Y; X P U ) Y P U.
Theorem 3.20. Let S = (F, A) be a soft set over U and P = (U, S) a soft approximation space. Then we have
aprP ðXÞ ¼
\ fY # U : X P Yg
for all X # U. Moreover, if S is a full soft set, then we have
aprP ðXÞ
[ fY # U : X _ P Yg;
for all X # U. Proof. Let u 2 aprP(X). If X P Y, then by definition aprP(X) = aprP(Y). But aprP(Y) # Y for any Y # U. It follows that
u 2 aprP ðXÞ ¼ aprP ðYÞ # Y: T T Hence u 2 fY # U : X P Yg, and so aprP ðXÞ # fY # U : X P Yg. Next, we show that the reverse inclusion T T
fY # U : X P Yg # apr P ðXÞ also holds. Let u 2 fY # U : X P Yg. Then by Proposition 3.7, we have aprP(aprP(X)) = aprP(X). T
Thus X P apr P ðXÞ, and it follows that u 2 aprP(X). Consequently, we conclude that aprP ðXÞ ¼ fY # U : X P Yg. S
Now assume that S is a full soft set and u 2 fY # U : X _ P Yg. Then there exists some Y0 # U such that u 2 Y0 and X _ P Y 0 . Since S is full, we have Y 0 # aprP ðY 0 Þ by Proposition 3.10. It follows that u 2 Y 0 # apr P ðY 0 Þ ¼ aprP ðXÞ. Hence we conclude that S apr P ðXÞ fY # U : X _ P Yg as required. h Note that for a full soft set S, the inclusion in Theorem 3.20 may be strict, i.e.,
aprP ðXÞ%
[ fY # U : X _ P Yg:
Moreover, if S is not a full soft set, this inclusion may not hold. For illustration, we consider the following example. Example 3.21. Let U = {u1, u2, . . ., u8} and E = {e1, e2, . . ., e6}. Now, we reconsider the full soft set S1 = (F, A) given in Example 3.8, and take the soft approximation space P = (U, S1). For X = {u4, u7, u8} # U, we already know that aprP(X) = X, and apr P ðXÞ ¼ Y ¼ fu2 ; u3 ; u4 ; u5 ; u7 ; u8 g. Then it is easy to see that
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\ fY # U : X P Yg ¼ X ¼ aprP ðXÞ
and
[ fY # U : X _ P Yg ¼ fu2 ; u3 ; u4 ; u7 ; u8 g$Y: Next, let B = {e1, e3, e5} and consider the soft set S2 = (G, B) over U given by Table 3. Note that S2 is not a full soft set. Let Q = (U, S2) be a new soft approximation space. For the same set X = {u4, u7, u8} # U, we have aprQ(X) = ;, and
aprQ ðXÞ ¼ Gðe1 Þ [ Gðe3 Þ ¼ fu1 ; u2 ; u4 ; u8 g ¼ W: Then it is clear that
\ fY # U : X Q Yg ¼ ; ¼ aprQ ðXÞ
and
[ fY # U : X _ Q Yg ¼ fu1 ; u2 ; u4 ; u5 ; u6 ; u7 ; u8 g W:
4. Relationships between soft rough sets and Pawlak’s rough sets In this section, we shall explore the relationships between soft rough sets and Pawlak’s rough sets. We start with the following basic notion. Definition 4.1. A soft set S = (F, A) over U is call a partition soft set if {F(a): a 2 A} forms a partition of U. By definition, we immediately have that every partition soft set is a full soft set. The following result indicates that soft sets and binary relations are closely related. Theorem 4.2. Let S = (F, A) be a soft set over U. Then S induces a binary relation qS # A U, which is defined by
ðx; yÞ 2 qS () y 2 FðxÞ for all x 2 A and y 2 U. Conversely, let q be a binary relation from A to U. Define a set-valued mapping F q : A ! PðUÞ by
F q ðxÞ ¼ fy 2 U : ðx; yÞ 2 qg for all x 2 A. Then Sq = (Fq, A) is a soft set over U. Moreover, we have that SqS ¼ S and qSq ¼ q. Proof. It is easy to see that the first part of the statement holds. Thus we only need to show that SqS ¼ S and qSq ¼ q. Suppose that S = (F, A) is a soft set over U and x 2 A. Then for any y 2 U, by definition we have that
y 2 F qS ðxÞ () ðx; yÞ 2 qS () y 2 FðxÞ: That is, F qS ðxÞ ¼ FðxÞ for all x 2 A. Thus F qS ¼ F, whence SqS ¼ S. Next, assume that q # A U, x 2 A and y 2 U. Then by definition, it is clear that
ðx; yÞ 2 qSq () y 2 F q ðxÞ () ðx; yÞ 2 q: Therefore we conclude that qSq ¼ q as required. h In what follows, qS is called the canonical relation of the soft set S, and Sq is referred to as the canonical soft set of the binary relation q. The following example shows that every quotient set can be viewed as a partition soft set. Example 4.3. Let q be an equivalence relation on U. Then the set-valued mapping F q : U ! PðUÞ in Theorem 4.2 coincides with the nature mapping of the equivalence relation q. That is, Fq(x) = [x]q for all x 2 U. Thus the canonical soft set of the equivalence relation q can be identified with the quotient set U/q. Moreover, it is easy to see that Sq is a partition soft set since {Fq(x): x 2 U} = U/q is a partition of U.
Table 3 Tabular representation of the soft set S2.
e1 e3 e5
u1
u2
u3
u4
u5
u6
u7
u8
1 0 0
0 1 0
0 0 1
1 1 0
0 0 0
0 0 0
0 0 0
0 1 0
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Theorem 4.4. Let R be an equivalence relation on U, SR = (FR, U) the canonical soft set of R and P = (U, SR) a soft approximation space. Then for all X # U,
R X ¼ aprP ðXÞ and R X ¼ aprP ðXÞ: Thus in this case, X # U is a (Pawlak) rough set if and only if X is a soft P-rough set. Proof. Let X # U and u 2 U. First, we show that R*X = aprP(X). If u 2 R*X, then [u]R # X. Thus there exists u 2 U such that u 2 [u]R = FR(u) # X. Hence u 2 aprP(X), and so R*X # aprP(X). Conversely, assume that u 2 aprP(X). By definition, there exists v 2 U such that u 2 FR(v) = [v]R # X. It follows that [u]R = [v]R # X, whence u 2 R*X. Therefore, we conclude that R*X = aprP(X). Now it remains to show that R X ¼ apr P ðXÞ. Suppose that u 2 R*X. Then we have that [u]R \ X – ;. Hence there exists u 2 U such that u 2 [u]R = FR(u) and FR(u) \ X – ;. Thus u 2 aprP ðXÞ, and so R X # apr P ðXÞ. Conversely, assume that u 2 aprP ðXÞ. Then there exists v 2 U such that u 2 FR(v) = [v]R and FR(v) \ X – ;. It follows that [u]R = [v]R and [u]R \ X – ;. Hence u 2 R*X, and so we conclude that R X ¼ apr P ðXÞ. h Theorem 4.5. Let S = (F, A) be a partition soft set over U and P = (U, S) a soft approximation space. Define an equivalence relation R on U by
ðx; yÞ 2 R () 9a 2 A;
fx; yg # FðaÞ
for all x, y 2 U. Then, for all X # U,
R X ¼ aprP ðXÞ and R X ¼ aprP ðXÞ: Proof. Let X # U and u 2 U. Note first that there exists a unique element au 2 A such that u 2 F(au) since S = (F, A) is a partition soft set over U. We claim that F(au) = [u]R, where R is the equivalence relation on U given as above. In fact, this is true since for all x 2 U
x 2 ½uR () ðx; uÞ 2 R () 9a 2 A;
fx; ug # FðaÞ () x 2 Fðau Þ:
Next, we show that R*X = aprP(X). If u 2 R*X, then [u]R # X. Thus there exists au 2 A such that u 2 F(au) = [u]R # X. Hence u 2 aprP(X), and so R*X # aprP(X). Conversely, assume that u 2 aprP(X). Then there exists a 2 A such that u 2 F(a) # X. But by uniqueness of the element au 2 A, we have that a = au. It follows that [u]R = F(au) = F(a) # X, and so u 2 R*X. Hence we conclude that R*X = aprP(X). To complete the proof, we only need to show that R X ¼ apr P ðXÞ. Assume that u 2 R*X. Then we have that [u]R \ X – ;. Hence there exists au 2 A such that u 2 F(au) = [u]R and F(au) \ X – ;. Thus u 2 apr P ðXÞ, and so R X # apr P ðXÞ. Conversely, suppose that u 2 apr P ðXÞ. Then there exists a 2 A such that u 2 F(a) and F(a) \ X – ;. By uniqueness of the element au 2 A, we have that a = au. Thus it follows that [u]R = F(au) = F(a) and [u]R \ X – ;. Hence u 2 R*X, and so we conclude that R X ¼ apr P ðXÞ. h The above results reveal that Pawlak’s rough set model can be viewed as a special instance of soft rough sets. Since one can consider F() as a basic granule of the universe, it follows that a soft set can be used to granulate the universe. The granulation structure induced by a soft set is referred to as a soft approximation space. Moreover, it is worth noting that in contrast to equivalence classes, there is no restriction on the subsets F(), whence the granulation structure induced by a soft set may not form a partition or even a covering of the universe. Soft rough sets are also associated with generalized rough sets based on covering or binary relation. Next, we consider the relationship between soft rough sets and generalized rough sets based on binary relation [10,15,27,28,32,33,36]. Let R # U U be a binary relation on U without any additional constraints. For x, y 2 U, if (x, y) 2 R, y is said to be R-related to x, and denoted by xRy. In this case, we say that x is a predecessor of y, and y is a successor of x. Note that a binary relation R can be equivalently defined by a mapping from U to the power set PðUÞ given by Rs(x) = {y 2 U: xRy} for all x 2 U. By Theorem 4.2, (Rs, U) is the canonical soft set of the binary relation R on U, and so we can discuss soft rough sets in the corresponding soft approximation space. According to [32], the set Rs(x) of R-related elements of x may be interpreted as a successor neighborhood of x, and Rs is called the successor operator. In general, by substituting [x]R with Rs(x), one may approximate a subset X # U by a pair of subsets of U with respect to the binary relation R. These facts indicate that rough sets constructed from arbitrary binary relations by using neighborhood operators can also be related to soft rough sets. Finally, we present a comparative analysis between rough sets and soft rough sets, by giving an interesting example. Example 4.6. Let us consider the following soft set S = (F, E) which describes ‘‘life expectancy’’. Suppose that the universe U = {u1, u2, u3, u4, u5, u6} consists of six persons and E = {e1, e2, e3, e4} is a set of decision parameters. The ei (i = 1, 2, 3, 4) stands for ‘‘under stress’’, ‘‘young’’, ‘‘drug addict’’ and ‘‘healthy’’. Set F(e1) = {u5}, F(e2) = {u1, u2}, F(e3) = ; and F(e4) = {u1, u2, u3, u6}. The soft set (F, E) can be viewed as the following collection of approximations:
ðF; EÞ ¼ fðunder stress; fu5 gÞ; ðyoung;fu1 ; u2 gÞ; ðdrug addict; ;Þ; ðhealthy;fu1 ; u2 ; u3 ; u6 gÞg:
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Table 4 An information table.
Sex Age category Living area Habits
u1
u2
u3
u4
u5
u6
Woman Young City NSND
Woman Young City NSND
Man Mature age City Smoke
Man Old Village SD
Man Mature age City Smoke
Man Baby Village NSND
On the other hand, ‘‘life expectancy’’ topic can also be described using rough sets as follows: The evaluation will be done in terms of attributes: ‘‘sex’’, ‘‘age category’’, ‘‘living area’’, ‘‘habits’’, characterized by the value sets ‘‘{man, woman}’’, ‘‘{baby, young, mature age, old}’’, ‘‘{village, city}’’, ‘‘{smoke, drinking, smoke and drinking, no smoke and no drinking}’’. We denote ‘‘smoke and drinking’’ by SD and ‘‘no smoke and no drinking’’ by NSND. The information will be given by Table 4, where the rows are labeled by attributes and the table entries are the attribute values for each person. From here we obtain the following equivalence classes, induced by the above mentioned attributes:
½u1 R ¼ ½u2 R ¼ fu1 ; u2 g;
½u3 R ¼ ½u5 R ¼ fu3 ; u5 g;
½u4 R ¼ fu4 g;
½u6 R ¼ fu6 g:
Let X be a target subset of U, that we wish to represent using the above equivalence classes. Hence we analyze the upper and lower approximations of X, in some particular cases: 1. Set X = {u1, u2, u3, u6}. It follows that
R ðXÞ ¼ fu1 ; u2 ; u6 g;
R ðXÞ ¼ fu1 ; u2 ; u3 ; u5 ; u6 g:
Let us calculate now the soft P-lower and P-upper approximations of X, where P = (U, S). By Proposition 3.4. we obtain
aprP ðXÞ ¼
[ fFðeÞ : FðeÞ # Xg ¼ fu1 ; u2 ; u3 ; u6 g ¼ X;
aprP ðXÞ ¼
[ fFðeÞ : FðeÞ \ X – ;g ¼ fu1 ; u2 ; u3 ; u6 g ¼ X;
e2E
e2E
hence X is soft P-definable. 2. Set X = {u5}. It follows that R*(X) = ;, R*(X) = {u3, u5}. On the other hand, apr P ðXÞ ¼ apr P ðXÞ ¼ X, hence X is soft P-definable. 3. Set X = {u4, u5}. It follows that R*(X) = {u4}, R*(X) = {u3, u4, u5}. On the other hand, apr P ðXÞ ¼ apr P ðXÞ ¼ fu5 g. In this case, X is not a subset of apr P ðXÞ. 4. Set X = {u2, u3}. Then R*(X) = aprP(X) = ;, while R*(X) = {u1, u2, u3, u5} and aprP ðXÞ ¼ fu1 ; u2 ; u3 ; u6 g. The above results show that soft rough set approximation is a worth considering alternative to the rough set approximation. Soft rough sets could provide a better approximation than rough sets do, depending on the structure of the equivalence classes and of the subsets F(e), where e 2 E. 5. Conclusion In this study, we have proposed the new concept of soft rough sets, which can be viewed as a soft set based generalization of the standard rough set model. We presented important properties of soft rough approximations based on soft approximation spaces, giving interesting examples. Several new types of soft sets were proposed, and soft rough equal relations were discussed as well. Furthermore, we examined the relationship between soft rough sets and Pawlak’s rough sets, and compared these two different models. As future work, connections between soft rough sets and various types of generalized rough set models could be explored. Acknowledgements The authors are highly grateful to the anonymous referees and Professor Witold Pedrycz, Editor-in-Chief, for their valuable comments and suggestions which greatly improve the quality of this paper. We are also indebted to Professor Yongming Li for helpful discussion and valuable suggestions. This work was supported by the National Natural Science Foundation of China (Nos. 10926031 and 10571112), the Natural Science Foundation of Education Department of Shaanxi Province of China (No. 2010JK831), and by the Shaanxi Provincial Research and Development Plan of Science and Technology (No. 2008K01-33). References [1] H. Aktasß, N. Çag˘man, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726–2735. [2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96.
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