# On Characterization of Rough Type-2 Fuzzy Sets.

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 of rough 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 a-plane representation of the general type-2 fuzzy sets and claimed that the a-plane representation can greatly reduce the computational workload. Currently, [alpha]-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 by [??], is expressed as

[mathematical expression not reproducible], (1)

where [J.sub.x] is the primary membership of x; [u.sub.A](x) = [mathematical expression not reproducible] is the secondary membership function; [f.sub.x](u) is a secondary membership grade.

In the following sections, the class of all type-2 fuzzy sets of the universe of discourse U is denoted as [F.sub.2](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]). An a-plane for type-2 fuzzy set A, which is denoted by [A.sub.[alpha]], is defined as follows:

[mathematical expression not reproducible], (2)

where [[S.sup.A..sub.L](x | [alpha]), [S.sup.A.sub.U](x | [alpha])] denote an [alpha]-cut of the secondary membership function [u.sub.A](x).

Definition 3 (see [32]). The a-plane representation (theorem) for type-2 fuzzy set A is

[mathematical expression not reproducible]. (3)

Theorem 4 (see [32]). Let [(A [union] B).sub.[alpha]] and [(A [intersection] B).sub.[alpha]] be [alpha]-plane of A [union] B and A [intersection] B, respectively; one has

[mathematical expression not reproducible], (4)

where

[mathematical expression not reproducible]. (5)

Obviously, [mathematical expression not reproducible] hold.

Theorem 5 (see [36]). Let [([A.sup.c]).sub.[alpha]] be [alpha]-plane of [A.sup.c]; one has

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible].

Definition 6 (see [36]). Let A, B [member of] [F.sub.2](U), and define A [subset or equal to] B if [S.sup.A.sub.L] (x | [alpha]) [less than or equal to] [S.sup.B.sub.L] (x | [alpha]) and [S.sup.A.sub.U] (x | [alpha]) [less than or equal to] [S.sup.B.sub.U] (x | [alpha]) hold for any [alpha] [member of] [0,1] and x [lambda] U. If A [subset or equal to] B and B [subset or equal to] 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. Let U be a nonempty universe of discourse and let R e P(U x U) be the clear equivalence relation on U. denotes the equivalence class. Then (U, R) is called the Pawlak approximation space. For any A [member of] [F.sub.2](U), define the upper and lower rough type-2 fuzzy approximation operators [bar.R](A) and [bar.R](A) about (U, R) by

[mathematical expression not reproducible], (7)

where [mathematical expression not reproducible] and

[mathematical expression not reproducible]. (8)

Clearly, if A 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 = [[integral].sub.x[member of]U][[[integral].sub.u[member of][1,1]]/x and type-2 fuzzy empty set 0 = [[integral].sub.x[member of]U][[[integral].sub.u[member of][0,0]] 1/u]/x. Obviously, [S.sup.U.sub.L](x | [alpha]) = [S.sup.U.sub.U] (x | [alpha]) = 1 and [S.sup.0.sub.L](x | [alpha]) = [S.sup.0.sub.U](x | [alpha]) = 0 for any x [member of] U and a [member of] [0,1].

Theorem 8. Let (U, R) be a Pawlak approximation space and let [R.bar] and [bar.R] be the rough type-2 fuzzy lower and upper approximation operators about (U, R); for any A [member of] [F.sub.2](U), the following properties hold:

(1) [R.bar](A) [subset or equal to] A [subset or equal to] [bar.R](A);

(2) [R.bar]([A.sup.C]) = [([bar.R](A)).sup.C], [bar.R]([A.sup.c]) = [([R.bar](A)).sup.C];

(3) [R.bar](A [intersection] B) = [R.bar](A) [intersection] [R.bar](B), [bar.R](A [union] B) = [bar.R](A) [union] [subset or equal to](B);

(4) A [subset or equal to] B [??] [R.bar](A) [subset or equal to] [R.bar](B), A [subset or equal to] B [??] [bar.R](A) [subset or equal to] [bar.R](B);

(5) [R.bar]([R.bar](A)) = [bar.R]([R.bar](A)) = [R.bar](A), [bar.R]([bar.R](A)) = [R.bar]([bar.R](A)) = [bar.R](A);

(6) [R.bar](U) = [bar.R](U) = U, [R.bar]([empty set]) = [bar.R]([empty set]) = [empty set].

Proof. (1) For any x [member of] U and a [member of] [0,1], we have

[mathematical expression not reproducible]. (9)

Thus, [R.bar](A) [subset or equal to] A. Similarly, A [subset or equal to] [bar.R](A). That is, [R.bar](A) [subset or equal to] A [subset or equal to] [bar.R](A).

(2) For any x [member of] U and a [member of] [0,1],

[mathematical expression not reproducible]. (10)

Furthermore,

[mathematical expression not reproducible]. (11)

Thus, [R.bar]([A.sup.C]) = [([bar.R](A)).sup.C]. Similarly, we can prove [bar.R]([A.sup.C]) = [([R.bar](A)).sup.C].

(3) For any x [member of] U and [alpha] [member of] [0,1],

[mathematical expression not reproducible]. (12)

Similarly, we have [S.sup.[R.bar](A [intersection] B).sub.U](x | [alpha]) = [S.sup.[R.bar](A)[R.bar](B).sub.U]. Thus, [R.bar](A [intersection] B) = [R.bar](A) [intersection] [R.bar](B). Similarly, we can prove [bar.R](A [union] B) = R(A) [union] [bar.R](B).

(4) For any x [member of] U and a [member of] [0,1], since A [subset or equal to] B, we have [S.sup.A.sub.l](x | [alpha]) [less than or equal to] [S.sup.B.sub.L](x | [alpha]) and [S.sup.A.sub.U] (x | [alpha]) [less than or equal to] [S.sup.B.sub.U] (x | [alpha]). Thus,

[mathematical expression not reproducible]. Furthermore, [mathematical expression not reproducible]. Hence, [R.bar](A) [subset or equal to] [R.bar](B). That is, A [subset or equal to] B [??] [R.bar](A) [subset or equal to] [R.bar](B). Similarly, we can prove A [subset or equal to] B [??] [bar.R](A) [subset or equal to] [bar.R](B).

(5) For any x [member of] U and a [member of] [0,1],

[mathematical expression not reproducible]. (13)

Similarly, we have [S.sup.[R.bar]([R.bar](A)).sub.U](x | [alpha]) = [S.sup.[R.bar](A).sub.U](x | [alpha]). Thus, [R.bar]([R.bar](A)) = [R.bar](A). In addition

[mathematical expression not reproducible]. (14)

Similarly, we can obtain [S.sup.[bar.R]([R.bar](A)).sub.U](x | [alpha]) = [S.sup.[R.bar](A).sub.U](x | [alpha]). Hence, [bar.R]([R.bar](A)) = [R.bar](A). That is, [R.bar]([R.bar](A)) = [R.bar]([R.bar](A)) = [R.bar](A). Similarly, we can prove [bar.R]([bar.R](A)) = [R.bar]([bar.R](A)) = [bar.R](A).

(6) For any x [member of] U and a [member of] [0,1], we have

[mathematical expression not reproducible]. (15)

Thus, [R.bar](U) = U. Similarly, [bar.R](U) = U. That is to say, [R.bar](U) = [bar.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.bar](A) = [R.bar](B).

(2) A and B are called upper rough equal denoted by A ~ B if [bar.R](A) = [bar.R](B).

(3) A and B are called rough equal denoted by A [approximately equal to] 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' [member of] [F.sub.2] (U):

(1) A [equivalent] B [??] [union] B [equivalent] A and A [union] B [equivalent] B;

(2) [mathematical expression not reproducible];

(3) if A [equivalent] A' and B [equivalent] B', then (A [intersection] B) [equivalent] (A' [intersection] B');

(4) [mathematical expression not reproducible];

(5) if A [equivalent] 0 or B [equivalent] 0, then (A [intersection] B) [equivalent] 0;

(6) [mathematical expression not reproducible];

(7) if A [subset or equal to] B and B [equivalent] [empty set], then A [equivalent] [empty set];

(8) if A [subset or equal to] B and A [??] U, then B [??] U;

(9) A [equivalent] U [??] A = U;

(10) [mathematical expression not reproducible].

Proof. By Definition 9, 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.bar](A) = [intersection] {B [member of] [F.sub.2] (U); B [equivalent] A};

(2) [bar.R](A) = [union] [B [member of] [F.sub.2](U); B [??] A}.

Proof. By Theorem 8(5), this theorem can be easily proved.

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

Definition 12. Let U be a nonempty finite universe of discourse and let R [member of] P(U x U) be an arbitrary binary relation on U; [R.sub.s](x) = {y [member of] U; (x, y) [member of] R}; then (U, R) is called the generalized Pawlak approximation space. For any A [member of] [F.sub.2](U), the upper and lower generalized rough type-2 fuzzy approximation operators [bar.app](A) and [app.bar](A) about (U, R) are, respectively, defined as follows:

[mathematical expression not reproducible], (16)

where [mathematical expression not reproducible], and

[mathematical expression not reproducible]. (17)

In particular, if R is an equivalence relation of the universe U, then [R.sub.s](x) degrades to the equivalence class The lower approximation [app.bar](A) and the upper approximation [bar.app](A) reduce to the lower approximation [R.bar](A) and the upper approximation [bar.R](A) in the classical Pawlak approximation space, respectively.

In [36], a type-2 fuzzy singleton set [1.sub.y] and its complement [1.sub.U-{y}] are, respectively, defined as follows:

[mathematical expression not reproducible]. (18)

Based on the above definition, we can obtain that [mathematical expression not reproducible] hold for any [alpha] [member of] [0,1] and x [member of] U.

Theorem 13. Let (U, R) be a generalized Pawlak approximation space and let [app.bar] and [bar.app] be the generalized rough type-2 fuzzy lower and upper approximation operators about (U, R); for any A [member of] [F.sub.2](U), the following properties hold:

(1) [app.bar]([A.sup.c]) = [([bar.app](A)).sup.c], [bar.app]([A.sup.c]) = [([app.bar](A)).sup.c];

(2) [app.bar](A [intersection] B) = [app.bar](A)[intersection][app.bar](B), [bar.app](A [union] B) = [bar.app](A) [union] [bar.app](B);

(3) A [subset or equal to] B [??] [app.bar] (A) [subset or equal to] [app.bar](B), A [subset or equal to] B [??] [bar.app](A) [subset or equal to] [bar.app](B);

(4) [app.bar](U) = [bar.app](U) = U, [app.bar](0) = [bar.app](0) = 0;

(5) [mathematical expression not reproducible] for any [alpha] [lambda] [0,1] and x, y [member of] U;

(6) [mathematical expression not reproducible] for any [alpha][member of] [0,1] and x, y [member of] U.

Proof. The proofs of (1)-(4) are analogous to Theorem 8, and therefore we omit them.

(5) For any [alpha] [member of] [0,1] and x, y [member of] U, we have

[mathematical expression not reproducible]. (19)

If R(x, y) = 1, then y [member of] [R.sub.s](x). Thus, [mathematical expression not reproducible]. On the other hand, if R(x, y) = 0.That is, [mathematical expression not reproducible], then y [not member of] [R.sub.s](x). Thus, [mathematical expression not reproducible]

a) = 1. Therefore, we have [mathematical expression not reproducible].

Similarly, we can obtain [mathematical expression not reproducible].

That is to say, [mathematical expression not reproducible].

(6) The proof procedure is similar to (5).

Theorem 14. Let (U, R) be a generalized Pawlak approximation space and let [app.bar] and [bar.app] be the generalized rough type-2 fuzzy lower and upper approximation operators about (U, R); for any A [member of] [F.sub.2](U), the following statements are equivalent:

(1) R is reflexive;

(2) [app.bar](A) [subset or equal to] A;

(3) A [subset or equal to] [bar.app] (A).

Proof.

(1)[??](2). If R is reflexive, then x [member of] [R.sub.s](x) for any x [member of] U. For any [alpha] [member of] [0,1],

[mathematical expression not reproducible]. (20)

Similarly, we can derive [S.sup.[app.bar](A).sub.L](x | [alpha]) [less than or equal to] [S.sup.A.sub.U](x | [alpha]). Thus, [app.bar](A) [subset or equal to] A.

(2)[??](3). We can obtain the conclusion according to Theorem 13(1).

(3)[mathematical expression not reproducible].

If A [subset or equal to] [bar.app](A), then [S.sup.A.sub.L](x | [alpha]) [less than or equal to] [S.sup.[bar.app](A).sub.L](x | [alpha]) and [S.sup.A.sub.U](x | [alpha]) [less than or equal to] [S.sup.[bar.app].sub.U](x | [alpha]) hold for any [alpha] [member of] [0,1] and x [member of] U. Hence, 1 [less than or equal to] [S.sup.[bar.app](A).sub.L](y | [alpha]) and 1 [less than or equal to] [S.sup.[bar.app](A).sub.U]](y | [alpha]). We can obtain 1 = [S.sup.[bar.app](A).sub.L] (y | [alpha]) and 1 = [S.sup.[bar.app](A).sub.U](y | [alpha]). Therefore, [mathematical expression not reproducible]. That is to say, there exists z [member of] [R.sub.s](y) such that [S.sup.A.sub.L](z | [alpha]) = [S.sup.A.sub.U](z | [alpha]) = 1. Thus, y [member of] [R.sub.s](y). We can conclude that R is reflexive.

Theorem 15. Let (U, R) be a generalized Pawlak approximation space and let [app.bar] and [bar.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 any [alpha] [member of] [0,1] and x, y [member of] U, [mathematical expression not reproducible] hold.

(3) For any [alpha] [member of] [0,1] and x, y [member of] U, [mathematical expression not reproducible] hold.

Proof.

(1)[??](2). For any [alpha] [member of] [0,1] and x, y [member of] U

[mathematical expression not reproducible]. (21)

If x [member of] [R.sup.s](y), then [mathematical expression not reproducible].

Since R is symmetric, we have y [member of] [R.sub.s](x). Hence, [mathematical expression not reproducible].

Similarly, we can prove [mathematical expression not reproducible].

(2)[??](3). For any [alpha] [member of] [0,1] and x, y [member of] U

[mathematical expression not reproducible]. (22)

Since [mathematical expression not reproducible], thus [mathematical expression not reproducible].

Similarly, we can prove [mathematical expression not reproducible].

(3)[??](1). For any [alpha] [member of] [0,1] and x, y [member of] U, if y [member of] [R.sub.s](x), in the following, we only should prove x [member of] [R.sub.s](y). Thus, [mathematical expression not reproducible].

We know that [mathematical expression not reproducible]. Hence, [mathematical expression not reproducible]. That is to say, there exists z [member of] [R.sub.s](y) such that (z | [alpha]) = 0. We can obtain x [member of] [R.sub.s](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 e F2(U), the following statements are equivalent:

(1) R is transitive;

(2) [bar.app]([bar.app](A)) [subset or equal to] [bar.app](A);

(3) [app.bar](A) [subset or equal to] app([app.bar](A)).

Proof.

(1)[??](2). For any [alpha] [member of] [0,1] and x [member of] U

[mathematical expression not reproducible]. (23)

Since R is transitive, we have y [member of] [R.sub.s](x) [conjunction] z [member of] [R.sub.s](y) [??] z [member of] [R.sub.s](x).Thus, [mathematical expression not reproducible]. Similarly, S[bar.app]([bar.app](A))(x | [alpha]) [less than or equal to] [S.sup.[bar.app](A).sub.U](x | [alpha]). Hence, [bar.app]([bar.app](A)) [subset or equal to] [bar.app](A).

(2) [??] (1). For any [alpha] [member of] [0,1] and x, y, z [member of] U, if y [member of] [R.sub.s](x) and z [member of] [R.sub.s](y), then

[mathematical expression not reproducible]. (24)

Furthermore, [mathematical expression not reproducible]. Since [bar.app] ([bar.app] (A)) [subset or equal to] [bar.app] (A), we have [mathematical expression not reproducible]. That is, [mathematical expression not reproducible]. Thus, there exists t [member of] [R.sub.s](x) such that [mathematical expression not reproducible]. That is to say, z [member of] [R.sub.s](x).

We can conclude that y [member of] [R.sub.s](x) [conjunction] z [member of] [R.sub.s](y) [??] z [member of] [R.sub.s](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 set [??] = [[integral].sub.x[member of]U] [u.sub.[??]] = [[integral].sub.x[member of]U] [beta]/x, where [beta] is secondary membership function and is not related to x. Clearly, [mathematical expression not reproducible] hold for any [alpha] [member of] [0,1] and x, y, [member of] U where l([alpha]) and r([alpha]) are functions related to [alpha].

Similar to [36], we define two special type-2 fuzzy sets denoted by [mathematical expression not reproducible] for any [alpha] [member of] [0,1], x [member of] U, and A [member of] [F.sub.2] (U), respectively. The two special type-2 fuzzy sets satisfy the following properties: [mathematical expression not reproducible]. Obviously, [mathematical expression not reproducible] are constant type-2 fuzzy sets.

Definition 17. Let L, H : [F.sub.2] (U) [right arrow] [F.sub.2] (U) be two operators. They are called dual operators if for all A [member of] [F.sub.2] (U)

(L1) [(L([A.sup.c])).sup.c] = H(A); (H1) [(H([A.sup.c])).sup.c] = L(A).

Lemma 18. Let (U, R) be a generalized Pawlak approximation space and let [app.bar] and [bar.app] be the generalized rough type-2 fuzzy lower and upper approximation operators about (U, R); for any A [member of] [F.sub.2] (U) and constant type-2 fuzzy set [??], the following statements hold:

(1) [bar.app](A [intersection] [??]) = [bar.app](A) [intersection] [??];

(2) [app.bar](A [union] [??]) = [app.bar](A) [union] [??].

Proof. (1) For any [alpha] [member of] [0,1] and x [member of] U,

[mathematical expression not reproducible]. (25)

Similarly, we can obtain [S.sup.[bar.aap](A[intersection][??]).sub.U] (x | [alpha]) = [S.sup.[bar.aap](A[intersection][??]).sub.U] (x | [alpha]).

Thus, [bar.app](A [intersection] [??]) = [bar.app](A) [intersection] [??].

(2) The proof procedure is similar to (1).

Lemma 19 (see [36]). For any A [member of] [F.sub.2] (U), [alpha] [member of] [0,1], and y [member of] U, the following statements hold:

(1) [mathematical expression not reproducible];

(2) [mathematical expression not reproducible];

(3) [mathematical expression not reproducible];

(4) [mathematical expression not reproducible].

Theorem 20. Let L, H : [F.sub.2] (U) [right arrow] [F.sub.2] (U) be two dual operators. Then there exists a crisp binary relation R on U such that, for all A [member of] [F.sub.2] (U), L(A) = [app.bar](A) and H(A) = [bar.app](A) hold if and only if L and H satisfy the axioms: for all A, B [member of] [F.sub.2] (U) and any constant type-2 fuzzy set [??],

(L2) L(A [intersection] B) = L(A) [intersection] L(B);

(L3) L(A [union] [??]) = L(A) [union] [??];

(L4) L([1.sub.U-{y}]) [member of] P(U) for any y [member of] 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:

[mathematical expression not reproducible]. (26)

Thus, [mathematical expression not reproducible]. Then, we can obtain

[mathematical expression not reproducible]

(According to Lemma 19(1))

[mathematical expression not reproducible]

(According to Theorem 13 (2))

[mathematical expression not reproducible]

(According to Lemma 18 (2))

[mathematical expression not reproducible] (27)

(According to Theorem 13 (5))

[mathematical expression not reproducible]

(According to (L3))

[mathematical expression not reproducible]

(According to (L2))

= [S.sup.L(A).sub.L] (x | [alpha]) (According to Lemma 19 (1))

Similarly, [S.sup.[app.bar](A).sub.U] (x | [alpha]) = [S.sup.L(A).sub.U] (x | [alpha]). Thus, L(A) = [app.bar](A).

We have H(A) = [bar.app] (A) according to Theorem 13(1).

Theorem 21. Let L, H : [F.sub.2] (U) [right arrow] [F.sub.2] (U) be two dual operators. Then there exists a crisp binary relation R on U such that, for all A [member of] [F.sub.2] (U), L(A) = [f.bar](A) and H(A) = [bar.f](A) hold if and only if L and H satisfy the axioms: for all A, B [member of] [F.sub.2] (U) and any constant type-2 fuzzy set [??],

(H2) H(A [union] B) = H(A) [union] H(B);

(H3) H(A [intersection] [??]) = H(A) [intersection] [??];

(H4) H([1.sub.y]) [member of] P(U) for any y [member of] 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:

[mathematical expression not reproducible]. (28)

Thus, [mathematical expression not reproducible]. Then, we can obtain

[mathematical expression not reproducible]

(According to Lemma 19 (3))

[mathematical expression not reproducible]

(According to Theorem 13 (2))

[mathematical expression not reproducible]

(According to Lemma 18(1))

[mathematical expression not reproducible]

(According to Theorem 13 (6))

[mathematical expression not reproducible] (According to (H3))

[mathematical expression not reproducible] (According to (H2))

= [S.sup.H(A).sub.L] (x | [alpha]) (According to Lemma 19 (3)). (29)

Similarly, [S.sup.[app.bar](A).sub.U])(x | [alpha]) = [S.sup.H(A).sub.U] (x | [alpha]). Thus, H(A) = [bar.app](A).

We have L(A) = [app.bar](A) according to Theorem 13(1).

Definition 22. Suppose that L,H : [F.sub.2] (U) [right arrow] [F.sub.2] (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 ([F.sub.2] (U), [intersection], [union], 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. Let L,H : [F.sub.2] (U) [right arrow] [F.sub.2] (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 [member of] [F.sub.2] (U), L(A) = [app.bar](A) and H(A) = [bar.app](A) hold if and only if L and H satisfy the following axioms:

(L5) L(A) [subset or equal to] A;

(H5) A [subset or equal to] H(A).

Proof. "[??]" It follows immediately from Theorem 14.

"[??]" It follows immediately from Theorems 14,20, and 21.

Theorem 24. Let L, H : [F.sub.2] (U) [right arrow] [F.sub.2] (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 e [F.sub.2] (U), L(A) = [app.bar](A) and H(A) = [bar.app](A) hold if and only if L and H satisfy the following axioms:

(L6) For any [mathematical expression not reproducible] hold.

(H6) For any [mathematical expression not reproducible] hold.

Proof. "[??]" It follows immediately from Theorem 15.

"[??]" It follows immediately from Theorems 15,20, and 21.

Theorem 25. Let L, H : [F.sub.2] (U) [right arrow] [F.sub.2] (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 [member of] [F.sub.2] (U), L(A) = [app.bar](A) and H(A) = [bar.app](A) hold if and only if L and H satisfy the following axioms:

(L7) L(A) [subset or equal to] L(L(A));

(H7) H(H(A)) [subset or equal to] H(A).

Proof. "[??]" It follows immediately from Theorem 16.

"[??]" It follows immediately from Theorems 16,20, and 21.

Theorem 26. Let L, H : [F.sub.2] (U) [right arrow] [F.sub.2] (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 [member of] [F.sub.2] (U) L(A) = [app.bar](A) and H(A) = [bar.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 = {[x.sub.1], [x.sub.2], ..., [x.sub.n]}. A is the attribute set; that is, A = {[a.sub.1], [a.sub.2]], ..., [a.sub.m]}. F is the relation set of U and A; that is, F = {[f.sub.j]; 1 [less than or equal to] j [less than or equal to] m}, ([f.sub.j] : U [right arrow] [V.sub.j]), and [V.sub.j] is the domain of the attribute [a.sub.j].

We call (U, A, F, A 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 = {[d.sub.1], [d.sub.2], ..., [d.sub.p]}. G is the relation set of U and D, G = {[g.sub.j]; 1 [less than or equal to] j [less than or equal to] p|, ([g.sub.j] : U [right arrow] [V'.sub.j]), and [V'.sub.j] is the domain of the attribute [d.sub.j].

(U, A, F, A G) is called a type-2 fuzzy information system, where (U, A, F) is the classical information system, and [d.sub.j] (1 [less than or equal to] j [less than or equal to] p) is the type-2 fuzzy sets on U.

Definition 27. Let (U, A, F, A G) be a type-2 fuzzy information system, and

[mathematical expression not reproducible], (30)

where [R.sub.B] is a crisp relation determined by attribute set B. If [L.sub.B] (x | [alpha]) = [L.sub.A] (x | [alpha]) for any [alpha] [member of] [0,1] and x [member of] U, then B is called lower approximation consistent set of A.

If B [subset or equal to] 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, A G) be a type-2 fuzzy information system; for any [alpha] [member of] [0,1] and x, j [member of] U,

[mathematical expression not reproducible] (31)

is called discernibility matrix of (U, A, F, A G).

Theorem 29. Let (U, A, F, A G) be the type-2 fuzzy information system. For any x, y [member of] U and [alpha] [member of] [0,1], B is the lower approximation consistent set of A if and only if [mathematical expression not reproducible].

Proof. If [mathematical expression not reproducible].

In the following, we only should prove [L.sub.A] (x | [alpha]) = [L.sub.A] (y | [alpha]).

For any [mathematical expression not reproducible]. Similarly, we have [mathematical expression not reproducible].

Since B is the lower approximation consistent set of A, we can obtain

[mathematical expression not reproducible]. (32)

Hence, [mathematical expression not reproducible]. That is, [L.sub.A] (x | [alpha]) = [L.sub.A] (y | [alpha]). That is to say, there must be [mathematical expression not reproducible] which hold for [L.sub.A] (x | [alpha]) [not equal to] [L.sub.A] (y | [alpha]).

"[??]" Since [mathematical expression not reproducible] forms a partition of [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (33)

If [L.sub.A] (x | [alpha]) [not equal to] [L.sub.A] (y | [alpha]), then [mathematical expression not reproducible]. If [mathematical expression not reproducible].

Furthermore, we can obtain [mathematical expression not reproducible].

Thus,

[mathematical expression not reproducible]. (34)

Similarly, [mathematical expression not reproducible]. That is, [L.sub.B] (x | [alpha]) = [L.sub.A] (x | [alpha]). 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 [member of] U and [alpha] [member of] [0,1], B is the lower approximation consistent set of A if and only if [D.sub.xy] [not equal to] 0 [??] B [intersection] [D.sub.xy] [not equal to] 0.

Proof. "[??]" If B is the lower approximation consistent set of A and [D.sub.xy] [not equal to] 0, then we have [L.sub.A] (x | [alpha]) [not equal to] [L.sub.A] (y | [alpha]). Thus, we can obtain [mathematical expression not reproducible] from Theorem 29. That is, there exists [a.sub.i] [member of] B such that [f.sub.i] (x) [not equal to] [f.sub.i] (y). Hence, B [intersection] [D.sub.xy] = 0.

"[??]" For any [alpha] [member of] [0, 1] and x, y [member of] U, if [L.sub.A] (x | [alpha]) [not equal to] [L.sub.A] (y | a) holds, in the following, we only should prove [mathematical expression not reproducible].

If [D.sub.xy] [not equal to] 0, then B [intersection] [D.sub.xy] [not equal to] 0. That is, there exists [a.sub.i] [member of] B such that [f.sub.i] (x) [not equal to] [f.sub.i] (y). Thus, [mathematical expression not reproducible]. That is to say, [mathematical expression not reproducible].

We have that B is the lower approximation consistent set of A from Theorem 29.

A discernibility function f(S) for S = (U, A, F, D, G) is a Boolean function of m Boolean variables [bar.[a.sub.1]], ..., [bar.[a.sub.m]] corresponding to the attributes [a.sub.1], ..., [a.sub.m], respectively, and is defined as follows:

f(S)([bar.[a.sub.1]], ..., [bar.[a.sub.m]]) = [conjunction] {[disjunction] ([D.sub.xy]) : [D.sub.xy] [not equal to] 0}, (35)

where [disjunction] ([D.sub.xy]) is the disjunction of all variables [bar.a] such that a [member of] [D.sub.xy]. In the sequel, we write [a.sub.i] instead of [bar.[a.sub.i]] when no confusion can arise.

Theorem 31. Suppose B [subset or equal to] A; then B is a lower approximation reduction of A if and only if B is the minimal set satisfying [D.sub.xy] [not equal to] 0 [??] B [intersection] [D.sub.xy] [not equal to] 0.

Proof. The proof is straightforward by Theorem 30.

Let g(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 I and [A.sub.k] [subset or equal to] A for k = 1, ..., l such that g(S) = ([conjunction] [A.sub.1]) [disjunction] ... [disjunction] ([A.sub.1], ..., [A.sub.l]), where every element in [A.sub.1] only appears one time. We have the following theorem.

Theorem 32. If Red(S) is the collection of all lower approximation reductions of A, then Red(S) = [[A.sub.1], ..., [A.sub.l]}.

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 many [alpha]-planes will be used. Call that number [DELTA] + 1. Regardless of [DELTA] + 1, [alpha] = 0 and [alpha] = 1 must always be used.

Step 2. For every [alpha], [x.sub.i] (1 [less than or equal to] i [less than or equal to] n), and [d.sub.j] (1 [less than or equal to] j [less than or equal to] p), compute Sp [mathematical expression not reproducible].

Step 3. For every [alpha], [x.sub.i] (1 [less than or equal to] i [less than or equal to] n), compute [L.sub.A] ([x.sub.i] | [alpha]).

Step 4. Compute [D.sub.xy] using Definition 28.

Step 5. Define f(S) = [conjunction] {[disjunction] ([D.sub.xy]) : [D.sub.xy] [not equal to] 0}.

Step 6. Compute g(S) = ([conjunction] [A.sub.1]) [disjunction] ... [disjunction] ([conjunction] [A.sub.l]) by f(S).

Step 7. Output all lower approximation reductions {[A.sub.1], ..., [A.sub.l]}.

Example 33. Table 1 gives a type-2 fuzzy information system, where the universe is U = {[x.sub.1], [x.sub.2], ..., [x.sub.8]}, the condition attribute set is A = {[a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4]}, and the decision attribute set is D = {[d.sub.1], [d.sub.2]}, where [d.sub.i] [member of] [F.sub.2] (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 universe U can be divided into three basic equivalence classes in accordance with the condition attribute set A = {[a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4]}:

U/[R.sub.A] = {[X.sub.1], [X.sub.2], [X.sub.3]} = {{[x.sub.1], [x.sub.3], [x.sub.8]}, {[x.sub.2], [x.sub.4], [x.sub.6]}, {[x.sub.5], [x.sub.7]}}. (36)

In the following, we should decide on how many [alpha]-planes will be used, where [alpha] [member of] [0,1]. Call that number [DELTA] + 1. Regardless of [DELTA] + 1, [alpha] = 0 and [alpha] = 1 must always be used. If [DELTA] + 1 = 11, then [alpha] = 0, 0.1, 0.2, 03, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. When [alpha] = 0.5, the approximation results of the type-2 fuzzy object are summarized in Table 2.

Similarly, we can compute the other approximation results for [alpha] [not equal to] 0.5. The discernibility matrix of (U, A, F, D, G) is presented in Table 3.

It is easy to compute {[a.sub.2], [a.sub.4]} and {[a.sub.2], [a.sub.3]} as the lower approximation reductions of (U, A, F, D, G).

6. Conclusions

The [alpha]-plane representation method of type-2 fuzzy sets has been extensively studied and can greatly reduce the computational workload. Based on a-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.

http://dx.doi.org/10.1155/2016/4819353

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Tao Zhao and Zhenbo Wei

School of Electrical Engineering and Information, Sichuan University, Chengdu 610065, China

Correspondence should be addressed to Zhenbo Wei; zhengbo_wei@126.com

Received 19 October 2015; Accepted 31 December 2015

```TABLE 1: A type-2 fuzzy information system.

Object      [a.sub.1]   [a.sub.2]   [a.sub.3]   [a.sub.4]

[x.sub.1]       1           2           3           1
[x.sub.2]       1           2           1           3
[x.sub.3]       1           2           3           1
[x.sub.4]       1           2           1           3
[x.sub.5]       1           3           1           2
[x.sub.6]       1           2           1           3
[x.sub.7]       1           3           1           2
[x.sub.8]       1           2           3           1

Object             [d.sub.1]                  [d.sub.2]

[x.sub.1]   T (0.7, 0.75, 0.8, 0.85)    T (0.2, 0.3, 0.4, 0.6)
[x.sub.2]      S (0.5, 0.6, 0.8)       T (0.2, 0.7, 0.8, 0.96)
[x.sub.3]   T (0.57, 0.7, 0.8, 0.85)      S (0.58, 0.7, 0.8)
[x.sub.4]     T (0.8, 0.85, 0.9,1)       S (0.86, 0.9, 0.96)
[x.sub.5]      S (0.8, 0.9, 0.95)      T (0.6, 0.75, 0.85, 0.9)
[x.sub.6]      S (0.4, 0.8, 0.95)         S (0.7, 0.86, 0.9)
[x.sub.7]    T (0.75, 0.85, 0.9,1)       S (0.75, 0.8, 0.96)
[x.sub.8]        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.sub.A]   [mathematical    [mathematical    [mathematical
expression not   expression not   expression not
reproducible]    reproducible]    reproducible]

[x.sub.1]         0.6350           0.8250           0.2500
[x.sub.2]         0.5500           0.7000           0.4500
[x.sub.3]         0.7800           0.8800           0.5600

U/[R.sub.A]   [mathematical
expression not
reproducible]

[x.sub.1]         0.5000
[x.sub.2]         0.8800
[x.sub.3]         0.8750

TABLE 3: The discernibility matrix.

Object                  [x.sub.1]                      [x.sub.2]

[x.sub.1]                   0                    {[a.sub.3], [a.sub.4]}
[x.sub.2]         {[a.sub.3], [a.sub.4]}                   0
[x.sub.3]                   0                    {[a.sub.3], [a.sub.4]}
[x.sub.4]         {[a.sub.3], [a.sub.4]}                   0
[x.sub.5]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.2], [a.sub.4]}
[x.sub.6]         {[a.sub.3], [a.sub.4]}                   0
[x.sub.7]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.2], [a.sub.4]}
[x.sub.8]                   0                    {[a.sub.3], [a.sub.4]}

Object                  [x.sub.3]                      [x.sub.4]

[x.sub.1]                   0                    {[a.sub.3], [a.sub.4]}
[x.sub.2]         {[a.sub.3], [a.sub.4]}                   0
[x.sub.3]                   0                    {[a.sub.3], [a.sub.4]}
[x.sub.4]         {[a.sub.3], [a.sub.4]}                   0
[x.sub.5]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.2], [a.sub.4]}
[x.sub.6]         {[a.sub.3], [a.sub.4]}                   0
[x.sub.7]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.2], [a.sub.4]}
[x.sub.8]                   0                    {[a.sub.3], [a.sub.4]}

Object                  [x.sub.5]                      [x.sub.6]

[x.sub.1]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.3], [a.sub.4]}
[x.sub.2]         {[a.sub.2], [a.sub.4]}                   0
[x.sub.3]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.3], [a.sub.4]}
[x.sub.4]         {[a.sub.2], [a.sub.4]}                   0
[x.sub.5]                   0                    {[a.sub.2], [a.sub.4]}
[x.sub.6]         {[a.sub.2], [a.sub.4]}                   0
[x.sub.7]                   0                    {[a.sub.2], [a.sub.4]}
[x.sub.8]   {[a.sub.2], [a.sub.3], [a.sub.4]}    {[a.sub.3], [a.sub.4]}

Object                  [x.sub.7]

[x.sub.1]   {[a.sub.2], [a.sub.3], [a.sub.4]}
[x.sub.2]         {[a.sub.2], [a.sub.4]}
[x.sub.3]   {[a.sub.2], [a.sub.3], [a.sub.4]}
[x.sub.4]         {[a.sub.2], [a.sub.4]}
[x.sub.5]                   0
[x.sub.6]         {[a.sub.2], [a.sub.4]}
[x.sub.7]                   0
[x.sub.8]   {[a.sub.2], [a.sub.3], [a.sub.4]}

Object                  [x.sub.8]

[x.sub.1]                   0
[x.sub.2]         {[a.sub.3], [a.sub.4]}
[x.sub.3]                   0
[x.sub.4]         {[a.sub.3], [a.sub.4]}
[x.sub.5]   {[a.sub.2], [a.sub.3], [a.sub.4]}
[x.sub.6]         {[a.sub.3], [a.sub.4]}
[x.sub.7]   {[a.sub.2], [a.sub.3], [a.sub.4]}
[x.sub.8]                   0
```