# On Characterization of Rough Type-2 Fuzzy Sets

Authors: Tao Zhao; Zhenbo Wei
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the generalized Pawlak approximation space are also introduced. Some basic properties of the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators are discussed. The connections between special crisp binary relations and generalized rough type-2 fuzzy approximation operators are further examined. The axiomatic characterization of generalized rough type-2 fuzzy approximation operators is also presented. Finally, the attribute reduction of type-2 fuzzy information systems is investigated.


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## Body

## 1. Introduction

Rough sets theory [1, 2] proposed by Pawlak is a mathematical tool to deal with uncertainties and imprecision. Currently, rough set theory has been found to have very successful applications in many fields such as expert systems, machine learning, pattern recognition, decision analysis, and knowledge discovery. In Pawlak rough set model, the equivalence relation plays an important role and is a very restrictive requirement, which may limit its applications in real applications. Hence, the generalization ofrough set model is an important branch of rough set theory study. Afterward, several authors proposed generalized rough set model, variable precision rough set model, covering rough set model, rough fuzzy set model and fuzzy rough set model, and so forth.Fuzzy set theory (type-1 fuzzy sets) was proposed by Zadeh in 1965 [3]. The type-1 fuzzy set theory has been widely applied in various fields [4–6]. Since rough set theory is highly complementary with fuzzy set theory, it is desirable to study the combination of fuzzy sets and rough sets. Dubois and Prade proposed rough fuzzy set model and fuzzy rough set model in [7]. Afterwards, many interesting achievements are obtained on the basis of [7]. Currently, the knowledge representation of fuzzy rough sets and the attribute reduction using fuzzy rough sets are two main branches on the researches of fuzzy rough sets. On the knowledge representation of fuzzy rough sets, researchers mainly focus on the construction of fuzzy rough approximation operators and the axiomatic characterization of fuzzy rough sets. At present, there exist many valuable results on the knowledge representation of fuzzy rough sets [8–12]. The definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication were introduced in [8]. The lower and upper approximations of rough sets and fuzzy rough sets were, respectively, characterized by using outer and inner products in [9]. A general framework for the study of fuzzy rough sets was presented in [10]. The lattice and topological structures of fuzzy rough sets were proposed in [11]. The axiomatic approach of fuzzy rough sets was shown in [12]. On the attribute reduction using fuzzy rough sets, researchers mainly focus on the development of fast reduction algorithms. Jensen and Shen did a pioneering work on attribute reduction using fuzzy rough sets in [13]. A fuzzy rough QuickReduct algorithm was presented in [13], and this novel approach has been applied to aid classification of web content, with very promising results. However, the QuickReduct algorithm has been proved to have some shortcomings in [14]. Hence, some improved algorithms were proposed in [14, 15].As a generalization of type-1 fuzzy sets, intuitionistic fuzzy sets were proposed by Atanassov [16, 17]. By setting the degree of membership, the degree of nonmembership, and the degree of hesitation, intuitionistic fuzzy sets depict the nature of ambiguity and solve the hesitant information in judging problems. Cornelis et al. [18] proposed intuitionistic fuzzy rough sets by combining intuitionistic fuzzy sets and rough sets. Afterwards, the theory and application studies of intuitionistic fuzzy rough sets have been expanded in [19–22]. The concept of interval-valued intuitionistic fuzzy sets was firstly proposed in [23]. Interval-valued intuitionistic fuzzy sets provide more flexible and effective way to deal with uncertainties because both membership degree and nonmembership degree of interval-valued intuitionistic fuzzy sets are denoted by interval numbers of [0,1]. In [24], an interval-valued intuitionistic fuzzy rough model has been proposed by combining the classical Pawlak rough set theory with the interval-valued intuitionistic fuzzy set theory. Furthermore, in [25], the rough approximations of an interval-valued intuitionistic fuzzy set in the classical Pawlak approximation space and the generalized Pawlak approximation space have been presented, respectively.Zadeh in 1975 proposed type-2 fuzzy sets which can enhance the system’s ability to deal with uncertainties [26]. The applications of type-2 fuzzy sets were limited due to the computational complexity. However, interval type-2 fuzzy sets, as a special case of general type-2 fuzzy sets, have been extensively studied in practical applications. There are some achievements about interval type-2 fuzzy rough sets by combining the characteristics of rough sets and interval type-2 fuzzy sets [27–29]. General type-2 fuzzy sets may be better than the interval type-2 fuzzy sets to deal with uncertainties because general type-2 fuzzy sets can obtain more degree of freedom [30]. In order to simplify the calculation for general type-2 fuzzy sets, the representation of general type-2 fuzzy sets is also an important research issue. Liu in [31] firstly proposed the α-plane representation of the general type-2 fuzzy sets and claimed that the α-plane representation can greatly reduce the computational workload. Currently, α-plane method has been extensively studied [32–35] because it can take advantage of the interval type-2 fuzzy sets theory to study the general type-2 fuzzy sets.Although general type-2 fuzzy sets and rough sets have important applications, a few results about the combination of general type-2 fuzzy set theory and rough set theory were presented [36–38]. In [36], we proposed general type-2 fuzzy rough sets based on general type-2 fuzzy relations. However, there are no contributions on the definitions of approximation operators of general type-2 fuzzy sets in the Pawlak approximation space and the generalized Pawlak approximation space. Moreover, it can be seen that the attribute reduction of information systems is an important application of rough set theory. The traditional rough set model can effectively handle these information systems in which both condition attributes and decision attributes are clear. In some cases, condition attributes may be clear and decision attributes may be interval type-2 fuzzy sets on objects set. That is to say, every decision attribute value is an interval on [0,1]. At this time, the rough approximations of an interval type-2 fuzzy set in the generalized Pawlak approximation space were also introduced in [39]. Therefore, the interval type-2 rough fuzzy set model proposed in [39] may be suitable to deal with these information systems with interval values. However, in highly uncertain situations, we would encounter these information systems where all condition attributes are clear and all decision attributes are general type-2 fuzzy sets on objects set; that is, every decision attribute value is a type-1 fuzzy set on [0,1]. To provide certain theoretical basis for dealing with such problem, we propose rough type-2 fuzzy set model by combining the classical Pawlak rough set theory with the general type-2 fuzzy set theory in this paper. The rest of our work is organized as follows. In Section 2, the basic definitions and terminologies on type-2 fuzzy sets are reviewed briefly. In Section 3, the rough type-2 fuzzy approximation operators based on the clear equivalence relation are firstly defined. The generalized rough type-2 fuzzy approximation operators based on the clear generalized binary relation are also derived in Section 3. In Section 4, the generalized rough type-2 fuzzy approximation operators are characterized by axioms. The attribute reduction method of type-2 fuzzy information systems is presented in Section 5. The last section concludes this paper.


## 2. Preliminaries

In this section, the basic definitions and terminologies on type-2 fuzzy sets with some modified notations are recalled.Definition 1 (see [40]).
A type-2 fuzzy set, denoted byA~, is expressed as(1)A=∫x∈UuAxx=∫x∈U∫u∈Jxfxu/uxJx⊆0,1,where Jx is the primary membership of x; uA(x)=∫u∈Jxfx(u)/u is the secondary membership function; fx(u) is a secondary membership grade.
In the following sections, the class of all type-2 fuzzy sets of the universe of discourseU is denoted as F2(U), and the class of all crisp sets of the universe of discourse U is denoted as P(U). Currently, it is difficult to deal with the type-2 fuzzy sets whose secondary membership functions are not normal and convex. For simplicity, we only study these type-2 fuzzy sets whose secondary membership function is normal and convex in this paper.Definition 2 (see [32]).
Anα-plane for type-2 fuzzy set A, which is denoted by Aα, is defined as follows:(2)Aα=∫x∈U∫u∈Jxx,u∣fxu≥α=∫x∈U∫u∈SLAx∣α,SUAx∣αuxα∈0,1,where [SLAx∣α,SUAx∣α] denote an α-cut of the secondary membership function uA(x).Definition 3 (see [32]).
Theα-plane representation (theorem) for type-2 fuzzy set A is(3)A=⋃α∈0,1αAα.Theorem 4 (see [32]).
Let(A∪B)α and (A∩B)α be α-plane of A∪B and A∩B, respectively; one has(4)A∪B=⋃α∈0,1αAα∪Bα;A∩B=⋃α∈0,1αAα∩Bα,where(5)Aα∪Bα=∫x∈U∫u∈SLAx∣α∨SLBx∣α,SUAx∣α∨SUBx∣αux;Aα∩Bα=∫x∈U∫u∈SLAx∣α∧SLBx∣α,SUAx∣α∧SUBx∣αux.
Obviously,SLA∪Bx∣α=SLAx∣α∨SLBx∣α, SUA∪Bx∣α=SUAx∣α∨SUBx∣α, SLA∩Bx∣α=SLAx∣α∧SLBx∣α, and SUA∩Bx∣α=SUAx∣α∧SUBx∣α hold.Theorem 5 (see [36]).
Let(Ac)α be α-plane of Ac; one has(6)Ac=⋃α∈0,1αAcα=⋃α∈0,1αAαc,where Aα=∫x∈U∫u∈SLAx∣α,SUAx∣αu/x.Definition 6 (see [36]).
LetA,B∈F2(U), and define A⊆B if SLAx∣α≤SLBx∣α and SUAx∣α≤SUBx∣α hold for any α∈[0,1] and x∈U. If A⊆B and B⊆A, then A=B.


## 3. Rough Type-2 Fuzzy Sets

In this section, we introduce rough type-2 fuzzy approximation operators and generalized rough type-2 fuzzy approximation operators induced from the Pawlak approximation space and the generalized Pawlak approximation space, respectively, and discuss their properties.


### 3.1. Rough Type-2 Fuzzy Approximation Operators Based on the Equivalence Relation

Definition 7.
LetU be a nonempty universe of discourse and let R∈P(U×U) be the clear equivalence relation on U. [x]R denotes the equivalence class. Then (U,R) is called the Pawlak approximation space. For any A∈F2(U), define the upper and lower rough type-2 fuzzy approximation operators R¯(A) and R_(A) about (U,R) by (7)R¯A≡⋃α∈0,1αR¯Aα,R_A≡⋃α∈0,1αR_Aα,where R_Aα=∫x∈U∫u∈[SLR_(A)x∣α,SUR_(A)x∣α]u/x,  (R¯(A))α=∫x∈U∫u∈[SLR¯(A)x∣α,SUR¯(A)x∣α]u/x, and (8)SLR_Ax∣α=⋀y∈xRSLAy∣α,SUR_Ax∣α=⋀y∈xRSUAy∣α,SLR¯Ax∣α=⋁y∈xRSLAy∣α,SUR¯Ax∣α=⋁y∈xRSUAy∣α.
Clearly, ifA is degraded to type-1 fuzzy set, then the rough type-2 fuzzy approximation operators defined in Definition 7 reduce to the rough type-1 fuzzy approximation operators. In the following, we define type-2 fuzzy universe set U=∫x∈U∫u∈[1,1]1/u/x and type-2 fuzzy empty set ∅=∫x∈U∫u∈[0,0]1/u/x. Obviously, SLUx∣α=SUUx∣α=1 and SL∅x∣α=SU∅x∣α=0 for any x∈U and α∈[0,1].Theorem 8.
Let(U,R) be a Pawlak approximation space and let R_ and R¯ be the rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following properties hold:(1)
R_(A)⊆A⊆R¯(A);(2)
R_(Ac)=(R¯(A))c,  R¯(Ac)=(R_(A))c;(3)
R_(A∩B)=R_(A)∩R_(B),  R¯(A∪B)=R¯(A)∪R¯(B);(4)
A⊆B⇒R_(A)⊆R_(B),  A⊆B⇒R¯(A)⊆R¯(B);(5)
R_(R_(A))=R¯(R_(A))=R_(A),  R¯(R¯(A))=R_(R¯(A))=R¯(A);(6)
R_(U)=R¯(U)=U,  R_(∅)=R¯(∅)=∅.Proof.
(1) For anyx∈U and α∈[0,1], we have (9)SLR_Ax∣α=⋀y∈xRSLAy∣α≤SLAx∣α,SUR_Ax∣α=⋀y∈xRSUAy∣α≤SUAx∣α.Thus, R_(A)⊆A. Similarly, A⊆R¯(A). That is, R_(A)⊆A⊆R¯(A).
(2) For anyx∈U and α∈[0,1],(10)SLR_Acx∣α=⋀y∈xRSLAcy∣α=⋀y∈xR1-SUAy∣α=1-⋁y∈xRSUAy∣α=1-⋁y∈xRSUAy∣α=1-SUR¯Ax∣α=SLR¯Acx∣α.Furthermore, (11)SUR_Acx∣α=⋀y∈xRSUAcy∣α=⋀y∈xR1-SLAy∣α=1-⋁y∈xRSLAy∣α=1-⋁y∈xRSLAy∣α=1-SLR¯Ax∣α=SUR¯Acx∣α.Thus, R_(Ac)=R¯(A)c. Similarly, we can prove R¯(Ac)=(R_(A))c.
(3) For anyx∈U and α∈[0,1], (12)SLR_A∩Bx∣α=⋀y∈xRSLA∩By∣α=⋀y∈xRSLAy∣α∧SLBy∣α=⋀y∈xRSLAy∣α∧⋀y∈xRSLBy∣α=SLR_Ax∣α∧SLR_Bx∣α=SLR_A∩R_Bx∣α.Similarly, we have SUR_(A∩B)x∣α=SUR_(A)∩R_(B)x∣α. Thus, R_(A∩B)=R_(A)∩R_(B). Similarly, we can prove R¯(A∪B)=R¯(A)∪R¯(B).
(4) For anyx∈U and α∈[0,1], since A⊆B, we have SLAx∣α≤SLBx∣α and SUAx∣α≤SUBx∣α. Thus, SLR_(A)x∣α=⋀y∈[x]RSLAy∣α≤⋀y∈[x]RSLBy∣α=SLR_(B)x∣α. Furthermore, SUR_(A)x∣α=⋀y∈[x]RSUAy∣α≤⋀y∈[x]RSUBy∣α=SUR_(B)x∣α. Hence, R_(A)⊆R_(B). That is, A⊆B⇒R_(A)⊆R_(B). Similarly, we can prove A⊆B⇒R¯(A)⊆R¯(B).
(5) For anyx∈U and α∈[0,1],(13)SLR_R_Ax∣α=⋀y∈xRSLR_Ay∣α=⋀y∈xR⋀z∈yRSLAz∣α=⋀z∈xRSLAz∣α=SLR_Ax∣α.Similarly, we have SUR_(R_(A))x∣α=SUR_(A)x∣α. Thus, R_(R_(A))=R_(A). In addition(14)SLR¯R_Ax∣α=⋁y∈xRSLR_Ay∣α=⋁y∈xR⋀z∈yRSLAz∣α=⋁y∈xR⋀z∈xRSLAz∣α=⋀z∈xRSLAz∣α=SLR_Ax∣α.Similarly, we can obtain SUR¯(R_(A))x∣α=SUR_(A)x∣α. Hence, R¯(R_(A))=R_(A). That is, R_(R_(A))=R¯(R_(A))=R_(A). Similarly, we can prove R¯(R¯(A))=R_(R¯(A))=R¯(A).
(6) For anyx∈U and α∈[0,1], we have(15)SLR_Ux∣α=⋀y∈xRSLUy∣α=1,SUR_Ux∣α=⋀y∈xRSUUy∣α=1.Thus, R_(U)=U. Similarly, R¯(U)=U. That is to say, R_(U)=R¯(U)=U. The second equation can be proved in a similar way.Definition 9.
Let(U,R) be a Pawlak approximation space and let A and B be the type-2 fuzzy sets on U.(1)
A and B are called lower rough equal denoted by A≃B if R_(A)=R_(B).(2)
A and B are called upper rough equal denoted by A≂B if R¯(A)=R¯(B).(3)
A and B are called rough equal denoted by A≈B if A and B are both lower rough equal and upper rough equal.Theorem 10.
Let(U,R) be a Pawlak approximation space. Then the following properties hold for any A,B,A′,B′∈F2(U):(1)
A≃B⇔A∩B≃A and A∩B≃B;(2)
A≂B⇔A∪B≂A and A∪B≂B;(3)
ifA≃A′ and B≃B′, then (A∩B)≃(A′∩B′);(4)
ifA≂A′ and B≂B′, then (A∪B)≂(A′∪B′);(5)
ifA≃∅ or B≃∅, then (A∩B)≃∅;(6)
ifA≂U or B≂U, then (A∪B)≂U;(7)
ifA⊆B and B≃∅, then A≃∅;(8)
ifA⊆B and A≂U, then B≂U;(9)
A≃U⇔A=U;(10)
A≂∅⇔A=∅.Proof.
By Definition9, the proof procedure is trivial.Theorem 11.
Let(U,R) be a Pawlak approximation space and let A be type-2 fuzzy set on U. Then, one has(1)
R_(A)=⋂{B∈F2(U);B≃A};(2)
R¯(A)=⋃{B∈F2(U);B≂A}.Proof.
By Theorem8(5), this theorem can be easily proved.


### 3.2. Generalized Rough Type-2 Fuzzy Approximation Operators Based on the Generalized Binary Relation

Definition 12.
LetU be a nonempty finite universe of discourse and let R∈P(U×U) be an arbitrary binary relation on U; Rs(x)={y∈U;(x,y)∈R}; then (U,R) is called the generalized Pawlak approximation space. For any A∈F2(U), the upper and lower generalized rough type-2 fuzzy approximation operators app¯(A) and app_(A) about (U,R) are, respectively, defined as follows:(16)app¯A≡⋃α∈0,1αapp¯Aα,app_A≡⋃α∈0,1αapp_Aα,where app_(A)α=∫x∈U∫u∈[SLapp_Ax∣α,SUapp_(A)x∣α]u/x, (app¯(A))α=∫x∈U∫u∈[SLapp¯(A)x∣α,SUapp¯(A)x∣α]u/x, and (17)SLapp_Ax∣α=⋀y∈RsxSLAy∣α,SUapp_Ax∣α=⋀y∈RsxSUAy∣α,SLapp¯Ax∣α=⋁y∈RsxSLAy∣α,SUapp¯Ax∣α=⋁y∈RsxSUAy∣α.
In particular, ifR is an equivalence relation of the universe U, then Rs(x) degrades to the equivalence class [x]R. The lower approximation app_(A) and the upper approximation app¯(A) reduce to the lower approximation R_(A) and the upper approximation R¯(A) in the classical Pawlak approximation space, respectively.
In [36], a type-2 fuzzy singleton set 1y and its complement 1U-{y} are, respectively, defined as follows: (18)
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Based on the above definition, we can obtain thatSL1yx∣α=SU1yx∣α=1x=y0x≠y and SL1U-{y}x∣α=SU1U-{y}x∣α=0x=y1x≠y hold for any α∈[0,1] and x∈U.Theorem 13.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following properties hold:(1)
app_(Ac)=(app¯(A))c,  app¯(Ac)=(app_(A))c;(2)
app_(A∩B)=app_(A)∩app_(B),  app¯(A∪B)=app¯(A)∪app¯(B);(3)
A⊆B⇒app_(A)⊆app_(B),  A⊆B⇒app¯(A)⊆app¯(B);(4)
app_(U)=app¯(U)=U,  app_(∅)=app¯(∅)=∅;(5)
SLapp_(1U-{y})x∣α=SUapp_(1U-{y})x∣α=1-R(x,y) for any α∈[0,1] and x,y∈U;(6)
SLapp¯(1y)x∣α=SUapp¯(1y)x∣α=R(x,y) for any α∈[0,1] and x,y∈U.Proof.
The proofs of (1)–(4) are analogous to Theorem8, and therefore we omit them.
(5) For anyα∈[0,1] and x,y∈U, we have(19)SLapp_1U-yx∣α=⋀z∈RsxSL1U-yz∣α.If R(x,y)=1, then y∈Rs(x). Thus, ⋀z∈Rs(x)SL1U-{y}z∣α≤SL1U-{y}y∣α=0. That is, SLapp_(1U-{y})x∣α=0. On the other hand, if R(x,y)=0, then y∉Rs(x). Thus, ⋀z∈Rs(x)SL1U-{y}z∣α=1. Therefore, we have SLapp_(1U-{y})x∣α=1-R(x,y).
Similarly, we can obtainSUapp_(1U-{y})x∣α=1-R(x,y). That is to say, SLapp_(1U-{y})x∣α=SUapp_(1U-{y})x∣α=1-R(x,y).
(6) The proof procedure is similar to (5).Theorem 14.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following statements are equivalent:(1)
R is reflexive;(2)
app_(A)⊆A;(3)
A⊆app¯(A).Proof.
(1)
⇒
(2). If R is reflexive, then x∈Rs(x) for any x∈U. For any α∈[0,1],(20)SLapp_Ax∣α=⋀y∈RsxSLAy∣α≤SLAx∣α.Similarly, we can derive SUapp_(A)x∣α≤SUAx∣α. Thus, app_(A)⊆A.
(2)
⇒
(3). We can obtain the conclusion according to Theorem 13(1).
(3)
⇒
(1). For any y∈U, let A=1y; thus SLAx∣α=SUAx∣α=1x=y0x≠y.
IfA⊆app¯(A), then SLAx∣α≤SLapp¯(A)x∣α and SUAx∣α≤SUapp¯(A)x∣α hold for any α∈[0,1] and x∈U. Hence, 1≤SLapp¯(A)y∣α and 1≤SUapp¯(A)y∣α. We can obtain 1=SLapp¯(A)y∣α and 1=SUapp¯(A)y∣α. Therefore, ⋁z∈Rs(y)SLAz∣α=⋁z∈Rs(y)SUAz∣α=1. That is to say, there exists z∈Rs(y) such that SLAz∣α=SUAz∣α=1. Thus, y∈Rs(y).
We can conclude thatR is reflexive.Theorem 15.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); the following statements are equivalent:(1)
R is symmetric.(2)
For anyα∈[0,1] and x,y∈U, SLapp¯(1x)y∣α=SLapp¯(1y)x∣α and SUapp¯(1x)y∣α=SUapp¯(1y)x∣α hold.(3)
For anyα∈0,1 and x,y∈U, SLapp_(1U-{y})x∣α=SLapp_(1U-{x})y∣α and SUapp_(1U-{y})x∣α=SUapp_(1U-{x})y∣α hold.Proof.
(1)
⇒
(2). For any α∈[0,1] and x,y∈U(21)SLapp¯1yx∣α=⋁z∈RsxSL1yz∣α,SLapp¯1xy∣α=⋁z∈RsySL1xz∣α.If x∈Rs(y), then SLapp¯(1x)y∣α=1.
SinceR is symmetric, we have y∈Rs(x). Hence, SLapp¯(1y)x∣α=1. That is, SLapp¯(1x)y∣α=SLapp¯(1y)x∣α.
Similarly, we can proveSUapp¯(1x)y∣α=SUapp¯(1y)x∣α.
(2)
⇒
(3). For any α∈[0,1] and x,y∈U(22)
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Similarly, we can proveSUapp_(1U-{y})x∣α=SUapp_(1U-{x})y∣α.
(3)
⇒
(1). For any α∈[0,1] and x,y∈U, if y∈Rs(x), in the following, we only should prove x∈Rs(y). Thus, SLapp_(1U-{y})x∣α=⋀z∈Rs(x)SL1U-{y}z∣α=0.
We know thatSLapp_(1U-{y})x∣α=SLapp_(1U-{x})y∣α. Hence, SLapp_(1U-{x})y∣α=⋀z∈Rs(y)SL1U-{x}z∣α=0. That is to say, there exists z∈Rs(y) such that SL1U-{x}z∣α=0.
We can obtainx∈Rs(y). Therefore, R is symmetric.Theorem 16.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following statements are equivalent:(1)
R is transitive;(2)
app¯app¯A⊆app¯(A);(3)
app_A⊆app_app_A.Proof.
(1)
⇒
(2). For any α∈[0,1] and x∈U(23)SLapp¯app¯Ax∣α=⋁y∈RsxSLapp¯Ay∣α=⋁y∈Rsx⋁z∈RsySLAz∣α=⋁y∈Rsx∧z∈RsySLAz∣α.Since R is transitive, we have y∈Rs(x)∧z∈Rs(y)⇒z∈Rs(x). Thus, SLapp¯(app¯(A))x∣α=⋁y∈Rs(x)∧z∈Rs(y)SLAz∣α≤⋁z∈Rs(x)SLAz∣α=SLapp¯(A)x∣α. Similarly, SUapp¯(app¯(A))x∣α≤SUapp¯(A)x∣α. Hence, app¯(app¯(A))⊆app¯(A).
(2)
⇒
(1). For any α∈[0,1] and x,y,z∈U, if y∈Rsx and z∈Rs(y), then (24)SLapp¯app¯1zx∣α=⋁t∈RsxSLapp¯1zt∣α=⋁t∈Rsx⋁p∈RstSL1zp∣α=⋁t∈Rsx∧t≠y⋁p∈RstSL1zp∣α∨⋁p∈RsySL1zp∣α]=⋁t∈Rsx∧t≠y⋁p∈RstSL1zp∣α∨⋁p∈Rsy∧p≠zSL1zp∣α∨SL1zz∣α=⋁t∈Rsx∧t≠y⋁p∈RstSL1zp∣α∨⋁p∈Rsy∧p≠zSL1zp∣α∨1=1.Furthermore, SLapp¯(1z)x∣α=⋁t∈Rs(x)SL1zt∣α. Since app¯(app¯(A))⊆app¯(A), we have SLapp¯(app¯(1z))x∣α≤SLapp¯(1z)x∣α. That is, 1≤⋁t∈Rs(x)SL1zt∣α. Thus, there exists t∈Rs(x) such that SL1zt∣α=1. That is to say, z∈Rs(x).
We can conclude thaty∈Rs(x)∧z∈Rs(y)⇒z∈Rs(x). Therefore, R is transitive.
(2)
⇔
(3). We can directly obtain the statement according to Theorem 13(1).


## 3.1. Rough Type-2 Fuzzy Approximation Operators Based on the Equivalence Relation

Definition 7.
LetU be a nonempty universe of discourse and let R∈P(U×U) be the clear equivalence relation on U. [x]R denotes the equivalence class. Then (U,R) is called the Pawlak approximation space. For any A∈F2(U), define the upper and lower rough type-2 fuzzy approximation operators R¯(A) and R_(A) about (U,R) by (7)R¯A≡⋃α∈0,1αR¯Aα,R_A≡⋃α∈0,1αR_Aα,where R_Aα=∫x∈U∫u∈[SLR_(A)x∣α,SUR_(A)x∣α]u/x,  (R¯(A))α=∫x∈U∫u∈[SLR¯(A)x∣α,SUR¯(A)x∣α]u/x, and (8)SLR_Ax∣α=⋀y∈xRSLAy∣α,SUR_Ax∣α=⋀y∈xRSUAy∣α,SLR¯Ax∣α=⋁y∈xRSLAy∣α,SUR¯Ax∣α=⋁y∈xRSUAy∣α.
Clearly, ifA is degraded to type-1 fuzzy set, then the rough type-2 fuzzy approximation operators defined in Definition 7 reduce to the rough type-1 fuzzy approximation operators. In the following, we define type-2 fuzzy universe set U=∫x∈U∫u∈[1,1]1/u/x and type-2 fuzzy empty set ∅=∫x∈U∫u∈[0,0]1/u/x. Obviously, SLUx∣α=SUUx∣α=1 and SL∅x∣α=SU∅x∣α=0 for any x∈U and α∈[0,1].Theorem 8.
Let(U,R) be a Pawlak approximation space and let R_ and R¯ be the rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following properties hold:(1)
R_(A)⊆A⊆R¯(A);(2)
R_(Ac)=(R¯(A))c,  R¯(Ac)=(R_(A))c;(3)
R_(A∩B)=R_(A)∩R_(B),  R¯(A∪B)=R¯(A)∪R¯(B);(4)
A⊆B⇒R_(A)⊆R_(B),  A⊆B⇒R¯(A)⊆R¯(B);(5)
R_(R_(A))=R¯(R_(A))=R_(A),  R¯(R¯(A))=R_(R¯(A))=R¯(A);(6)
R_(U)=R¯(U)=U,  R_(∅)=R¯(∅)=∅.Proof.
(1) For anyx∈U and α∈[0,1], we have (9)SLR_Ax∣α=⋀y∈xRSLAy∣α≤SLAx∣α,SUR_Ax∣α=⋀y∈xRSUAy∣α≤SUAx∣α.Thus, R_(A)⊆A. Similarly, A⊆R¯(A). That is, R_(A)⊆A⊆R¯(A).
(2) For anyx∈U and α∈[0,1],(10)SLR_Acx∣α=⋀y∈xRSLAcy∣α=⋀y∈xR1-SUAy∣α=1-⋁y∈xRSUAy∣α=1-⋁y∈xRSUAy∣α=1-SUR¯Ax∣α=SLR¯Acx∣α.Furthermore, (11)SUR_Acx∣α=⋀y∈xRSUAcy∣α=⋀y∈xR1-SLAy∣α=1-⋁y∈xRSLAy∣α=1-⋁y∈xRSLAy∣α=1-SLR¯Ax∣α=SUR¯Acx∣α.Thus, R_(Ac)=R¯(A)c. Similarly, we can prove R¯(Ac)=(R_(A))c.
(3) For anyx∈U and α∈[0,1], (12)SLR_A∩Bx∣α=⋀y∈xRSLA∩By∣α=⋀y∈xRSLAy∣α∧SLBy∣α=⋀y∈xRSLAy∣α∧⋀y∈xRSLBy∣α=SLR_Ax∣α∧SLR_Bx∣α=SLR_A∩R_Bx∣α.Similarly, we have SUR_(A∩B)x∣α=SUR_(A)∩R_(B)x∣α. Thus, R_(A∩B)=R_(A)∩R_(B). Similarly, we can prove R¯(A∪B)=R¯(A)∪R¯(B).
(4) For anyx∈U and α∈[0,1], since A⊆B, we have SLAx∣α≤SLBx∣α and SUAx∣α≤SUBx∣α. Thus, SLR_(A)x∣α=⋀y∈[x]RSLAy∣α≤⋀y∈[x]RSLBy∣α=SLR_(B)x∣α. Furthermore, SUR_(A)x∣α=⋀y∈[x]RSUAy∣α≤⋀y∈[x]RSUBy∣α=SUR_(B)x∣α. Hence, R_(A)⊆R_(B). That is, A⊆B⇒R_(A)⊆R_(B). Similarly, we can prove A⊆B⇒R¯(A)⊆R¯(B).
(5) For anyx∈U and α∈[0,1],(13)SLR_R_Ax∣α=⋀y∈xRSLR_Ay∣α=⋀y∈xR⋀z∈yRSLAz∣α=⋀z∈xRSLAz∣α=SLR_Ax∣α.Similarly, we have SUR_(R_(A))x∣α=SUR_(A)x∣α. Thus, R_(R_(A))=R_(A). In addition(14)SLR¯R_Ax∣α=⋁y∈xRSLR_Ay∣α=⋁y∈xR⋀z∈yRSLAz∣α=⋁y∈xR⋀z∈xRSLAz∣α=⋀z∈xRSLAz∣α=SLR_Ax∣α.Similarly, we can obtain SUR¯(R_(A))x∣α=SUR_(A)x∣α. Hence, R¯(R_(A))=R_(A). That is, R_(R_(A))=R¯(R_(A))=R_(A). Similarly, we can prove R¯(R¯(A))=R_(R¯(A))=R¯(A).
(6) For anyx∈U and α∈[0,1], we have(15)SLR_Ux∣α=⋀y∈xRSLUy∣α=1,SUR_Ux∣α=⋀y∈xRSUUy∣α=1.Thus, R_(U)=U. Similarly, R¯(U)=U. That is to say, R_(U)=R¯(U)=U. The second equation can be proved in a similar way.Definition 9.
Let(U,R) be a Pawlak approximation space and let A and B be the type-2 fuzzy sets on U.(1)
A and B are called lower rough equal denoted by A≃B if R_(A)=R_(B).(2)
A and B are called upper rough equal denoted by A≂B if R¯(A)=R¯(B).(3)
A and B are called rough equal denoted by A≈B if A and B are both lower rough equal and upper rough equal.Theorem 10.
Let(U,R) be a Pawlak approximation space. Then the following properties hold for any A,B,A′,B′∈F2(U):(1)
A≃B⇔A∩B≃A and A∩B≃B;(2)
A≂B⇔A∪B≂A and A∪B≂B;(3)
ifA≃A′ and B≃B′, then (A∩B)≃(A′∩B′);(4)
ifA≂A′ and B≂B′, then (A∪B)≂(A′∪B′);(5)
ifA≃∅ or B≃∅, then (A∩B)≃∅;(6)
ifA≂U or B≂U, then (A∪B)≂U;(7)
ifA⊆B and B≃∅, then A≃∅;(8)
ifA⊆B and A≂U, then B≂U;(9)
A≃U⇔A=U;(10)
A≂∅⇔A=∅.Proof.
By Definition9, the proof procedure is trivial.Theorem 11.
Let(U,R) be a Pawlak approximation space and let A be type-2 fuzzy set on U. Then, one has(1)
R_(A)=⋂{B∈F2(U);B≃A};(2)
R¯(A)=⋃{B∈F2(U);B≂A}.Proof.
By Theorem8(5), this theorem can be easily proved.


## 3.2. Generalized Rough Type-2 Fuzzy Approximation Operators Based on the Generalized Binary Relation

Definition 12.
LetU be a nonempty finite universe of discourse and let R∈P(U×U) be an arbitrary binary relation on U; Rs(x)={y∈U;(x,y)∈R}; then (U,R) is called the generalized Pawlak approximation space. For any A∈F2(U), the upper and lower generalized rough type-2 fuzzy approximation operators app¯(A) and app_(A) about (U,R) are, respectively, defined as follows:(16)app¯A≡⋃α∈0,1αapp¯Aα,app_A≡⋃α∈0,1αapp_Aα,where app_(A)α=∫x∈U∫u∈[SLapp_Ax∣α,SUapp_(A)x∣α]u/x, (app¯(A))α=∫x∈U∫u∈[SLapp¯(A)x∣α,SUapp¯(A)x∣α]u/x, and (17)SLapp_Ax∣α=⋀y∈RsxSLAy∣α,SUapp_Ax∣α=⋀y∈RsxSUAy∣α,SLapp¯Ax∣α=⋁y∈RsxSLAy∣α,SUapp¯Ax∣α=⋁y∈RsxSUAy∣α.
In particular, ifR is an equivalence relation of the universe U, then Rs(x) degrades to the equivalence class [x]R. The lower approximation app_(A) and the upper approximation app¯(A) reduce to the lower approximation R_(A) and the upper approximation R¯(A) in the classical Pawlak approximation space, respectively.
In [36], a type-2 fuzzy singleton set 1y and its complement 1U-{y} are, respectively, defined as follows: (18)
u
1
y
x
=
1
1
x
=
y
1
0
x
≠
y
,
u
1
U
-
y
x
=
1
0
x
=
y
1
1
x
≠
y
.
Based on the above definition, we can obtain thatSL1yx∣α=SU1yx∣α=1x=y0x≠y and SL1U-{y}x∣α=SU1U-{y}x∣α=0x=y1x≠y hold for any α∈[0,1] and x∈U.Theorem 13.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following properties hold:(1)
app_(Ac)=(app¯(A))c,  app¯(Ac)=(app_(A))c;(2)
app_(A∩B)=app_(A)∩app_(B),  app¯(A∪B)=app¯(A)∪app¯(B);(3)
A⊆B⇒app_(A)⊆app_(B),  A⊆B⇒app¯(A)⊆app¯(B);(4)
app_(U)=app¯(U)=U,  app_(∅)=app¯(∅)=∅;(5)
SLapp_(1U-{y})x∣α=SUapp_(1U-{y})x∣α=1-R(x,y) for any α∈[0,1] and x,y∈U;(6)
SLapp¯(1y)x∣α=SUapp¯(1y)x∣α=R(x,y) for any α∈[0,1] and x,y∈U.Proof.
The proofs of (1)–(4) are analogous to Theorem8, and therefore we omit them.
(5) For anyα∈[0,1] and x,y∈U, we have(19)SLapp_1U-yx∣α=⋀z∈RsxSL1U-yz∣α.If R(x,y)=1, then y∈Rs(x). Thus, ⋀z∈Rs(x)SL1U-{y}z∣α≤SL1U-{y}y∣α=0. That is, SLapp_(1U-{y})x∣α=0. On the other hand, if R(x,y)=0, then y∉Rs(x). Thus, ⋀z∈Rs(x)SL1U-{y}z∣α=1. Therefore, we have SLapp_(1U-{y})x∣α=1-R(x,y).
Similarly, we can obtainSUapp_(1U-{y})x∣α=1-R(x,y). That is to say, SLapp_(1U-{y})x∣α=SUapp_(1U-{y})x∣α=1-R(x,y).
(6) The proof procedure is similar to (5).Theorem 14.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following statements are equivalent:(1)
R is reflexive;(2)
app_(A)⊆A;(3)
A⊆app¯(A).Proof.
(1)
⇒
(2). If R is reflexive, then x∈Rs(x) for any x∈U. For any α∈[0,1],(20)SLapp_Ax∣α=⋀y∈RsxSLAy∣α≤SLAx∣α.Similarly, we can derive SUapp_(A)x∣α≤SUAx∣α. Thus, app_(A)⊆A.
(2)
⇒
(3). We can obtain the conclusion according to Theorem 13(1).
(3)
⇒
(1). For any y∈U, let A=1y; thus SLAx∣α=SUAx∣α=1x=y0x≠y.
IfA⊆app¯(A), then SLAx∣α≤SLapp¯(A)x∣α and SUAx∣α≤SUapp¯(A)x∣α hold for any α∈[0,1] and x∈U. Hence, 1≤SLapp¯(A)y∣α and 1≤SUapp¯(A)y∣α. We can obtain 1=SLapp¯(A)y∣α and 1=SUapp¯(A)y∣α. Therefore, ⋁z∈Rs(y)SLAz∣α=⋁z∈Rs(y)SUAz∣α=1. That is to say, there exists z∈Rs(y) such that SLAz∣α=SUAz∣α=1. Thus, y∈Rs(y).
We can conclude thatR is reflexive.Theorem 15.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); the following statements are equivalent:(1)
R is symmetric.(2)
For anyα∈[0,1] and x,y∈U, SLapp¯(1x)y∣α=SLapp¯(1y)x∣α and SUapp¯(1x)y∣α=SUapp¯(1y)x∣α hold.(3)
For anyα∈0,1 and x,y∈U, SLapp_(1U-{y})x∣α=SLapp_(1U-{x})y∣α and SUapp_(1U-{y})x∣α=SUapp_(1U-{x})y∣α hold.Proof.
(1)
⇒
(2). For any α∈[0,1] and x,y∈U(21)SLapp¯1yx∣α=⋁z∈RsxSL1yz∣α,SLapp¯1xy∣α=⋁z∈RsySL1xz∣α.If x∈Rs(y), then SLapp¯(1x)y∣α=1.
SinceR is symmetric, we have y∈Rs(x). Hence, SLapp¯(1y)x∣α=1. That is, SLapp¯(1x)y∣α=SLapp¯(1y)x∣α.
Similarly, we can proveSUapp¯(1x)y∣α=SUapp¯(1y)x∣α.
(2)
⇒
(3). For any α∈[0,1] and x,y∈U(22)
S
L
a
p
p
_
1
U
-
y
x
∣
α
=
⋀
z
∈
R
s
x
S
L
1
U
-
y
z
∣
α
=
⋀
z
∈
R
s
x
S
L
1
y
c
z
∣
α
=
⋀
z
∈
R
s
x
1
-
S
U
1
y
z
∣
α
=
1
-
⋁
z
∈
R
s
x
S
U
1
y
z
∣
α
=
1
-
S
U
a
p
p
¯
1
y
x
∣
α
,
S
L
a
p
p
_
1
U
-
x
y
∣
α
=
⋀
z
∈
R
s
y
S
L
1
U
-
x
z
∣
α
=
⋀
z
∈
R
s
y
S
L
1
x
c
z
∣
α
=
⋀
z
∈
R
s
y
1
-
S
U
1
x
z
∣
α
=
1
-
⋁
z
∈
R
s
y
S
U
1
x
z
∣
α
=
1
-
S
U
a
p
p
¯
1
x
y
∣
α
.Since SUapp¯(1x)y∣α=SUapp¯(1y)x∣α, thus SLapp_(1U-{y})x∣α=SLapp_(1U-{x})y∣α.
Similarly, we can proveSUapp_(1U-{y})x∣α=SUapp_(1U-{x})y∣α.
(3)
⇒
(1). For any α∈[0,1] and x,y∈U, if y∈Rs(x), in the following, we only should prove x∈Rs(y). Thus, SLapp_(1U-{y})x∣α=⋀z∈Rs(x)SL1U-{y}z∣α=0.
We know thatSLapp_(1U-{y})x∣α=SLapp_(1U-{x})y∣α. Hence, SLapp_(1U-{x})y∣α=⋀z∈Rs(y)SL1U-{x}z∣α=0. That is to say, there exists z∈Rs(y) such that SL1U-{x}z∣α=0.
We can obtainx∈Rs(y). Therefore, R is symmetric.Theorem 16.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U), the following statements are equivalent:(1)
R is transitive;(2)
app¯app¯A⊆app¯(A);(3)
app_A⊆app_app_A.Proof.
(1)
⇒
(2). For any α∈[0,1] and x∈U(23)SLapp¯app¯Ax∣α=⋁y∈RsxSLapp¯Ay∣α=⋁y∈Rsx⋁z∈RsySLAz∣α=⋁y∈Rsx∧z∈RsySLAz∣α.Since R is transitive, we have y∈Rs(x)∧z∈Rs(y)⇒z∈Rs(x). Thus, SLapp¯(app¯(A))x∣α=⋁y∈Rs(x)∧z∈Rs(y)SLAz∣α≤⋁z∈Rs(x)SLAz∣α=SLapp¯(A)x∣α. Similarly, SUapp¯(app¯(A))x∣α≤SUapp¯(A)x∣α. Hence, app¯(app¯(A))⊆app¯(A).
(2)
⇒
(1). For any α∈[0,1] and x,y,z∈U, if y∈Rsx and z∈Rs(y), then (24)SLapp¯app¯1zx∣α=⋁t∈RsxSLapp¯1zt∣α=⋁t∈Rsx⋁p∈RstSL1zp∣α=⋁t∈Rsx∧t≠y⋁p∈RstSL1zp∣α∨⋁p∈RsySL1zp∣α]=⋁t∈Rsx∧t≠y⋁p∈RstSL1zp∣α∨⋁p∈Rsy∧p≠zSL1zp∣α∨SL1zz∣α=⋁t∈Rsx∧t≠y⋁p∈RstSL1zp∣α∨⋁p∈Rsy∧p≠zSL1zp∣α∨1=1.Furthermore, SLapp¯(1z)x∣α=⋁t∈Rs(x)SL1zt∣α. Since app¯(app¯(A))⊆app¯(A), we have SLapp¯(app¯(1z))x∣α≤SLapp¯(1z)x∣α. That is, 1≤⋁t∈Rs(x)SL1zt∣α. Thus, there exists t∈Rs(x) such that SL1zt∣α=1. That is to say, z∈Rs(x).
We can conclude thaty∈Rs(x)∧z∈Rs(y)⇒z∈Rs(x). Therefore, R is transitive.
(2)
⇔
(3). We can directly obtain the statement according to Theorem 13(1).


## 4. Axiomatic Characterization of Generalized Rough Type-2 Fuzzy Sets

In this section, the axiomatic characterization of generalized rough type-2 fuzzy approximation operators is presented. To characterize the generalized rough type-2 fuzzy approximation operators by axioms, we define a constant type-2 fuzzy setβ⃡=∫x∈Uuβ⃡(x)/x=∫x∈Uβ/x, where β is secondary membership function and is not related to x. Clearly, SLβ⃡x∣α=SLβ⃡y∣α=l(α) and SUβ⃡x∣α=SUβ⃡y∣α=r(α) hold for any α∈[0,1]andx,y∈U, where l(α) and r(α) are functions related to α.Similar to [36], we define two special type-2 fuzzy sets denoted by SLAx∣α⃡ and SUAx∣α⃡ for any α∈[0,1], x∈U, and A∈F2(U), respectively. The two special type-2 fuzzy sets satisfy the following properties: SLSLAx∣α⃡y∣η=SLAx∣η,  SUSLAx∣α⃡y∣η=SUAx∣η,  SLSUAx∣α⃡y∣η=SLAx∣η, and SUSUAx∣α⃡y∣η=SUAx∣η for any η∈[0,1] and y∈U. Obviously, SLAx∣α⃡ and SUAx∣α⃡ are constant type-2 fuzzy sets.Definition 17.
LetL,H:F2(U)→F2(U) be two operators. They are called dual operators if for all A∈F2(U)(L1)
(L(Ac))c=H(A);(H1)
(H(Ac))c=L(A).Lemma 18.
Let(U,R) be a generalized Pawlak approximation space and let app_ and app¯ be the generalized rough type-2 fuzzy lower and upper approximation operators about (U,R); for any A∈F2(U) and constant type-2 fuzzy set β⃡, the following statements hold:(1)
app¯(A∩β⃡)=app¯(A)∩β⃡;(2)
app_(A∪β⃡)=app_(A)∪β⃡.Proof.
(1) For anyα∈[0,1] and x∈U,(25)SLapp¯A∩β⃡x∣α=⋁y∈RsxSLA∩β⃡y∣α=⋁y∈RsxSLAy∣α∧SLβ⃡y∣α=⋁y∈RsxSLAy∣α∧lα=⋁y∈RsxSLAy∣α∧lα=SLapp-Ax∣α∧SLβ⃡x∣α=SLapp¯A∩β⃡x∣α.Similarly, we can obtain SUapp¯(A∩β⃡)x∣α=SUapp¯(A)∩β⃡x∣α. Thus, app¯(A∩β⃡)=app¯(A)∩β⃡.
(2) The proof procedure is similar to (1).Lemma 19 (see [36]).
For anyA∈F2(U), α∈[0,1], and y∈U, the following statements hold:(1)
A=⋂y∈U[1U-{y}∪SLAy∣α⃡];(2)
A=⋂y∈U[1U-{y}∪SUAy∣α⃡];(3)
A=⋃y∈U[1y∩SLAy∣α⃡];(4)
A=⋃y∈U[1y∩SUAy∣α⃡].Theorem 20.
LetL,H:F2(U)→F2(U) be two dual operators. Then there exists a crisp binary relation R on U such that, for all A∈F2(U),  L(A)=app_(A) and H(A)=app¯(A) hold if and only if L and H satisfy the axioms: for all A,B∈F2(U) and any constant type-2 fuzzy set β⃡,(L2)
L(A∩B)=L(A)∩L(B);(L3)
L(A∪β⃡)=L(A)∪β⃡;(L4)
L(1U-{y})∈P(U) for any y∈U.Proof.
“⇒” It follows immediately from Theorem 13 and Lemma 18.
“⇐” By employing L and axiom (L4), we can define a crisp relation R on U:(26)y∈Rsx⟺Rx,y=1⟺SLL1U-yx∣α=SUL1U-yx∣α=0for anyα∈0,1,y∉Rsx⟺Rx,y=0⟺SLL1U-yx∣α=SUL1U-yx∣α=1for anyα∈0,1.Thus, 1-R(x,y)=SLL(1U-{y})(x∣α)=SUL(1U-{y})(x∣α). Then, we can obtain(27)
S
L
a
p
p
_
A
x
∣
α
=
S
L
a
p
p
_
⋂
y
∈
U
1
U
-
y
∪
S
L
A
y
∣
α
⃡
x
∣
α
According to Lemma
19
1
=
S
L
⋂
y
∈
U
a
p
p
_
1
U
-
y
∪
S
L
A
y
∣
α
⃡
x
∣
α
According to Theorem
13
2
=
S
L
⋂
y
∈
U
a
p
p
_
1
U
-
y
∪
S
L
A
y
∣
α
⃡
x
∣
α
According to Lemma
18
2
=
⋀
y
∈
U
S
L
a
p
p
_
1
U
-
y
x
∣
α
∨
S
L
S
L
A
y
∣
α
⃡
x
∣
α
=
⋀
y
∈
U
1
-
R
x
,
y
∨
S
L
S
L
A
y
∣
α
⃡
x
∣
α
According to Theorem
13
5
=
⋀
y
∈
U
S
L
L
1
U
-
y
x
∣
α
∨
S
L
S
L
A
y
∣
α
⃡
x
∣
α
=
S
L
⋂
y
∈
U
L
1
U
-
y
∪
S
L
A
y
∣
α
⃡
x
∣
α
=
S
L
⋂
y
∈
U
L
1
U
-
y
∪
S
L
A
y
∣
α
⃡
x
∣
α
According to
L
3
=
S
L
L
⋂
y
∈
U
1
U
-
y
∪
S
L
A
y
∣
α
⃡
x
∣
α
According to
L
2
=
S
L
L
A
x
∣
α
According to Lemma
19
1
.Similarly, SUapp_(A)(x∣α)=SUL(A)(x∣α). Thus, L(A)=app_(A).
We haveH(A)=app¯(A) according to Theorem 13(1).Theorem 21.
LetL,H:F2(U)→F2(U) be two dual operators. Then there exists a crisp binary relation R on U such that, for all A∈F2(U),  L(A)=f_(A) and H(A)=f¯(A) hold if and only if L and H satisfy the axioms: for all A,B∈F2(U) and any constant type-2 fuzzy set β⃡,(H2)
H(A∪B)=H(A)∪H(B);(H3)
H(A∩β⃡)=H(A)∩β⃡;(H4)
H(1y)∈P(U) for any y∈U.Proof.
“⇒” It follows immediately from Theorem 13 and Lemma 18.
“⇐” By employing H and axiom (H4), we can define a crisp relation R on U:(28)y∈Rsx⟺Rx,y=1⟺SLH1yx∣α=SUH1yx∣α=1for anyα∈0,1,y∉Rsx⟺Rx,y=0⟺SLH1yx∣α=SUH1yx∣α=0for anyα∈0,1.Thus, R(x,y)=SLH(1y)x∣α=SUH(1y)x∣α. Then, we can obtain(29)
S
L
a
p
p
¯
A
x
∣
α
=
S
L
a
p
p
¯
⋃
y
∈
U
1
y
∩
S
L
A
y
∣
α
⃡
x
∣
α
According to Lemma
19
3
=
S
L
⋃
y
∈
U
a
p
p
¯
1
y
∩
S
L
A
y
∣
α
⃡
x
∣
α
According to Theorem
13
2
=
S
L
⋃
y
∈
U
a
p
p
¯
1
y
∩
S
L
A
y
∣
α
⃡
x
∣
α
According to Lemma
18
1
=
⋁
y
∈
U
S
L
a
p
p
¯
1
y
x
∣
α
∧
S
L
S
L
A
y
∣
α
⃡
x
∣
α
=
⋁
y
∈
U
R
x
,
y
∧
S
L
S
L
A
y
∣
α
⃡
x
∣
α
According to Theorem
13
6
=
⋁
y
∈
U
S
L
H
1
y
x
∣
α
∧
S
L
S
L
A
y
∣
α
⃡
x
∣
α
=
S
L
⋃
y
∈
U
H
1
y
∩
S
L
A
y
∣
α
⃡
x
∣
α
=
S
L
⋃
y
∈
U
H
1
y
∩
S
L
A
y
∣
α
⃡
x
∣
α
According to
H
3
=
S
L
H
⋃
y
∈
U
1
y
∩
S
L
A
y
∣
α
⃡
x
∣
α
According to
H
2
=
S
L
H
A
x
∣
α
According to Lemma
19
3
.Similarly, SUapp¯(A)x∣α=SUH(A)x∣α. Thus, H(A)=app¯(A).
We haveL(A)=app_(A) according to Theorem 13(1).Definition 22.
Suppose thatL,H:F2(U)→F2(U) are two dual operators. If L satisfies axioms (L2), (L3), and (L4) or equivalently H satisfies axioms (H2), (H3), and (H4), then the system (F2(U),∩,∪,c,L,H) is called a generalized rough type-2 fuzzy set algebra, and L and H are called generalized type-2 fuzzy approximation operators.Theorem 23.
LetL,H:F2(U)→F2(U) be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists a reflexive relation R on U such that, for all A∈F2(U),  L(A)=app_(A) and H(A)=app¯(A) hold if and only if L and H satisfy the following axioms: (L5)
L(A)⊆A;(H5)
A⊆H(A).Proof.
“⇒” It follows immediately from Theorem 14.
“⇐” It follows immediately from Theorems 14, 20, and 21.Theorem 24.
LetL,H:F2(U)→F2(U) be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists a symmetric relation R on U such that, for all A∈F2(U),  L(A)=app_(A) and H(A)=app¯(A) hold if and only if L and H satisfy the following axioms: (L6)
For anyα∈[0,1] and x,y∈U, SLL(1U-{y})x∣α=SLL(1U-{x})y∣α and SUL(1U-{y})x∣α=SUL(1U-{x})y∣α hold.(H6)
For anyα∈[0,1] and x,y∈U,  SLH1xy∣α=SLH(1y)x∣α and SUH(1x)y∣α=SUH(1y)x∣α hold.Proof.
“⇒” It follows immediately from Theorem 15.
“⇐” It follows immediately from Theorems 15, 20, and 21.Theorem 25.
LetL,H:F2(U)→F2(U) be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists a transitive relation R on U such that, for all A∈F2(U),  L(A)=app_(A) and H(A)=app¯(A) hold if and only if L and H satisfy the following axioms: (L7)
L(A)⊆L(L(A));(H7)
H(H(A))⊆H(A).Proof.
“⇒” It follows immediately from Theorem 16.
“⇐” It follows immediately from Theorems 16, 20, and 21.Theorem 26.
LetL,H:F2(U)→F2(U) be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists an equivalence relation R on U such that, for all A∈F2(U),  L(A)=app_(A) and H(A)=app¯(A) hold if and only if L satisfies the axioms (L5)–(L7) and H satisfies the axioms (H5)–(H7).Proof.
“⇒” It follows immediately from Theorems 14, 15, and 16.
“⇐” It follows immediately from Theorems 14, 15, 16, 20, and 21.


## 5. The Attribute Reduction of Type-2 Fuzzy Information System

Let(U,A,F) be an information system. Here, U is the set of objects; that is, U={x1,x2,…,xn}. A is the attribute set; that is, A={a1,a2,…,am}. F is the relation set of U and A; that is, F={fj;1≤j≤m},  (fj:U→Vj), and Vj is the domain of the attribute aj.We call(U,A,F,D,G) a decision information system, where (U,A,F) is the classical information system, A is the condition attribute set, and D is the decision attribute set; that is, D={d1,d2,…,dp}. G is the relation set of U and D, G={gj;1≤j≤p},  (gj:U→Vj′), and Vj′ is the domain of the attribute dj.(
U
,
A
,
F
,
D
,
G
) is called a type-2 fuzzy information system, where (U,A,F) is the classical information system, and dj  (1≤j≤p) is the type-2 fuzzy sets on U.Definition 27.
Let(U,A,F,D,G) be a type-2 fuzzy information system, and(30)
L
B
x
∣
α
=
S
L
R
B
_
d
1
x
∣
α
,
S
L
R
B
_
d
2
x
∣
α
,
…
,
S
L
R
B
_
d
p
x
∣
α
,
S
U
R
B
_
d
1
x
∣
α
,
S
U
R
B
_
d
2
x
∣
α
,
…
,
S
U
R
B
_
d
p
x
∣
α
,where RB is a crisp relation determined by attribute set B. If LBx∣α=LAx∣α for any α∈[0,1] and x∈U, then B is called lower approximation consistent set of A.
IfB⊆A is the lower approximation consistent set of A and any subset of B are not the lower approximation consistent sets of A, then B is called the lower approximation reduction of A.Definition 28.
Let(U,A,F,D,G) be a type-2 fuzzy information system; for any α∈[0,1] and x,y∈U,(31)Dxy=ai∈A;fix≠fiyLAx∣α≠LAy∣α∅LAx∣α=LAy∣αis called discernibility matrix of (U,A,F,D,G).Theorem 29.
Let(U,A,F,D,G) be the type-2 fuzzy information system. For any x,y∈U and α∈[0,1], B is the lower approximation consistent set of A if and only if LAx∣α≠LAy∣α⇒[x]RB∩[y]RB=∅.Proof.
“⇒” If [x]RB∩[y]RB≠∅, then [x]RB=[y]RB.
In the following, we only should proveLAx∣α=LAy∣α.
For anyj≤p,  SLRB_(dj)x∣α=⋀t∈[x]RBSLdjt∣α=⋀t∈[y]RBSLdjt∣α=SLRB_djy∣α. Similarly, we have SURB_(dj)x∣α=SURB_(dj)y∣α.
SinceB is the lower approximation consistent set of A, we can obtain (32)SLRB_djx∣α=SLRA_djx∣α,SURB_djx∣α=SURA_djx∣α,SLRB_djy∣α=SLRA_djy∣α,SURB_djy∣α=SURA_djy∣α.Hence, SLRA_(dj)x∣α=SLRA_(dj)y∣α and SURA_(dj)x∣α=SURA_(dj)y∣α. That is, LAx∣α=LAy∣α. That is to say, there must be [x]RB∩[y]RB=∅ which hold for LAx∣α≠LAy∣α.
“⇐” Since J([x]RB)={[z]RA;[z]RA⊆[x]RB} forms a partition of [x]RB, we have(33)xRB=⋃zRA;z∈xRB=⋃zRA;zRA∈JxRB.If LAx∣α≠LAy∣α, then [x]RB∩[y]RB=∅. If [x]RB∩[y]RB≠∅, then LAx∣α=LAy∣α.
Furthermore, we can obtain[x]RB=[y]RB from [x]RB∩[y]RB≠∅.
Thus,(34)SLRB_djx∣α=⋀t∈xRBSLdjt∣α=⋀t∈⋃zRA;zRA∈JxRBSLdjt∣α=⋀t∈zRA∧zRA∈JxRBSLdjt∣α=⋀zRA∈JxRB⋀t∈zRASLdjt∣α=⋀zRA∈JxRBSLRA_djz∣α=SLRA_djx∣α.Similarly, SURB_(dj)x∣α=SURA_(dj)x∣α. That is, LBx∣α=LAx∣α. Hence, B is the lower approximation consistent set of A.Theorem 30.
Let(U,A,F,D,G) be the type-2 fuzzy information system. For any x,y∈U and α∈[0,1], B is the lower approximation consistent set of A if and only if Dxy≠∅⇒B∩Dxy≠∅.Proof.
“⇒” If B is the lower approximation consistent set of A and Dxy≠∅, then we have LAx∣α≠LAy∣α. Thus, we can obtain [x]RB∩[y]RB=∅ from Theorem 29. That is, there exists ai∈B such that fi(x)≠fi(y). Hence, B∩Dxy≠∅.
“⇐” For any α∈[0,1] and x,y∈U, if LAx∣α≠LAy∣α holds, in the following, we only should prove [x]RB∩[y]RB=∅.
IfDxy≠∅, then B∩Dxy≠∅. That is, there exists ai∈B such that fi(x)≠fi(y). Thus, [x]RB≠[y]RB. That is to say, [x]RB∩[y]RB=∅.
We have thatB is the lower approximation consistent set of A from Theorem 29.
A discernibility functionf(S) for S=(U,A,F,D,G) is a Boolean function of m Boolean variables a1¯,…,am¯ corresponding to the attributes a1,…,am, respectively, and is defined as follows:(35)fSa1¯,…,am¯=⋀⋁Dxy:Dxy≠∅,where ⋁(Dxy) is the disjunction of all variables a¯ such that a∈Dxy. In the sequel, we write ai instead of ai¯ when no confusion can arise.Theorem 31.
SupposeB⊆A; then B is a lower approximation reduction of A if and only if B is the minimal set satisfying Dxy≠∅⇒B∩Dxy≠∅.Proof.
The proof is straightforward by Theorem30.Letg(S) be the reduced disjunctive form of f(S) obtained from f(S) by applying the multiplication and absorption laws as many times as possible. Then, there exist l and Ak⊆A for k=1,…,l such that gS=⋀A1∨⋯∨(⋀Al), where every element in A1 only appears one time. We have the following theorem.Theorem 32.
IfRed(S) is the collection of all lower approximation reductions of A, then Red(S)={A1,…,Al}.Proof.
The proof is omitted since this theorem is similar to the one in [14].The all lower approximation reductions of type-2 fuzzy information systems can be computed by discernibility matrix approach. In the following, we develop an algorithm to find all lower approximation reductions of type-2 fuzzy information system(U,A,F,D,G).Step 1.
Decide on how manyα-planes will be used. Call that number Δ+1. Regardless of Δ+1, α=0 and α=1 must always be used.Step 2.
For everyα, xi  (1≤i≤n), and dj  (1≤j≤p), compute SLRA_djxi∣α and SURA_djxi∣α.Step 3.
For everyα,  xi  (1≤i≤n), compute LAxi∣α.Step 4.
ComputeDxy using Definition 28.Step 5.
Definef(S)=⋀{⋁(Dxy):Dxy≠∅}.Step 6.
ComputegS=⋀A1∨⋯∨(⋀Al) by f(S).Step 7.
Output all lower approximation reductions{A1,…,Al}.Example 33.
Table1 gives a type-2 fuzzy information system, where the universe is U={x1,x2,…,x8}, the condition attribute set is A={a1,a2,a3,a4}, and the decision attribute set is D={d1,d2}, where di∈F2(U)  (i=1,2). T(·,·,·,·) denotes trapezoid function, the first parameter and fourth parameter of () denote bottom left and right endpoint, respectively, and the second parameter and third parameter of () denote top left and right endpoint, respectively. Furthermore, S(·,·,·) denotes triangular function, the first parameter and third parameter of () denote bottom left and right endpoint, respectively, and the second parameter of () denotes apex.
Obviously, the universeU can be divided into three basic equivalence classes in accordance with the condition attribute set A={a1,a2,a3,a4}:(36)URA=X1,X2,X3=x1,x3,x8,x2,x4,x6,x5,x7.
In the following, we should decide on how manyα-planes will be used, where α∈[0,1]. Call that number Δ+1. Regardless of Δ+1,  α=0 and α=1 must always be used. If Δ+1=11, then α=0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1. When α=0.5, the approximation results of the type-2 fuzzy object are summarized in Table 2.
Similarly, we can compute the other approximation results forα≠0.5. The discernibility matrix of (U,A,F,D,G) is presented in Table 3.
It is easy to compute{a2,a4} and {a2,a3} as the lower approximation reductions of (U,A,F,D,G).Table 1
A type-2 fuzzy information system.
Object
a
1
a
2
a
3
a
4
d
1
d
2
x
1
1
2
3
1
T (0.7, 0.75, 0.8, 0.85)
T (0.2, 0.3, 0.4, 0.6)
x
2
1
2
1
3
S (0.5, 0.6, 0.8)
T (0.2, 0.7, 0.8, 0.96)
x
3
1
2
3
1
T (0.57, 0.7, 0.8, 0.85)
S (0.58, 0.7, 0.8)
x
4
1
2
1
3
T (0.8, 0.85, 0.9, 1)
S (0.86, 0.9, 0.96)
x
5
1
3
1
2
S (0.8, 0.9, 0.95)
T (0.6, 0.75, 0.85, 0.9)
x
6
1
2
1
3
S (0.4, 0.8, 0.95)
S (0.7, 0.86, 0.9)
x
7
1
3
1
2
T (0.75, 0.85, 0.9, 1)
S (0.75, 0.8, 0.96)
x
8
1
2
3
1
S (0.8, 0.9, 1)
T (0.47, 0.65, 0.8, 1)Table 2
The approximation results of the type-2 fuzzy object.
U
/
R
A
S
L
R
A
_
d
1
(
x
∣
0.5
)
S
U
R
A
_
(
d
1
)
(
x
∣
0.5
)
S
L
R
A
_
(
d
2
)
(
x
∣
0.5
)
S
U
R
A
_
(
d
2
)
(
x
∣
0.5
)
X
1
0.6350
0.8250
0.2500
0.5000
X
2
0.5500
0.7000
0.4500
0.8800
X
3
0.7800
0.8800
0.5600
0.8750Table 3
The discernibility matrix.
Object
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
1
∅
{
a
3
,
a
4
}
∅
{
a
3
,
a
4
}
{
a
2
,
a
3
,
a
4
}
{
a
3
,
a
4
}
{
a
2
,
a
3
,
a
4
}
∅
x
2
{
a
3
,
a
4
}
∅
{
a
3
,
a
4
}
∅
{
a
2
,
a
4
}
∅
{
a
2
,
a
4
}
{
a
3
,
a
4
}
x
3
∅
{
a
3
,
a
4
}
∅
{
a
3
,
a
4
}
{
a
2
,
a
3
,
a
4
}
{
a
3
,
a
4
}
{
a
2
,
a
3
,
a
4
}
∅
x
4
{
a
3
,
a
4
}
∅
{
a
3
,
a
4
}
∅
{
a
2
,
a
4
}
∅
{
a
2
,
a
4
}
{
a
3
,
a
4
}
x
5
{
a
2
,
a
3
,
a
4
}
{
a
2
,
a
4
}
{
a
2
,
a
3
,
a
4
}
{
a
2
,
a
4
}
∅
{
a
2
,
a
4
}
∅
{
a
2
,
a
3
,
a
4
}
x
6
{
a
3
,
a
4
}
∅
{
a
3
,
a
4
}
∅
{
a
2
,
a
4
}
∅
{
a
2
,
a
4
}
{
a
3
,
a
4
}
x
7
{
a
2
,
a
3
,
a
4
}
{
a
2
,
a
4
}
{
a
2
,
a
3
,
a
4
}
{
a
2
,
a
4
}
∅
{
a
2
,
a
4
}
∅
{
a
2
,
a
3
,
a
4
}
x
8
∅
{
a
3
,
a
4
}
∅
{
a
3
,
a
4
}
{
a
2
,
a
3
,
a
4
}
{
a
3
,
a
4
}
{
a
2
,
a
3
,
a
4
}
∅


## 6. Conclusions

Theα-plane representation method of type-2 fuzzy sets has been extensively studied and can greatly reduce the computational workload. Based on α-plane theory, the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators have been defined, and their properties have been derived. The connections between special binary relations and generalized rough type-2 fuzzy upper and lower approximation operators are examined. The axiomatics are provided to fully characterize the generalized rough type-2 fuzzy upper and lower approximation operators. The attribute reduction of the type-2 fuzzy information system has been investigated, and an attribute reduction algorithm to find all reductions of the type-2 fuzzy information system has also been developed. Further research will focus on the applications of the proposed rough type-2 fuzzy set model.



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*Source: 4819353-2016-02-02.xml*