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Int. J. Fuzzy Log. Intell. Syst. 2015; 15(3): 208-215

Published online September 30, 2015

https://doi.org/10.5391/IJFIS.2015.15.3.208

© The Korean Institute of Intelligent Systems

Intuitionistic Fuzzy Rough Approximation Operators

Sang Min Yun, and Seok Jong Lee

Department of Mathematics, Chungbuk National University, Cheongju, Korea

Correspondence to :
Seok Jong Lee (sjl@cbnu.ac.kr)

Received: July 3, 2015; Revised: September 20, 2015; Accepted: September 24, 2015

Since upper and lower approximations could be induced from the rough set structures, rough sets are considered as approximations. The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations by Dubois and Prade. In this paper, we introduce and investigate some properties of intuitionistic fuzzy rough approximation operators and intuitionistic fuzzy relations by means of topology.

Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy approximation space

A Chang’s fuzzy topology [1] is a crisp subfamily of fuzzy sets, and hence fuzziness in the notion of openness of a fuzzy set has not been considered, which seems to be a drawback in the process of fuzzification of the concept of topological spaces. In order to give fuzziness of the fuzzy sets, ?oker [2] introduced intuitionistic fuzzy topological spaces using the idea of intuitionistic fuzzy sets which was proposed by Atanassov [3]. Also ?oker and Demirci [4] defined intuitionistic fuzzy topological spaces in ?ostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces. Since then, many researchers [5?9] investigated such intuitionistic fuzzy topological spaces.

On the other hand, the theory of rough sets was proposed by Z. Pawlak [10]. It is a new mathematical tool for the data reasoning, and it is an extension of set theory for the research of intelligent systems characterized by insufficient and incomplete informations. The fundamental structure of rough set theory is an approximation space. Based on rough set theory, upper and lower approximations could be induced. By using these approximations, knowledge hidden in information systems may be exposed and expressed in the form of decision rules(see [10, 11]). The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations by Dubois and Prade [12]. The relations between fuzzy rough sets and fuzzy topological spaces have been studied in some papers [13?15].

The main interest of this paper is to investigate characteristic properties of intuitionistic fuzzy rough approximation operators and intuitionistic fuzzy relations by means of topology. We prove that the upper approximation of a set is the set itself if and only if the set is a lower set whenever the intuitionistic fuzzy relation is reflexive. Also we have the result that if an intuitionistic fuzzy upper approximation operator is a closure operator or an intuitionistic fuzzy lower approximation operator is an interior operator in the intuitionistic fuzzy topology, then the order is an preorder.

Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair

A=(μA,νA)

where the functions μA : XI and νA : XI denote the degree of membership and the degree of nonmembership respectively and μA + νA ≤ 1(see [3]). Obviously, every fuzzy set μ in X is an intuitionistic fuzzy set of the form (μ, 1? ? μ).

Throughout this paper, ‘IF’ stands for ‘intuitionistic fuzzy.’ I ? I denotes the family of all intuitionistic fuzzy numbers (a, b) such that a, b ∈ [0, 1] and a + b ≤ 1, with the order relation defined by

(a,b)(c,d)?iff?ac?and?bd.

For any IF set A = (μA, νA) of X, the value

πA(x)=1-μA(x)-νA(x)

is called an indeterminancy degree(or hesitancy degree) of x to A(see [3]). Szmidt and Kacprzyk call πA(x) an intuitionistic index of x in A(see [16]). Obviously

0πA(x)1,?????????xX.

Note πA(x) = 0 iff νA(x) = 1 ? μA(x). Hence any fuzzy set μA can be regarded as an IF set (μA, νA) with πA = 0.

IF(X) denotes the family of all intuitionistic fuzzy sets in X, and cIF(X) denotes the family of all intuitionistic fuzzy sets in X with constant hesitancy degree, i.e., if A ∈ cIF(X), then πA = c for some constant c ∈ [0, 1). When we process basic operations on IF(X), we do as in [3].

Definition 2.1

( [2, 17]) Any subfamily of IF(X) is called an intuitionistic fuzzy topology on X in the sense of Lowen ( [18]), if

  • (1) for each (a, b) ∈ I ? I, (a,b)?T,

  • (2) A, B implies AB,

  • (3) {Aj | jJ} ⊆ implies ?jJAj.

The pair (X, ) is called an intuitionistic fuzzy topological space. Every member of is called an intuitionistic fuzzy open set in X. Its complement is called an intuitionistic fuzzy closed set in X. We denote = {A ∈ IF(X) | AC }. The interior and closure of A denoted by int(A) and cl(A) respectively for each A ∈ IF(X) are defined as follows:

int(A)?or?intT(A)=?{BT?BA},cl(A)?or?clT(A)=?{BTC?AB}.

An IF topology is called an Alexandrov topology [19] if (2) in Definition 2.1 is replaced by

{Aj?jJ}T?implies??jJAjT.

Definition 2.2

( [20]) An IF set R on X × X is called an intuitionistic fuzzy relation on X. Moreover, R is called

  • (i) reflexive if R(x, x) = (1, 0) for all xX,

  • (ii) symmetric if R(x, y) = R(y, x) for all x, yX,

  • (iii) transitive if R(x, y) ∧ R(y, z) ≤ R(x, z) for all x, y, zX,

A reflexive and transitive IF relation is called an intuitionistic fuzzy preorder. A symmetric IF preorder is called an intuitionistic fuzzy equivalence. An IF preorder on X is called an intuitionistic fuzzy partial order if for any x, yX, R(x, y) = R(y, x) = (1, 0) implies that x = y.

Let R be an IF relation on X. R?1 is called the inverse relation of R if R?1(x, y) = R(y, x) for any x, yX. Also, RC is called the complement of R if RC (x, y) = (νR(x,y), μR(x,y)) for any x, yX when R(x, y) = (μR(x,y), νR(x,y)). It is obvious that R?1 ≠ = RC.

Definition 2.3

( [21]) Let R be an IF relation on X. The pair (X, R) is called an intuitionistic fuzzy approximation space. The intuitionistic fuzzy lower approximation of A ∈ IF(X) with respect to (X, R), denoted by R(A), is defined as follows:

R_(A)(x)=?yX(RC(x,y)A(y)).

Similarly, the intuitionistic fuzzy upper approximation of A ∈ IF(X) with respect to (X, R), denoted by R?(A), is defined as follows:

R?(A)(x)=?yX(R(x,y)A(y)).

The pair (R(A), R?(A)) is called the intuitionistic fuzzy rough set of A with respect to (X, R).

R : IF(X) → IF(X) and R? : IF(X) → IF(X) are called the intuitionistic fuzzy lower approximation operator and the intuitionistic fuzzy upper approximation operator, respectively. In general, we refer to R and R? as the intuitionistic fuzzy rough approximation operators.

Proposition 2.4

( [17, 21]) Let (X, R) be an IF approximation space. Then for any A, B ∈ IF(X), {Aj | jJ} ⊆ IF(X) and (a, b) ∈ I ? I,

  • R_((1,0)?)=(1,0)?,?????????R?((0,1)?)=(0,1)?,

  • ABR(A) ⊆ R(B), R?(A) ⊆ R?(B),

  • R(AC) = (R?(A))C, R?(AC) = (R(A))C,

  • R(AB) = R(A) ∩ R(B), R?(AB) = R?(A) ∪ R?(B),

  • R(∩jJAj) = ∩jJ (R(Aj)), R?(∪jJAj) = ∪jJ (R?(Aj)),

  • R_((a,b)?A)=(a,b)?R_(A),R?((a,b)?A)=(a,b)?R?(A).

Remark 2.5

Let (X, R) be an IF approximation space. Then

R?(x(1,0))(y)=?zX(R(y,z)x(1,0)(z))=R(y,x),R_(x(1,0)C)(y)=?zX(R(C)(y,z)x(1,0)C(z))=R(C)(y,x).

Let (X, R) be an IF approximation space. (X, R) is called areflexive(resp., preordered) intuitionistic fuzzy approximation space, if R is a reflexive intuitionistic fuzzy relation(resp., an intuitionistic fuzzy preorder). If R is an intuitionistic fuzzy partial order, then (X, R) is called a partially ordered intuitionistic fuzzy approximation space. An intuitionistic fuzzy preorder R is called an intuitionistic fuzzy equality, if R is both an intuitionistic fuzzy equivalence and an intuitionistic fuzzy partial order.

Theorem 2.6

( [17, 21]) Let (X, R) be an IF approximation space. Then

  • R is reflexive

    AIF(X),???R_(A)AAIF(X),???AR?(A).

  • R is transitive

    AIF(X),???R_(A)R_(R_(A))AIF(X),???R?(R?(A))R?(A).

Definition 3.1

( [22]) Let (X, R) be an IF approximation space. Then A ∈ IF(X) is called an intuitionistic fuzzy upper set in (X, R) if

A(x)R(x,y)A(y),?????????x,yX.

Dually, A is called an intuitionistic fuzzy lower set in (X, R) if A(y) ∧ R(x, y) ≤ A(x) for all x, yX.

Let R be an IF preorder on X. For x, yX, the real number R(x, y) can be interpreted as the degree to which ‘xy’ holds true. The condition A(x) ∧ R(x, y) ≤ A(y) can be interpreted as the statement that if x is in A and xy, then y is in A. Particularly, if R is an IF equivalence, then an IF set A is an upper set in (X, R) if and only if it is a lower set in (X, R).

The classical preorder xy can be naturally extended to R(x, y) = (1, 0) in an IF preorder. Obviously, the notion of IF upper sets and IF lower sets agrees with that of upper sets and lower sets in classical preordered space.

Proposition 3.2

Let (X, R) be an IF approximation space and A ∈ IF(X). Then the following are equivalent:

  • R?(A) ⊆ A.

  • A is a lower set in (X, R).

  • A is an upper set in (X, R?1).

Proof

(1) ⇒ (2). Suppose that R?(A) ⊆ A. Since for each xX,

?yX(A(y)R(x,y))=R?(A)(x)A(x),

we have

A(y)R(x,y)A(x).

Thus A is a lower set in (X, R).

(2) ⇒ (3). This is obvious.

(3)⇒(1). Suppose that A is an upper set in (X, R?1). Then for any x, yX, A(x) ∧ R?1 (x, y) ≤ A(y). So A(x) ∧ R(y, x) ≤ A(y). Thus

R?(A)(y)=?xX(A(x)R(y,x))A(y).

Hence R?(A) ⊆ A.

Corollary 3.3

Let (X, R) be an IF approximation space and A ∈ IF(X). If R is reflexive, then the following are equivalent:

  • R?(A) = A.

  • A is a lower set in (X, R).

  • A is an upper set in (X, R?1).

Proof

This holds by Theorem 2.6 and Proposition 3.2.

Proposition 3.4

Let (X, R) be an IF approximation space and A ∈ IF(X). Then the following are equivalent:

  • R?(A) ⊇ A.

  • AC is a lower set in (X, R).

  • AC is an upper set in (X, R?1).

Proof

(1) ⇒ (2). Suppose that R?(A) ⊇ A. Since for each xX,

?yX(A(y)RC(x,y))=R_(A)(x)A(x),

we have

A(y)RC(x,y)A(x),AC(y)R(x,y)AC(x).

Thus AC is a lower set in (X, R).

(2) ⇒ (3). This is obvious.

(3) ⇒ (1). Suppose that AC is an upper set in (X, R?1).

Then for any x, yX, AC(x) ∧ R?1(x, y) ≤ AC(y). So AC(x) ∧ R(y, x) ≤ AC(y). Thus

A(x)RC(y,x)A(y),x,yX.

So

R_(A)(y)=?xX(A(x)RC(y,x))A(y).

Hence R(A) ⊇ A.

Corollary 3.5

Let (X, R) be an IF approximation space and A ∈ IF(X). If R is reflexive, then the following are equivalent:

  • R(A) = A.

  • AC is a lower set in (X, R).

  • AC is an upper set in (X, R?1).

Proof

This holds by Theorem 2.6 and the above proposition.

For each zX, we define IF sets [z]R : XI ? I by [z]R(x) = R(z, x), and [z]R : XI ? I by [z]R(x) = R(x, z).

Theorem 3.6

Let (X, R) be an IF approximation space. Then

  • (1) R is reflexive

    xX,???[x]R(x)=(1,0)xX,???[x]R(x)=(1,0).

  • R is symmetric

    xX,???[x]R=[x]RAIF(X),???A?is?a?lower?set?iff?A?is?an?upper?set.

  • R is transitive

    xX,???[x]R?is?a?lower?set?in?(X,R)xX,???[x]R?is?an?upper?set?in?(X,R)AIF(X),???R?(A)?is?a?lower?set?in?(X,R).

Proof

(1) and (2) are obvious. (3) By Proposition 3.2,

A ∈ IF(X), R?(A) is a lower set

AIF(X),???R?(R?(A))R?(A)R?is?transitivex,y,zX,???R(x,y)R(y,z)R(x,z)x,y,zX,???R(x,y)[z]R(y)[z]R(x)xX,???[x]R?is?a?lower?set.

Also,

R?(A) is a lower set

x,y,zX,???R(x,y)R(y,z)R(x,z)x,y,zX,???[x]R(y)R(y,z)[x]R(z)xX,???[x]R?is?an?upper?set.

Proposition 3.7

Let (X, R) be an IF approximation space. Then

R is symmetric

(x,y)X×X,???R_(x(1,0)C)(y)=R_(y(1,0)C)(x)(x,y)X×X,???R?(x(1,0))(y)=R?(y(1,0))(x).
Proof

By Remark 2.5, R_(x(1,0)C)(y)=RC(y,x)=RC(x,y)=R_(y(1,0)C)(x), because R is symmetric. Similarly we have that R?(x(1,0))(y) = R(y, x) = R(x, y) = R?(y(1,0))(x).

Theorem 3.8

Let R be an IF relation on X and let be an IF topology on X. If one of the following conditions is satisfied, then R is an IF preorder.

  • R? is a closure operator of .

  • R? is an interior operator of .

Proof

Suppose that satisfies (1). By Remark 2.5, R?(x(1,0)) (y) = R(y, x) for each xX. Since R? is a closure operator of , for each xX,

R(x,x)=R?(x(1,0))(x)=clT(x(1,0))(x)(x(1,0))(x)=(1,0).

Thus R is reflexive. For any x, y, zX, let cl (z(1,0)) (y) = (a, b). Then by Remark 2.5 and Proposition 2.4,

R(x,y)R(y,z)=R?(y(1,0))(x)R?(z(1,0))(y)=R?(y(1,0))(x)clT(z(1,0))(y)=R?(y(1,0))(x)(a,b)=R?((a,b)y(1,0))(x)=clT((a,b)y(1,0))(x)=clT(clT(z(1,0))(y)y(1,0))(x)clT(?yX[clT(z(1,0))(y)y(1,0)])(x)=clT(clT(z(1,0)))(x)=clT(z(1,0))(x)=R(x,z).

Hence R is transitive. Therefore R is an IF preorder.

Similarly we can prove for the case of (2).

Definition 3.9

For each A ∈ IF(X), we define

RA={(x,y)X×X?A(x)>A(y)}.

Obviously, RA = ?? iff A=(a,b)? for some (a, b) ∈ I ? I or A(x) and A(y) are non-comparable for all x, yX.

Proposition 3.10

Let (X, R) be an IF approximation space. Let A be an IF set with constant hesitancy degree, i.e., A ∈ cIF(X) with RA = ??. Then we have

  • (1) R(A) ⊇ ARC(x, y) ≥ A(x) ∨ A(y) for all (x, y) ∈ RA,

  • (2) R?(A) ⊆ AR(y, x) ≤ A(x)∧A(y) for all (x, y) ∈ RA.

Proof

(1) (⇒) Suppose that R(A) ⊇ A. Note that for each xX,

?yX(A(y)RC(x,y))=R_(A)(x)A(x).

Then A(y) ∨ RC(x, y) ≥ A(x) for any x, yX. Since A(x) > A(y) for each (x, y) ∈ RA, we have

RC(x,y)A(x)=A(x)A(y)?for?all?(x,y)RA.

(?) Suppose that for each (x, y) ∈ RA, RC(x, y) ≥ A(x) ∨ A(y). Let zX.

  • (i) If A(z) > A(y), then

    A(y)RC(z,y)A(y)(A(z)A(y))A(z).

  • (ii) If A(z) ≤ A(y), then

    A(y)RC(z,y)A(y)(A(z)A(y))A(y)A(z).

Hence R(A)(z) = ∧yX (A(y) ∨ RC (z, y)) ≥ A(z) for any zX. Thus R(A) ⊇ A.

(2) (⇒) Suppose that R?(A) ⊆ A. Note that for each yX,

?xX(A(x)R(y,x))=R?(A)(y)A(y).

Then A(x) ∧R(y, x) ≤ A(y) for any x, yX. Since A(x) > A(y) for each (x, y) ∈ RA, we have

R(y,x)A(y)=A(x)A(y).

(?) Suppose that for any (x, y) ∈ RA, R(y, x) ≤ A(x)∧A(y).

Let zX.

  • (i) If A(x) > A(z), then

    A(x)R(z,x)A(x)(A(x)A(z))A(z).

  • (ii) If A(x) ≤ A(z), then

    A(x)R(z,x)A(x)(A(x)A(z))A(x)A(z).

Thus R?(A)(z) = ∧xX (A(x) ∧ R(z, x)) ≤ A(z). Hence R?(A) ⊆ A.

Corollary 3.11

Let (X,R) be a reflexive IF approximation space. Then for each A ∈ cIF(X) with RA ≠ = ??,

  • R(A) = ARC(x, y) ≥ A(x) ∨ A(y) for all (x, y) ∈ RA,

  • R?(A) = AR(y, x) ≤ A(x)∧A(y) for all (x, y) ∈ RA.

Proof

By the above proposition and the reflexivity of R, it can be easily proved.

Let R1 and R2 be two IF relations on X. We denote R1R2 if R1(x, y) ≤ R2(x, y) for any x, yX. And R1 = R2 if R1R2 and R2R1.

Proposition 3.12

Let (X, R1) and (X, R2) be two IF approximation spaces. Then for each A ∈ IF(X),

  • R1R2R1?(A)R2?(A) and R1(A) ⊇ R2(A).

  • (R1R2)?(A)=R1?(A)R2?(A), (R1R2)(A) = R1(A) ∩ R2(A).

Proof

(1) For each xX,

R1?(A)(x)=?yX(A(y)(R1)(x,y))?yX(A(y)(R2)(x,y))=R2?(A)(x).

Thus we have R1?(A)R2?(A). Dually,

R1?(AC)R2?(AC)(R1?(AC))C(R2?(AC))CR1_(A)R2_(A).

(2) For each xX,

(R1R2)?(A)(x)=?yX(A(y)(R1R2)(x,y))=?yX(A(y)(R1(x,y)R2(x,y)))=?yX((A(y)R1(x,y))(A(y)R2(x,y)))(?yX(A(y)R1(x,y)))(?yX(A(y)R2(x,y)))=R1?(A)(x)R2?(A)(x)=(R1?(A)R2?(A))(x).

Thus we have (R1R2)?(A)R1?(A)R2?(A). Moreover, since R1R1R2 and R2R1R2, we have R1?(A)(R1R2)?(A)

and R1?(A)(R1R2)?(A). Thus R1?(A)R2?(A)(R1R2)?(A). Hence we have (R1R2)?(A)=R1?(A)R2?(A). By Proposition 2.4,

R1_(A)R2_(A)=(R1?(AC))C(R2?(AC))C=(R1?(AC)R2?(AC))C=((R1R2)?(AC))C=(R1R2)_(A).

Proposition 3.13

Let (X, R1) and (X, R2) be two reflexive IF approximation spaces. Then for each A ∈ IF(X),

  • R2 (R1(A)) ⊆ (R1R2)(A) and R1 (R2(A)) ⊆ (R1R2)(A).

  • R2?(R1?(A))(R1R2)?(A) and R1?(R2?(A))(R1R2)?(A).

Proof

(1) By Theorem 2.6, R2 (R1(A)) ⊆ R2(A) and R2 (R1(A)) ⊆ R1(A). Thus we have

R2_(R1_(A))R1_(A)R2_(A)(R1R2)_(A).

Similarly, we can prove that R1(R2(A)) ⊆ (R1R2)(A).

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

Proposition 3.14

Let (X, R1) and (X, R2) be two IF approximation spaces. If R1 is reflexive, R2 is transitive and R1R2, then

R1_(R2_(A))=R2_(A)?and?R1?(R2?(A))=R2?(A).
Proof

By Theorem 2.6, R1?(R2?(A))R2?(A). For each xX, by R1R2 and the transitivity of R2, we have

R1?(R2?(A))(x)=?yX(R2?(A)(y)R1(x,y))=?yX((?zX(A(z)R2(y,z))))R1(x,y)=?yX(?zX((A(z)R2(y,z))R1(x,y)))=?yX(?zX(A(z)(R2(y,z)R1(x,y))))?yX(?zX(A(z)(R2(y,z))R2(x,y)))?yX(?zX(A(z)R2(x,z)))=?zX(A(z)R2(x,z))=R2?(A)(x).

Thus R1?(R2?(A))R2?(A). So R1?(R2?(A))=R2?(A). By Proposition 2.4,

R1_(R2_(A))=R1_((R2?(AC))C)=(R1?(R2?(AC)))C=(R2?(AC))C=R2_(A).

We obtained characteristic properties of intuitionistic fuzzy rough approximation operator and intuitionistic fuzzy relation by means of topology. Particularly, we proved that the upper approximation of a set is the set itself if and only if the set is a lower set whenever the intuitionistic fuzzy relation is reflexive. Also we had the result that if an intuitionistic fuzzy upper approximation operator is a closure operator or an intuitionistic fuzzy lower approximation operator is an interior operator in the intuitionistic fuzzy topology, then the order is an preorder.

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Sang Min Yun received the Ph. D. degree from Chungbuk National University in 2015. His research interests include general topology and fuzzy topology. He is a member of KIIS and KMS.

E-mail: jivesm@naver.com

Seok Jong Lee received the M. S. and Ph. D. degrees from Yonsei University in 1986 and 1990, respectively. He is a professor at the Department of Mathematics, Chungbuk National University since 1989. He was a visiting scholar in Carleton University from 1995 to 1996, and Wayne State University from 2003 to 2004. His research interests include general topology and fuzzy topology. He is a member of KIIS, KMS, and CMS.

E-mail: sjl@cbnu.ac.kr

Article

Original Article

Int. J. Fuzzy Log. Intell. Syst. 2015; 15(3): 208-215

Published online September 30, 2015 https://doi.org/10.5391/IJFIS.2015.15.3.208

Copyright © The Korean Institute of Intelligent Systems.

Intuitionistic Fuzzy Rough Approximation Operators

Sang Min Yun, and Seok Jong Lee

Department of Mathematics, Chungbuk National University, Cheongju, Korea

Correspondence to:Seok Jong Lee (sjl@cbnu.ac.kr)

Received: July 3, 2015; Revised: September 20, 2015; Accepted: September 24, 2015

Abstract

Since upper and lower approximations could be induced from the rough set structures, rough sets are considered as approximations. The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations by Dubois and Prade. In this paper, we introduce and investigate some properties of intuitionistic fuzzy rough approximation operators and intuitionistic fuzzy relations by means of topology.

Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy approximation space

1. Introduction

A Chang’s fuzzy topology [1] is a crisp subfamily of fuzzy sets, and hence fuzziness in the notion of openness of a fuzzy set has not been considered, which seems to be a drawback in the process of fuzzification of the concept of topological spaces. In order to give fuzziness of the fuzzy sets, ?oker [2] introduced intuitionistic fuzzy topological spaces using the idea of intuitionistic fuzzy sets which was proposed by Atanassov [3]. Also ?oker and Demirci [4] defined intuitionistic fuzzy topological spaces in ?ostak’s sense as a generalization of smooth topological spaces and intuitionistic fuzzy topological spaces. Since then, many researchers [5?9] investigated such intuitionistic fuzzy topological spaces.

On the other hand, the theory of rough sets was proposed by Z. Pawlak [10]. It is a new mathematical tool for the data reasoning, and it is an extension of set theory for the research of intelligent systems characterized by insufficient and incomplete informations. The fundamental structure of rough set theory is an approximation space. Based on rough set theory, upper and lower approximations could be induced. By using these approximations, knowledge hidden in information systems may be exposed and expressed in the form of decision rules(see [10, 11]). The concept of fuzzy rough sets was proposed by replacing crisp binary relations with fuzzy relations by Dubois and Prade [12]. The relations between fuzzy rough sets and fuzzy topological spaces have been studied in some papers [13?15].

The main interest of this paper is to investigate characteristic properties of intuitionistic fuzzy rough approximation operators and intuitionistic fuzzy relations by means of topology. We prove that the upper approximation of a set is the set itself if and only if the set is a lower set whenever the intuitionistic fuzzy relation is reflexive. Also we have the result that if an intuitionistic fuzzy upper approximation operator is a closure operator or an intuitionistic fuzzy lower approximation operator is an interior operator in the intuitionistic fuzzy topology, then the order is an preorder.

2. Preliminaries

Let X be a nonempty set. An intuitionistic fuzzy set A is an ordered pair

A=(μA,νA)

where the functions μA : XI and νA : XI denote the degree of membership and the degree of nonmembership respectively and μA + νA ≤ 1(see [3]). Obviously, every fuzzy set μ in X is an intuitionistic fuzzy set of the form (μ, 1? ? μ).

Throughout this paper, ‘IF’ stands for ‘intuitionistic fuzzy.’ I ? I denotes the family of all intuitionistic fuzzy numbers (a, b) such that a, b ∈ [0, 1] and a + b ≤ 1, with the order relation defined by

(a,b)(c,d)?iff?ac?and?bd.

For any IF set A = (μA, νA) of X, the value

πA(x)=1-μA(x)-νA(x)

is called an indeterminancy degree(or hesitancy degree) of x to A(see [3]). Szmidt and Kacprzyk call πA(x) an intuitionistic index of x in A(see [16]). Obviously

0πA(x)1,?????????xX.

Note πA(x) = 0 iff νA(x) = 1 ? μA(x). Hence any fuzzy set μA can be regarded as an IF set (μA, νA) with πA = 0.

IF(X) denotes the family of all intuitionistic fuzzy sets in X, and cIF(X) denotes the family of all intuitionistic fuzzy sets in X with constant hesitancy degree, i.e., if A ∈ cIF(X), then πA = c for some constant c ∈ [0, 1). When we process basic operations on IF(X), we do as in [3].

Definition 2.1

( [2, 17]) Any subfamily of IF(X) is called an intuitionistic fuzzy topology on X in the sense of Lowen ( [18]), if

  • (1) for each (a, b) ∈ I ? I, (a,b)?T,

  • (2) A, B implies AB,

  • (3) {Aj | jJ} ⊆ implies ?jJAj.

The pair (X, ) is called an intuitionistic fuzzy topological space. Every member of is called an intuitionistic fuzzy open set in X. Its complement is called an intuitionistic fuzzy closed set in X. We denote = {A ∈ IF(X) | AC }. The interior and closure of A denoted by int(A) and cl(A) respectively for each A ∈ IF(X) are defined as follows:

int(A)?or?intT(A)=?{BT?BA},cl(A)?or?clT(A)=?{BTC?AB}.

An IF topology is called an Alexandrov topology [19] if (2) in Definition 2.1 is replaced by

{Aj?jJ}T?implies??jJAjT.

Definition 2.2

( [20]) An IF set R on X × X is called an intuitionistic fuzzy relation on X. Moreover, R is called

  • (i) reflexive if R(x, x) = (1, 0) for all xX,

  • (ii) symmetric if R(x, y) = R(y, x) for all x, yX,

  • (iii) transitive if R(x, y) ∧ R(y, z) ≤ R(x, z) for all x, y, zX,

A reflexive and transitive IF relation is called an intuitionistic fuzzy preorder. A symmetric IF preorder is called an intuitionistic fuzzy equivalence. An IF preorder on X is called an intuitionistic fuzzy partial order if for any x, yX, R(x, y) = R(y, x) = (1, 0) implies that x = y.

Let R be an IF relation on X. R?1 is called the inverse relation of R if R?1(x, y) = R(y, x) for any x, yX. Also, RC is called the complement of R if RC (x, y) = (νR(x,y), μR(x,y)) for any x, yX when R(x, y) = (μR(x,y), νR(x,y)). It is obvious that R?1 ≠ = RC.

Definition 2.3

( [21]) Let R be an IF relation on X. The pair (X, R) is called an intuitionistic fuzzy approximation space. The intuitionistic fuzzy lower approximation of A ∈ IF(X) with respect to (X, R), denoted by R(A), is defined as follows:

R_(A)(x)=?yX(RC(x,y)A(y)).

Similarly, the intuitionistic fuzzy upper approximation of A ∈ IF(X) with respect to (X, R), denoted by R?(A), is defined as follows:

R?(A)(x)=?yX(R(x,y)A(y)).

The pair (R(A), R?(A)) is called the intuitionistic fuzzy rough set of A with respect to (X, R).

R : IF(X) → IF(X) and R? : IF(X) → IF(X) are called the intuitionistic fuzzy lower approximation operator and the intuitionistic fuzzy upper approximation operator, respectively. In general, we refer to R and R? as the intuitionistic fuzzy rough approximation operators.

Proposition 2.4

( [17, 21]) Let (X, R) be an IF approximation space. Then for any A, B ∈ IF(X), {Aj | jJ} ⊆ IF(X) and (a, b) ∈ I ? I,

  • R_((1,0)?)=(1,0)?,?????????R?((0,1)?)=(0,1)?,

  • ABR(A) ⊆ R(B), R?(A) ⊆ R?(B),

  • R(AC) = (R?(A))C, R?(AC) = (R(A))C,

  • R(AB) = R(A) ∩ R(B), R?(AB) = R?(A) ∪ R?(B),

  • R(∩jJAj) = ∩jJ (R(Aj)), R?(∪jJAj) = ∪jJ (R?(Aj)),

  • R_((a,b)?A)=(a,b)?R_(A),R?((a,b)?A)=(a,b)?R?(A).

Remark 2.5

Let (X, R) be an IF approximation space. Then

R?(x(1,0))(y)=?zX(R(y,z)x(1,0)(z))=R(y,x),R_(x(1,0)C)(y)=?zX(R(C)(y,z)x(1,0)C(z))=R(C)(y,x).

Let (X, R) be an IF approximation space. (X, R) is called areflexive(resp., preordered) intuitionistic fuzzy approximation space, if R is a reflexive intuitionistic fuzzy relation(resp., an intuitionistic fuzzy preorder). If R is an intuitionistic fuzzy partial order, then (X, R) is called a partially ordered intuitionistic fuzzy approximation space. An intuitionistic fuzzy preorder R is called an intuitionistic fuzzy equality, if R is both an intuitionistic fuzzy equivalence and an intuitionistic fuzzy partial order.

Theorem 2.6

( [17, 21]) Let (X, R) be an IF approximation space. Then

  • R is reflexive

    AIF(X),???R_(A)AAIF(X),???AR?(A).

  • R is transitive

    AIF(X),???R_(A)R_(R_(A))AIF(X),???R?(R?(A))R?(A).

3. IF Rough Approximation Operator

Definition 3.1

( [22]) Let (X, R) be an IF approximation space. Then A ∈ IF(X) is called an intuitionistic fuzzy upper set in (X, R) if

A(x)R(x,y)A(y),?????????x,yX.

Dually, A is called an intuitionistic fuzzy lower set in (X, R) if A(y) ∧ R(x, y) ≤ A(x) for all x, yX.

Let R be an IF preorder on X. For x, yX, the real number R(x, y) can be interpreted as the degree to which ‘xy’ holds true. The condition A(x) ∧ R(x, y) ≤ A(y) can be interpreted as the statement that if x is in A and xy, then y is in A. Particularly, if R is an IF equivalence, then an IF set A is an upper set in (X, R) if and only if it is a lower set in (X, R).

The classical preorder xy can be naturally extended to R(x, y) = (1, 0) in an IF preorder. Obviously, the notion of IF upper sets and IF lower sets agrees with that of upper sets and lower sets in classical preordered space.

Proposition 3.2

Let (X, R) be an IF approximation space and A ∈ IF(X). Then the following are equivalent:

  • R?(A) ⊆ A.

  • A is a lower set in (X, R).

  • A is an upper set in (X, R?1).

Proof

(1) ⇒ (2). Suppose that R?(A) ⊆ A. Since for each xX,

?yX(A(y)R(x,y))=R?(A)(x)A(x),

we have

A(y)R(x,y)A(x).

Thus A is a lower set in (X, R).

(2) ⇒ (3). This is obvious.

(3)⇒(1). Suppose that A is an upper set in (X, R?1). Then for any x, yX, A(x) ∧ R?1 (x, y) ≤ A(y). So A(x) ∧ R(y, x) ≤ A(y). Thus

R?(A)(y)=?xX(A(x)R(y,x))A(y).

Hence R?(A) ⊆ A.

Corollary 3.3

Let (X, R) be an IF approximation space and A ∈ IF(X). If R is reflexive, then the following are equivalent:

  • R?(A) = A.

  • A is a lower set in (X, R).

  • A is an upper set in (X, R?1).

Proof

This holds by Theorem 2.6 and Proposition 3.2.

Proposition 3.4

Let (X, R) be an IF approximation space and A ∈ IF(X). Then the following are equivalent:

  • R?(A) ⊇ A.

  • AC is a lower set in (X, R).

  • AC is an upper set in (X, R?1).

Proof

(1) ⇒ (2). Suppose that R?(A) ⊇ A. Since for each xX,

?yX(A(y)RC(x,y))=R_(A)(x)A(x),

we have

A(y)RC(x,y)A(x),AC(y)R(x,y)AC(x).

Thus AC is a lower set in (X, R).

(2) ⇒ (3). This is obvious.

(3) ⇒ (1). Suppose that AC is an upper set in (X, R?1).

Then for any x, yX, AC(x) ∧ R?1(x, y) ≤ AC(y). So AC(x) ∧ R(y, x) ≤ AC(y). Thus

A(x)RC(y,x)A(y),x,yX.

So

R_(A)(y)=?xX(A(x)RC(y,x))A(y).

Hence R(A) ⊇ A.

Corollary 3.5

Let (X, R) be an IF approximation space and A ∈ IF(X). If R is reflexive, then the following are equivalent:

  • R(A) = A.

  • AC is a lower set in (X, R).

  • AC is an upper set in (X, R?1).

Proof

This holds by Theorem 2.6 and the above proposition.

For each zX, we define IF sets [z]R : XI ? I by [z]R(x) = R(z, x), and [z]R : XI ? I by [z]R(x) = R(x, z).

Theorem 3.6

Let (X, R) be an IF approximation space. Then

  • (1) R is reflexive

    xX,???[x]R(x)=(1,0)xX,???[x]R(x)=(1,0).

  • R is symmetric

    xX,???[x]R=[x]RAIF(X),???A?is?a?lower?set?iff?A?is?an?upper?set.

  • R is transitive

    xX,???[x]R?is?a?lower?set?in?(X,R)xX,???[x]R?is?an?upper?set?in?(X,R)AIF(X),???R?(A)?is?a?lower?set?in?(X,R).

Proof

(1) and (2) are obvious. (3) By Proposition 3.2,

A ∈ IF(X), R?(A) is a lower set

AIF(X),???R?(R?(A))R?(A)R?is?transitivex,y,zX,???R(x,y)R(y,z)R(x,z)x,y,zX,???R(x,y)[z]R(y)[z]R(x)xX,???[x]R?is?a?lower?set.

Also,

R?(A) is a lower set

x,y,zX,???R(x,y)R(y,z)R(x,z)x,y,zX,???[x]R(y)R(y,z)[x]R(z)xX,???[x]R?is?an?upper?set.

Proposition 3.7

Let (X, R) be an IF approximation space. Then

R is symmetric

(x,y)X×X,???R_(x(1,0)C)(y)=R_(y(1,0)C)(x)(x,y)X×X,???R?(x(1,0))(y)=R?(y(1,0))(x).
Proof

By Remark 2.5, R_(x(1,0)C)(y)=RC(y,x)=RC(x,y)=R_(y(1,0)C)(x), because R is symmetric. Similarly we have that R?(x(1,0))(y) = R(y, x) = R(x, y) = R?(y(1,0))(x).

Theorem 3.8

Let R be an IF relation on X and let be an IF topology on X. If one of the following conditions is satisfied, then R is an IF preorder.

  • R? is a closure operator of .

  • R? is an interior operator of .

Proof

Suppose that satisfies (1). By Remark 2.5, R?(x(1,0)) (y) = R(y, x) for each xX. Since R? is a closure operator of , for each xX,

R(x,x)=R?(x(1,0))(x)=clT(x(1,0))(x)(x(1,0))(x)=(1,0).

Thus R is reflexive. For any x, y, zX, let cl (z(1,0)) (y) = (a, b). Then by Remark 2.5 and Proposition 2.4,

R(x,y)R(y,z)=R?(y(1,0))(x)R?(z(1,0))(y)=R?(y(1,0))(x)clT(z(1,0))(y)=R?(y(1,0))(x)(a,b)=R?((a,b)y(1,0))(x)=clT((a,b)y(1,0))(x)=clT(clT(z(1,0))(y)y(1,0))(x)clT(?yX[clT(z(1,0))(y)y(1,0)])(x)=clT(clT(z(1,0)))(x)=clT(z(1,0))(x)=R(x,z).

Hence R is transitive. Therefore R is an IF preorder.

Similarly we can prove for the case of (2).

Definition 3.9

For each A ∈ IF(X), we define

RA={(x,y)X×X?A(x)>A(y)}.

Obviously, RA = ?? iff A=(a,b)? for some (a, b) ∈ I ? I or A(x) and A(y) are non-comparable for all x, yX.

Proposition 3.10

Let (X, R) be an IF approximation space. Let A be an IF set with constant hesitancy degree, i.e., A ∈ cIF(X) with RA = ??. Then we have

  • (1) R(A) ⊇ ARC(x, y) ≥ A(x) ∨ A(y) for all (x, y) ∈ RA,

  • (2) R?(A) ⊆ AR(y, x) ≤ A(x)∧A(y) for all (x, y) ∈ RA.

Proof

(1) (⇒) Suppose that R(A) ⊇ A. Note that for each xX,

?yX(A(y)RC(x,y))=R_(A)(x)A(x).

Then A(y) ∨ RC(x, y) ≥ A(x) for any x, yX. Since A(x) > A(y) for each (x, y) ∈ RA, we have

RC(x,y)A(x)=A(x)A(y)?for?all?(x,y)RA.

(?) Suppose that for each (x, y) ∈ RA, RC(x, y) ≥ A(x) ∨ A(y). Let zX.

  • (i) If A(z) > A(y), then

    A(y)RC(z,y)A(y)(A(z)A(y))A(z).

  • (ii) If A(z) ≤ A(y), then

    A(y)RC(z,y)A(y)(A(z)A(y))A(y)A(z).

Hence R(A)(z) = ∧yX (A(y) ∨ RC (z, y)) ≥ A(z) for any zX. Thus R(A) ⊇ A.

(2) (⇒) Suppose that R?(A) ⊆ A. Note that for each yX,

?xX(A(x)R(y,x))=R?(A)(y)A(y).

Then A(x) ∧R(y, x) ≤ A(y) for any x, yX. Since A(x) > A(y) for each (x, y) ∈ RA, we have

R(y,x)A(y)=A(x)A(y).

(?) Suppose that for any (x, y) ∈ RA, R(y, x) ≤ A(x)∧A(y).

Let zX.

  • (i) If A(x) > A(z), then

    A(x)R(z,x)A(x)(A(x)A(z))A(z).

  • (ii) If A(x) ≤ A(z), then

    A(x)R(z,x)A(x)(A(x)A(z))A(x)A(z).

Thus R?(A)(z) = ∧xX (A(x) ∧ R(z, x)) ≤ A(z). Hence R?(A) ⊆ A.

Corollary 3.11

Let (X,R) be a reflexive IF approximation space. Then for each A ∈ cIF(X) with RA ≠ = ??,

  • R(A) = ARC(x, y) ≥ A(x) ∨ A(y) for all (x, y) ∈ RA,

  • R?(A) = AR(y, x) ≤ A(x)∧A(y) for all (x, y) ∈ RA.

Proof

By the above proposition and the reflexivity of R, it can be easily proved.

Let R1 and R2 be two IF relations on X. We denote R1R2 if R1(x, y) ≤ R2(x, y) for any x, yX. And R1 = R2 if R1R2 and R2R1.

Proposition 3.12

Let (X, R1) and (X, R2) be two IF approximation spaces. Then for each A ∈ IF(X),

  • R1R2R1?(A)R2?(A) and R1(A) ⊇ R2(A).

  • (R1R2)?(A)=R1?(A)R2?(A), (R1R2)(A) = R1(A) ∩ R2(A).

Proof

(1) For each xX,

R1?(A)(x)=?yX(A(y)(R1)(x,y))?yX(A(y)(R2)(x,y))=R2?(A)(x).

Thus we have R1?(A)R2?(A). Dually,

R1?(AC)R2?(AC)(R1?(AC))C(R2?(AC))CR1_(A)R2_(A).

(2) For each xX,

(R1R2)?(A)(x)=?yX(A(y)(R1R2)(x,y))=?yX(A(y)(R1(x,y)R2(x,y)))=?yX((A(y)R1(x,y))(A(y)R2(x,y)))(?yX(A(y)R1(x,y)))(?yX(A(y)R2(x,y)))=R1?(A)(x)R2?(A)(x)=(R1?(A)R2?(A))(x).

Thus we have (R1R2)?(A)R1?(A)R2?(A). Moreover, since R1R1R2 and R2R1R2, we have R1?(A)(R1R2)?(A)

and R1?(A)(R1R2)?(A). Thus R1?(A)R2?(A)(R1R2)?(A). Hence we have (R1R2)?(A)=R1?(A)R2?(A). By Proposition 2.4,

R1_(A)R2_(A)=(R1?(AC))C(R2?(AC))C=(R1?(AC)R2?(AC))C=((R1R2)?(AC))C=(R1R2)_(A).

Proposition 3.13

Let (X, R1) and (X, R2) be two reflexive IF approximation spaces. Then for each A ∈ IF(X),

  • R2 (R1(A)) ⊆ (R1R2)(A) and R1 (R2(A)) ⊆ (R1R2)(A).

  • R2?(R1?(A))(R1R2)?(A) and R1?(R2?(A))(R1R2)?(A).

Proof

(1) By Theorem 2.6, R2 (R1(A)) ⊆ R2(A) and R2 (R1(A)) ⊆ R1(A). Thus we have

R2_(R1_(A))R1_(A)R2_(A)(R1R2)_(A).

Similarly, we can prove that R1(R2(A)) ⊆ (R1R2)(A).

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

Proposition 3.14

Let (X, R1) and (X, R2) be two IF approximation spaces. If R1 is reflexive, R2 is transitive and R1R2, then

R1_(R2_(A))=R2_(A)?and?R1?(R2?(A))=R2?(A).
Proof

By Theorem 2.6, R1?(R2?(A))R2?(A). For each xX, by R1R2 and the transitivity of R2, we have

R1?(R2?(A))(x)=?yX(R2?(A)(y)R1(x,y))=?yX((?zX(A(z)R2(y,z))))R1(x,y)=?yX(?zX((A(z)R2(y,z))R1(x,y)))=?yX(?zX(A(z)(R2(y,z)R1(x,y))))?yX(?zX(A(z)(R2(y,z))R2(x,y)))?yX(?zX(A(z)R2(x,z)))=?zX(A(z)R2(x,z))=R2?(A)(x).

Thus R1?(R2?(A))R2?(A). So R1?(R2?(A))=R2?(A). By Proposition 2.4,

R1_(R2_(A))=R1_((R2?(AC))C)=(R1?(R2?(AC)))C=(R2?(AC))C=R2_(A).

4. Conclusion

We obtained characteristic properties of intuitionistic fuzzy rough approximation operator and intuitionistic fuzzy relation by means of topology. Particularly, we proved that the upper approximation of a set is the set itself if and only if the set is a lower set whenever the intuitionistic fuzzy relation is reflexive. Also we had the result that if an intuitionistic fuzzy upper approximation operator is a closure operator or an intuitionistic fuzzy lower approximation operator is an interior operator in the intuitionistic fuzzy topology, then the order is an preorder.

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