Fuzzy Rough Sets Based on DominanceRelationsXiaoyan ZhangDepartment of Mathematics and Information ScienceGuangdong Ocean UniversityZhanjiang, P. R. China 524088datongzhangxiaoyan@AbstractThis model for fuzzy rough sets is one of the most important parts in rough set theory.Moreover, it is based on an equivalence relation (indiscernibility relation). However,many systems are not only concerned with fuzzy sets, but also based on a dominancerelation because of various factors in practice. To acquire knowledge from the systems,construction of model for fuzzy rough sets based on dominance relations is verynecessary. The main aim to this paper is to study this issue. Concepts of the lower andthe upper approximations of fuzzy rough sets based on dominance relations areproposed. Furthermore, model for fuzzy rough sets based on dominance relations isconstructed, and some properties are discussed.Keywords: Rough sets; Dominance relations; fuzzy sets.1IntroductionThe rough set theory [10,11], proposed by Pawlak in the early 1980s, is an extension of set theory for the study of intelligent systems. It can serve as a new mathematical tool to soft computing, and deal with inexact, uncertain or vague information. Moreover, this theory has been applied successfully in discovering hidden patterns in data, recognizing partial or total dependencies in systems, removing redundant knowledge, and many others [7,12,13,15,16]. Since its introduction, the theory has received wide attention on the research areas in both of the real-life applications and the theory itself.Theory of fuzzy sets initiated by Zedeh [9] also provides useful ways of describing and modeling vagueness in ill-defined environment. Naturally, Doubois and Prade [8] combined fuzzy sets and rough sets. Attempts to combine these two theories lead to some new notions [1,5,7], and some progresses were made [2,3,4,5,6,14]. The combination involves many types of approximations and the construction of fuzzy rough sets give a good model for solving this problem [5]. However, most of systems are not only concerned with fuzzy data, but also based on a dominance relation because of various factors. In order to obtain the succinct knowledge from the systems, construction of model for fuzzy rough sets based on dominance relations is needed.The main aim of the paper is to discuss the issue. In present paper, a dominance relation is introduced and instead of the equivalence relation (discernibility relation) in the standard fuzzy rough set theory. The lower and the upper approximation of a fuzzy rough set based on dominance relations are proposed. Thus a model for fuzzy rough sets based on dominance relations is constructed, and some properties are studied. Finally, we conclude the paper and look ahead the further research.2 Preliminaries This section recalls necessary some concepts used in the paper. Detailed description can be found in [15]. Definition 2.1 Let U be a set called universe and let R be an equivalence relation (indiscernibility) on U .The pair (,)S U R =is called a Pawlak approximation space. Then for any non-empty subset X of U , the sets){:[]}R R X x U x X =∈⊆and(){:[]}R R X x U x X =∈∩≠∅are respectively, called the lower and the upper approximations of X in S , where []R x denotes theequivalence class of the relation R containing the elementx .X is said to be definable set, if())R X R X =. Otherwise X is said to be rough set. Pawlak approximation space (,)S U R = derivesmainly from an equivalence relation. However, there exist a great of systems based on dominance relations in practice.Definition 2.2 If we denote{(,):()(),}B i j l i l j l R x x U U f x f x a B ≤=∈×≤∀∈where B is subset of attributes set, and ()l f x is the value of attribute l a , then B R ≤ is referred to asdominance relation of information system S . Moreover, we denote approximation space based ondominance relations by (,)S U R ≤≤=.For any non-empty subset X of U , denote(){:[]}R R X x U x X ≤≤=∈⊆and){:[]}R R X x U x X ≤≤=∈∩≠∅R ≤and R ≤are r espectively said to be the lower and the upper approximations of X with respect to a dominance relation R ≤.If we denote[]{:(,)}{:()(),}i j i j B B j l i l j l x x U x x R x U f x f x a B ≤≤=∈∈=∈≤∀∈then the following properties of a dominance relation are trivial . Proposition Let B R ≤ be a dominance relation. (1)B R ≤ is reflexive and transitive, but not symmetric, so it isn't an equivalence relation generally.(2) If 12B B A ⊆⊆, then 21A B B R R R ≤≤≤⊆⊆.(3) If 12B B A ⊆⊆, then 21[][][]i A i B i B x x x ≤≤≤⊆⊆.(4) If[]j i B x x ≤∈, then [][]j B i B x x ≤≤⊆.Next, we will review some notions of fuzzy sets. The notion of fuzzy sets provides a convenient tool forrepresenting vague concepts by allowing partial membership. In fuzzy systems, a fuzzy set can be defined using standard set operators.Definition2.3 Let U be a finite and non-empty set called universe. A fuzzy set A of U is defined by a membership function:[0,1]A U →.The membership value may be interpreted in term of the membership degree. In generally, let ()F U denote the set of all fuzzy sets, i.e., the set of all functions from U to [0,1].From above description, we can find that a crisp set can be regarded as a generated fuzzy set in which the membership is restricted to the extreme points {0,1} of [0,1].Definition 2.4 Let ,()A B F U ∈. For any x U ∈, if ()()A x B X ≤ is true, then we say that B containA or A is contained byB , which denoted by A B ⊆.If both A B ⊆and B A ⊆ are all true, then we say A is equal to B , denoted by A B =. Empty set ∅denotes the fuzzy set whose mem bership function is 0, and set U denotes the fuzzy set whose membership function is 1.Let denote intersection, union of A and B by A B ∩, A B ∪ respectively. Moreover, denotecomplement ofA by A : or C A . Membership functions of these fuzzy sets are defined asA B ∩=()()A x B x ∧ =min{(),()}A x B x ; A B ∪=()()A x B x ∨ =max{(),()}A x B x ;1()C A A A x ==−:.There many properties of these operators in fuzzy sets which similar with crisp sets. Detailed description can be found easily.Definition 2.5 The λ-level set orλ-cut, denoted by A λ, A λof a fuzzy set A in U comprise all elementsof U whose degree membership in A are all greater than or equal to λ, where 0<λ≤1. In other words,{:()}A x U A x λλ=∈≥is a non-fuzzy set, and be called the λ-level set or λ-cut. Moreover, the set {:()0}x U A x ∈> is defined the supports of fuzzy set A , and denoted by supp A .3 Fuzzy rough sets based on dominance relationsModel for fuzzy rough sets is generalize d by the standard Pawlak approximation space, and it is concerned with fuzzy sets on universe U. But the model is still depended on an equivalence relation. In practice, most of systems are not only related to fuzzy sets, but also based on dominance relations. In order to deal with this problem, the model of fuzzy rough sets based on dominance relations is proposed. Definition 3.1 Let (,)S U R ≤≤= be an approximation space based on dominance relation R ≤. For a fuzzyset A of U , the lower and the upper approximation of A denoted by )R A ≤ and ()R A ≤, are definedrespectively, by two fuzzy sets, whose membership functions ar e)()min{():[]},R R A x A y y x x U ≤≤=∈∈and()()max{():[]},R R A x A y y x x U ≤≤=∈∈where []R x ≤ denotes the dominance class of the relation R ≤.Fuzzy setA is called fuzzy definable set, if R R ≤≤=. Otherwise, A is called fuzzy rough set. R ≤ iscalled positive field ofA in (,)S U R ≤≤=, and R ≤: is called negative field of A in (,)S U R ≤≤=. Inaddition, ()(())R A R A ≤≤∩: is called boundary of A in (,)S U R ≤≤=.It can be easily verified that ()R A ≤ and ()R A ≤ will become the lower and the upper approximation of standard approximation space based on dominance relation, when A is a crisp set. Theorem 3.1 Let(),)A F U R A ≤∈and ()R A ≤ be the lower and the upper approximation ofA respectively. The following always hold. (1) )()R A A R A ≤≤⊆⊆.(2) ()()();R A B R A R B ≤≤≤∪=∪())().R A B R A R B ≤≤≤∩=∩(3) ())();R A R B R A B ≤≤≤∪⊆∪()()()R A B R A R B ≤≤≤⊆I I .(4)()();()();R A R A R A R A ≤≤≤≤==::::(5) ();().R U U R ≤≤=∅=∅(6)()(());R A R R A ≤≤≤⊆ (())()R R A R A ≤≤≤⊆.(7) IfA B ⊆, then )()R A R B ≤≤⊆ and ()().R A R B ≤≤⊆Proof. We need only to prove one hand, the other hand is similar completely. (1) It is clear.(2) Since for x U ∀∈,()()R A B x ≤∪sup{()():[]}sup{()():[]}sup{():[]} sup{():[]})())() ()()R R R R A B y y x A y B y y x A y y x B y y x R A x R B x R A R B ≤≤≤≤≤≤≤≤=∪∈=∨∈=∈∨∈=∨=∪thus()()()R A B R A R B ≤≤≤∪=∪ (3) Since for x U ∀∈,(()())()R A R B x ≤≤∪)())()inf{():[]}inf{():[]}inf{()():[]}inf{()():[]})(),R R R R R A x R B x A y y x B y y x A y B y y x A B y y x R A B x ≤≤≤≤≤≤≤=∨=∈∨∈≤∨∈=∪∈=∪thus()()().R A R B R A B ≤≤≤∪⊆∪(4) Since for x U ∀∈,(~)()R A x ≤inf{(~)():[]}inf{1():[]}1sup{():[]}1)()~()(),R R R A y y x A y y x A y y x R A x R A x ≤≤≤≤≤=∈=−∈=−∈=−= thus )~().R A R A ≤≤=(5) Because of for any ,()1x U U x ∀∈=, we have()()inf{():[]}1R R U x U y y x ≤≤=∈=.Hence ()R U U ≤=.(6) According to Proposition 1, if[]R y x ≤∈ ,then. [][]R R y x ≤≤⊆ Thus())()R R A x ≤≤min{()():[]}min{min{():[]}:[]}min{min{():[]}:[]}min{():[]}()()R R R R R R R A y y x A z z y y x A z z x y x A z z x R A x ≤≤≤≤≤≤≤≤=∈=∈∈≥∈∈=∈= Hence ()(())R A R R A ≤≤≤⊆.(7) It is obvious from definitions. Definition 3.2 Let(,)S U R ≤≤=be an approximation space based dominance relation R ≤, and,()AB FU ∈.If ))R A R B ≤≤=, then we say that fuzzy set A is equal to fuzzy set B with respect to the lowerapproximation, which denoted by A B :. If()()R A R B ≤≤=, then we say that fuzzy set A is equal to fuzzy set B with respect to the upper approximation, which denoted by A B ;. If)()R A R B ≤≤=both ()()R A R B ≤≤= and, then we say that fuzzy set A is fuzzy rough equal to setB , which denoted by A B ≈.Theorem 3.2 Let(,)S U R ≤≤=be an approximation space based on dominance relation R ≤, and,()A B F U ∈. The following always hold.(1) A B : if and only if A B A ∩: and A B B ∩:. (2) A B ; if and only if A B A ∪;and A B B ∪;. (3) If 'A A ; and 'B B ;, then ''A B A B ∪∪;. (4) If 'A A : and 'B B :, then ''A B A B ∩∩:. (5) If A B ⊆and B ∅; , then A ∅;.(6) IfA B ⊆and A U : , then B U :.(7) If A ∅:or B ∅:, then A B ∩∅:.(8) If A U ;or B U ;, then A B U ∪;. (9) A U :if only if A U =.(10) A ∅; if and only if A =∅.Proof . They can be obtained at once from above definition.Definition 3.3 Let(,)S U R ≤≤= be an approximation space based on dominance relation R ≤. For()A F U ∈, the lower and the upper approximation with respect to parameters ,(01)αβαβ<≤≤,denoted by()R A α≤ and ()R A β≤, are defined by(){:()()}R A x U R A x αα≤≤=∈≥;(){:()()}R A x U R A x ββ≤≤=∈≥From above definition, we can know that ()R A α≤ is the crisp set of some elements of U whose degree of membership in A certainly are not less than α, ()R A β≤ is the crisp set of some elements of U whose degree of membership in A possibly are not less than β. Remark For ,[0,1]αβ∀∈,above definition ()R A α≤ and )R A β≤will become )R A ≤ and ()R A ≤respectively, when A is crisp set.In fact, ,[0,1]αβ∀∈, since A is a crisp set, thus ()[0,1]A x ∈. So we have)R A α≤{:()()}{:()()1}{:()()1,[]}{:[]}()R R x U R A x x U R A x x U R A y y x x U x A R A α≤≤≤≤≤≤=∈≥=∈==∈=∀∈=∈⊆= Similarly, we can show that )()R A R A β≤≤=, when A is crisp set.Theorem 3.3 Let(,)S U R ≤≤= be an approximation space based on dominance relation R ≤, and,()A B F U ∈. For ,(01)αβαβ<≤≤, we have(1) ))),R A B R A R B βββ≤≤≤∪=∪ ()()().R A B R A R B ααα≤≤≤∩=∩ (2)))),R A R B R A B ααα≤≤≤∪⊆∪ ()()().R A B R A R B βββ≤≤≤∩⊆∩(3) If A B ⊆, then ()(),R A R B ββ≤≤⊆ and ()()R A R B αα≤≤⊆.(4) )()R A R B αβ≤≤⊆.(5)1)~(),R A R A ββ≤≤−=1(~),~).R A R A αα≤≤−=Proof . It can be achieved obviously by definitions and Theorem 3.1.Definition 3.4 Let (,)S U R ≤≤=be an approximation space based dominance relation R ≤, and()A F U ∈. Roughness measure of A in S ≤, denoted by ()R A ρ≤, is be defined as|()|()1|()|R R A A R A ρ≤≤≤=−when |()|0R A ≤=,()0R A ρ≤= is ordered.Clearly, there is 0()1R A ρ≤≤≤. Moreover,()0R A ρ≤=, if A is definable fuzzy set.Definition 3.5 Let (,)S U R ≤≤= be an approximation space based dominance relation R ≤, and ()A F U ∈.Roughness measure with respect to parameters ,(01)αβαβ<≤≤ of A in S ≤ ,denoted by ,()R A αβρ≤, is be defined as,|()|()1|()|R R A A R A αβαβρ≤≤≤=−when |()|0R A β≤=, ,()0R A αβρ≤= is ordered.From the definition, we have easily following properties. Theorem 3.4 (1) ,0()1R A αβρ≤≤≤.(2) If β is fixed, then we have that |()|R A α≤ increases. Thus, ,()R A αβρ≤ is to increase, when αincreases.(3) If α is fixed, then we have that |()|R A β≤ is to decrease, when βincreases. Thus, ,()R A αβρ≤ is toincrease, when β increases.Theorem 3.5 If membership function of fuzzy set A is a constant, i.e., there exists 0δ> such that()A x δ= for all x U ∈, then we have ,()1R A αβρ≤= when ,(01)αβαβ<≤≤. Otherwise,,()0R A αβρ≤=.Proof. When ,(01)αβαβ<≤≤, it is clear that |()|R A α≤=∅ , |()|1R A β≤=. So ,()1R A αβρ≤=.Otherwise, there exist two cases.Case 1. When δβα<≤, we can know |()|R A α≤=|()|R A β≤=∅. So ,()0R A αβρ≤= is true.Case 2. When βαδ≤≤, we can know |()|R A α≤=|()|R A U β≤=. So from the definition,,()0R A αβρ≤= is true.Theorem 3.6 Let ,()A B F U ∈, and A B ⊆, ,(01)αβαβ<≤≤. Followings are true.(1) If ())RA RB ββ≤≤=, then ,,()()R R B A αβαβρρ≤≤≤.(2) If()()R A R B αα≤≤=, then ,,()()R R A B αβαβρρ≤≤≤.(3) If there exis ts 0r >, such that ()A x r ≥ is true for all x U ∈, then ,,()()R R B A αβαβρρ≤≤≤ holdswhen r β≥.(4) If there exists 0r >, such that ()A x r ≥ is true for all x U ∈, then ,,()()0R R A B αβαβρρ≤≤== holdswhen r α≤. Proof. Since A B ⊆, we have ()()R A R B ββ≤≤⊆ and ))R A R B αα≤≤⊆. Hence, (1) and (2) can beobtained. (3) Fromr β≥, we have ())R A R B U ββ≤≤== . So ()()R A R B ββ≤≤⊆ can be get by (1).(4) From r α≤, we have ()()R A R B U αα≤≤== and ()()R A R B U ββ≤≤==. Thus ,,()()0R R A B αβαβρρ≤≤==. ?Theorem 3.7 Let ,()A B F U ∈. If A B ≈, then ,,()()R R A B αβαβρρ≤≤= for ,(01)αβαβ<≤≤.Proof. It can be obtained at once by the definition.4 ConclusionsIt is well know that there exist most of systems, which are not only concerned with fuzzy sets but alsobased on dominance relations in practice. Therefore, it is meaningful to study the fuzzy rough set based on dominance relations. In this paper, we discussed this problem mainly. We introduced concepts of the lower and the upper approximations of fuzzy rough sets based on dominance relations, and constructed the model for fuzzy rough sets based on dominance relation additionally. Furthermore some properties are obtained. In next work, we will consider the information systems which are based on dominance relations and with fuzzy decisions.R e f e r e n c e s[1] Cornelis. C., Cock, M.D. and Kerre. E.E., Intuitionistic Fuzzy Rough Sets: At the Crossroads ofImperfect Knowledge, Expert Systems, Vol.20, No.5, pp.260-270, Nov. (2003).[2] Banerjee. M.,Miltra. S., Pal. 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Science Press,Beijing, (2003).基于优势关系下的模糊粗糙集模型　张晓燕　广东海洋大学理学院数学与信息科学系　广东　湛江　５２４０８８　Ｅｍａｉｌ：　ｄａｔｏｎｇｚｈａｎｇｘｉａｏｙａｎ＠１２６．ｃｏｍ　摘　要：　本文给出了基于优势关系下模糊粗糙集上、下近似的定义，　并进一步对其性质做了详细的研究，　从而建立了基于优势关系下的模糊粗糙集模型．　这对解决现实生活中存在的基于优势关系的信息系统提供了有力的理论根据．　关键词：　粗糙集；　模糊集；　优势关系；　信息系统．　。