Intuitionistic Fuzzy Rough Approximation Operators
 Author: Yun Sang Min, Lee Seok Jong
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 15, Issue3, p208~215, 25 Sep 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.

KEYWORD
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, Çoker [2] introduced intuitionistic fuzzy topological spaces using the idea of intuitionistic fuzzy sets which was proposed by Atanassov [3]. Also Çoker and Demirci [4] defined intuitionistic fuzzy topological spaces in Šostak’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. Anintuitionistic fuzzy set A is an ordered pairwhere the functions 𝜇
_{A} :X →I and 𝜈_{A} :X →I denote the degree of membership and the degree of nonmembership respectively and 𝜇_{A} + 𝜈_{A} ≤ 1(see [3]). Obviously, every fuzzy set 𝜇 inX is an intuitionistic fuzzy set of the form .Throughout this paper, ‘IF’ stands for ‘intuitionistic fuzzy.’
I ⊗I denotes the family of all intuitionistic fuzzy numbers (a ,b ) such thata ,b ∈ [0, 1] anda +b ≤ 1, with the order relation defined byFor any IF set
A = (𝜇_{A} , 𝜈_{A} ) ofX , the valueis called an
indeterminancy degree (orhesitancy degree ) of 𝑥 toA (see [3]). Szmidt and Kacprzyk call 𝜋_{A} (𝑥) anintuitionistic index of 𝑥 inA (see [16]). ObviouslyNote 𝜋
_{A} (𝑥) = 0 iff 𝜈_{A} (𝑥) = 1 – 𝜇_{A} (𝑥). Hence any fuzzy set 𝜇_{A} can be regarded as an IF set (𝜇_{A} , 𝜈_{A} ) with 𝜋_{A} = 0.IF(
X ) denotes the family of all intuitionistic fuzzy sets inX , and cIF(X ) denotes the family of all intuitionistic fuzzy sets inX with constant hesitancy degree, i.e., ifA ∈ cIF(X ), then 𝜋_{A} =c for some constantc ∈ [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 anintuitionistic fuzzy topology onX in the sense of Lowen ([18]), if(1) for each (a, b) ∈ I ⊗ I, , (2) A, B ∈ implies A ᑎ B ∈ , (3) {Aj ∣ j ∈ J} ⊆ implies ∪j∈J Aj ∈ .
The pair (
X , ) is called anintuitionistic fuzzy topological space . Every member of is called anintuitionistic fuzzy open set inX . Its complement is called anintuitionistic fuzzy closed set inX . We denote . The interior and closure ofA denoted by int(A ) and cl(A ) respectively for eachA ∈ IF(X ) are defined as follows:An IF topology is called an
Alexandrov topology [19] if (2) in Definition 2.1 is replaced byDefinition 2.2. ( [20]) An IF setR onX ×X is called anintuitionistic fuzzy relation onX . Moreover,R is called(i) reflexive if R(𝑥, 𝑥) = (1, 0) for all 𝑥 ∈ X, (ii) symmetric if R(𝑥, 𝓎) = R(𝓎, 𝑥) for all 𝑥, 𝓎 ∈ X, (iii) transitive if R(𝑥, 𝓎) Λ R(𝓎, z) ≤ R(𝑥, z) for all 𝑥, 𝓎, z ∈ X,
A reflexive and transitive IF relation is called an
intuitionistic fuzzy preorder . A symmetric IF preorder is called anintuitionistic fuzzy equivalence . An IF preorder onX is called anintuitionistic fuzzy partial order if for any 𝑥, 𝓎 ∈X, R (𝑥, 𝓎) =R (𝓎, 𝑥) = (1, 0) implies that 𝑥 = 𝓎.Let
R be an IF relation onX .R ^{–1} is called theinverse relation ofR ifR ^{–1}(𝑥, 𝓎) =R (𝓎, 𝑥) for any 𝑥, 𝓎 ∈X . Also,R^{C} is called thecomplement ofR ifR^{C} (𝑥, 𝓎) = (𝜈_{R}_{(𝑥, 𝓎)}, 𝜇_{R(𝑥, 𝓎)}) for any 𝑥, 𝓎 ∈X whenR (𝑥, 𝓎) = (𝜇_{R(𝑥, 𝓎)}, 𝜈_{R(𝑥, 𝓎)}). It is obvious thatR ^{–1} ≠R^{C} .Definition 2.3. ( [21]) LetR be an IF relation onX . The pair (X, R ) is called anintuitionistic fuzzy approximation space . Theintuitionistic fuzzy lower approximation ofA ∈ IF(X ) with respect to (X, R ), denoted by , is defined as follows:Similarly, the
intuitionistic fuzzy upper approximation ofA ∈ IF(X ) with respect to (X, R ), denoted by , is defined as follows:The pair is called the
intuitionistic fuzzy rough set ofA with respect to (X ,R ).Proposition 2.4. ( [17, 21]) Let (X, R ) be an IF approximation space. Then for anyA, B ∈ IF(X ), {A_{j} ∣j ∈J } ⊆ IF(X ) and (a, b ) ∈I ⊗I ,Remark 2.5. Let (X, R ) be an IF approximation space. ThenLet (
X, R ) be an IF approximation space. (X, R ) is called areflexive (resp.,preordered ) intuitionistic fuzzy approximation space, ifR is a reflexive intuitionistic fuzzy relation (resp., an intuitionistic fuzzy preorder). IfR is an intuitionistic fuzzy partial order, then (X ,R ) is called apartially ordered intuitionistic fuzzy approximation space. An intuitionistic fuzzy preorderR is called anintuitionistic fuzzy equality , ifR 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(1)
R is reflexive(2)
R is transitive3. IF Rough Approximation Operator
Definition 3.1. ( [22]) Let (X, R ) be an IF approximation space. ThenA ∈ IF(X ) is called anintuitionistic fuzzy upper set in (X, R ) ifDually,
A is called anintuitionistic fuzzy lower set in (X, R ) ifA (𝓎) ΛR (𝑥, 𝓎) ≤A (𝑥) for all 𝑥, 𝓎 ∈X .Let
R be an IF preorder onX . For 𝑥, 𝓎 ∈X , the real numberR (𝑥, 𝓎) can be interpreted as the degree to which ‘𝑥 ≤ 𝓎 ’ holds true. The conditionA (𝑥) ΛR (𝑥, 𝓎) ≤A (𝓎) can be interpreted as the statement that if 𝑥 is inA and 𝑥 ≤ 𝓎, then 𝓎 is inA . Particularly, ifR is an IF equivalence, then an IF setA is an upper set in (X, R ) if and only if it is a lower set in (X, R ).The classical preorder 𝑥 ≤ 𝓎 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 andA ∈ IF(X ). Then the following are equivalent:Proof. (1) ⇒ (2). Suppose that . Since for each 𝑥 ∈X ,we have
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, 𝓎 ∈ X, A(𝑥) Λ R–1(𝑥, 𝓎) ≤ A(𝓎). So A(𝑥) Λ R(𝓎, 𝑥) ≤ A(𝓎). Thus
Hence .
Corollary 3.3. Let (X, R ) be an IF approximation space andA ∈ IF(X ). IfR is reflexive, then the following are equivalent:Proof. This holds by Theorem 2.6 and Proposition 3.2.Proposition 3.4. Let (X, R ) be an IF approximation space andA ∈ IF(X ). Then the following are equivalent:Proof. (1) ⇒ (2). Suppose that . Since for each 𝑥 ∈X ,we have
Thus
A^{C} 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, 𝓎 ∈
X, A^{C} (𝑥) ΛR ^{–1}(𝑥, 𝓎) ≤A^{C} (𝓎). SoA^{C} (𝑥) ΛR (𝓎, 𝑥) ≤A^{C} (𝓎). ThusSo
Hence .
Corollary 3.5. Let (X, R ) be an IF approximation space andA ∈ IF(X ). IfR is reflexive, then the following are equivalent:Proof. This holds by Theorem 2.6 and the above proposition.For each
z ∈X , we define IF sets [z ]^{R} :X →I ⊗I by [z ]^{R}(𝑥) =R (z , 𝑥), and [z ]_{R} :X →I ⊗I by [z ]_{R}(𝑥) =R (𝑥,z ).Theorem 3.6. Let (X, R ) be an IF approximation space. Then(1)
R is reflexive(2)
R is symmetric(3)
R is transitiveProof. (1) and (2) are obvious. (3) By Proposition 3.2,Also,
Proposition 3.7. Let (X, R ) be an IF approximation space. ThenR is symmetricProof. By Remark 2.5, , becauseR is symmetric. Similarly we have that .Theorem 3.8. LetR be an IF relation onX and let be an IF topology onX . If one of the following conditions is satisfied, thenR is an IF preorder.(1) is a closure operator of . (2) is an interior operator of .
Proof. Suppose that satisfies (1). By Remark 2.5, for each 𝑥 ∈X . Since is a closure operator of , for each 𝑥 ∈X ,Thus
R is reflexive. For any 𝑥, 𝓎,z ∈X , . Then by Remark 2.5 and Proposition 2.4,Hence
R is transitive. ThereforeR is an IF preorder.Similarly we can prove for the case of (2).
Definition 3.9. For eachA ∈ IF(X ), we defineObviously,
R_{A} = ∅ iff for some (a, b ) ∈I ⊗I orA (𝑥) andA (𝓎) are noncomparable for all 𝑥, 𝓎 ∈X .Proposition 3.10. Let (X, R ) be an IF approximation space. LetA be an IF set with constant hesitancy degree, i.e.,A ∈ cIF(X ) withR_{A} ≠ ∅. Then we haveProof. (1) (⇒) Suppose that . Note that for each 𝑥 ∈X ,Then
A (𝓎) ∨R^{C} (𝑥, 𝓎) ≥A (𝑥) for any 𝑥, 𝓎 ∈X . SinceA (𝑥) >A (𝓎) for each (𝑥, 𝓎) ∈R_{A} , we have(⇐) Suppose that for each (𝑥, 𝓎) ∈
R_{A}, R^{C} (𝑥, 𝓎) ≥ A(𝑥) ∨A (𝓎). Letz ∈X .(i) If
A (z) >A (𝓎), then(ii) If
A (z ) ≤A (𝓎), thenA (𝓎) ∨R^{C} (z , 𝓎) ≥A (𝓎) ∨ (A (z ) ∨A (𝓎)) ≥A (𝓎) ≥A (z ).Hence for any
z ∈X . Thus .Then
A (𝑥) ΛR (𝓎, 𝑥) ≤A (𝓎) for any 𝑥, 𝓎 ∈X . SinceA (𝑥) >A (𝓎) for each (𝑥, 𝓎) ∈R_{A} , we have(⇐) Suppose that for any (𝑥, 𝓎) ∈
R_{A}, R (𝓎, 𝑥) ≤A (𝑥) ΛA (𝓎). Letz ∈X .(i) If
A (𝑥) >A (z ), then(ii) If
A (𝑥) ≤A (z ), thenA (𝑥) ΛR (z , 𝑥) ≤A (𝑥) Λ (A (𝑥) ΛA (z )) ≤A (𝑥) ≤A (z ).Thus . Hence .
Corollary 3.11. Let (X, R ) be a reflexive IF approximation space. Then for eachA ∈ cIF(X ) withR_{A} ≠ ∅,Proof. By the above proposition and the reflexivity ofR , it can be easily proved.Let
R _{1} andR _{2} be two IF relations onX . We denoteR _{1} ⊆R _{2} ifR _{1}(𝑥, 𝓎) ≤R _{2}(𝑥, 𝓎) for any 𝑥, 𝓎 ∈X . AndR _{1} =R _{2} ifR _{1} ⊆R _{2} andR _{2} ⊆R _{1}.Proposition 3.12. Let (X, R _{1}) and (X, R _{2}) be two IF approximation spaces. Then for eachA ∈ IF(X ),Proof. (1) For each 𝑥 ∈X ,Thus we have . Dually,
Thus we have . Moreover, since
R _{1} ⊆R _{1} ∪R _{2} andR _{2} ⊆R _{1} ∪R _{2}, we have and Thus Hence we have . By Proposition 2.4,Proposition 3.13. Let (X, R _{1}) and (X, R _{2}) be two reflexive IF approximation spaces. Then for eachA ∈ IF(X ),Proof. (1) By Theorem 2.6, and . Thus we haveSimilarly, we can prove that
(2) The proof is similar to (1).
Proposition 3.14. Let (X, R _{1}) and (X, R _{2}) be two IF approximation spaces. IfR _{1} is reflexive,R _{2} is transitive andR _{1} ⊆R _{2}, thenProof. By Theorem 2.6, For each 𝑥 ∈X , byR _{1} ⊆R _{2} and the transitivity ofR _{2}, we haveThus . So . By Proposition 2.4,
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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