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Intuitionistic Fuzzy Rough Approximation Operators
Intuitionistic Fuzzy Rough Approximation Operators
International Journal of Fuzzy Logic and Intelligent Systems. 2015. Sep, 15(3): 208-215
Copyright © 2015, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : July 03, 2015
  • Accepted : September 24, 2015
  • Published : September 25, 2015
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Sang Min Yun
Seok Jong Lee

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
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. An intuitionistic fuzzy set A is an ordered pair
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where 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 in X is an intuitionistic fuzzy set of the form
PPT Slide
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.
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
PPT Slide
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For any IF set A = (𝜇 A , 𝜈 A ) of X , the value
PPT Slide
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is called an indeterminancy degree (or hesitancy degree ) of 𝑥 to A (see [3] ). Szmidt and Kacprzyk call 𝜋 A (𝑥) an intuitionistic index of 𝑥 in A (see [16] ). Obviously
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Note 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 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 𝜋 A = 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
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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,,
  • (2)A, B∈impliesAᑎB∈,
  • (3) {Aj∣j∈J} ⊆implies ∪j∈JAj∈.
The pair ( X ,
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) is called an intuitionistic fuzzy topological space . Every member of
PPT Slide
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is called an intuitionistic fuzzy open set in X . Its complement is called an intuitionistic fuzzy closed set in X . We denote
PPT Slide
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. The interior and closure of A denoted by int( A ) and cl( A ) respectively for each A ∈ IF( X ) are defined as follows:
PPT Slide
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An IF topology
PPT Slide
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is called an Alexandrov topology [19] if (2) in Definition 2.1 is replaced by
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Definition 2.2. ( [20] ) An IF set R on X × X is called an intuitionistic fuzzy relation on X . Moreover, R is called
  • (i)reflexiveifR(𝑥, 𝑥) = (1, 0) for all 𝑥 ∈X,
  • (ii)symmetricifR(𝑥, 𝓎) =R(𝓎, 𝑥) for all 𝑥, 𝓎 ∈X,
  • (iii)transitiveifR(𝑥, 𝓎) ΛR(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 an intuitionistic fuzzy equivalence . An IF preorder on X is called an intuitionistic 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 on X . R –1 is called the inverse relation of R if R –1 (𝑥, 𝓎) = R (𝓎, 𝑥) for any 𝑥, 𝓎 ∈ X . Also, RC is called the complement of R if RC (𝑥, 𝓎) = (𝜈 R (𝑥, 𝓎) , 𝜇 R(𝑥, 𝓎) ) for any 𝑥, 𝓎 ∈ X when R (𝑥, 𝓎) = (𝜇 R(𝑥, 𝓎) , 𝜈 R(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
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, is defined as follows:
PPT Slide
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Similarly, the intuitionistic fuzzy upper approximation of A ∈ IF( X ) with respect to ( X, R ), denoted by
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, is defined as follows:
PPT Slide
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The pair
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is called the intuitionistic fuzzy rough set of A with respect to ( X , R ).
PPT Slide
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and
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are called the intuitionistic fuzzy lower approximation operator and the intuitionistic fuzzy upper approximation operator , respectively. In general, we refer to
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and
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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 j J } ⊆ IF( X ) and ( a, b ) ∈ I I ,
PPT Slide
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Remark 2.5. Let ( X, R ) be an IF approximation space. Then
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Let ( X, R ) be an IF approximation space. ( X, R ) is called a reflexive (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
(1) R is reflexive
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(2) R is transitive
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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
PPT Slide
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Dually, A is called an intuitionistic fuzzy lower set in ( X, R ) if A (𝓎) Λ R (𝑥, 𝓎) ≤ A (𝑥) for all 𝑥, 𝓎 ∈ X .
Let R be an IF preorder on X . For 𝑥, 𝓎 ∈ X , the real number R (𝑥, 𝓎) can be interpreted as the degree to which ‘𝑥 ≤ 𝓎 ’ holds true. The condition A (𝑥) Λ R (𝑥, 𝓎) ≤ A (𝓎) can be interpreted as the statement that if 𝑥 is in A and 𝑥 ≤ 𝓎, 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 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 and A ∈ IF( X ). Then the following are equivalent:
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Proof. (1) ⇒ (2). Suppose that
PPT Slide
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. Since for each 𝑥 ∈ X ,
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we have
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Thus A is a lower set in ( X, R ).
  • (2) ⇒ (3). This is obvious.
  • (3) ⇒ (1). Suppose thatAis an upper set in (X, R–1). Then for any 𝑥, 𝓎 ∈X,A(𝑥) ΛR–1(𝑥, 𝓎) ≤A(𝓎). SoA(𝑥) ΛR(𝓎, 𝑥) ≤A(𝓎). Thus
PPT Slide
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Hence
PPT Slide
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.
Corollary 3.3. Let ( X, R ) be an IF approximation space and A ∈ IF( X ). If R is reflexive, then the following are equivalent:
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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:
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Proof. (1) ⇒ (2). Suppose that
PPT Slide
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. Since for each 𝑥 ∈ X ,
PPT Slide
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we have
PPT Slide
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Thus AC is a lower set in ( X, R ).
  • (2) ⇒ (3). This is obvious.
  • (3) ⇒ (1). Suppose thatACis an upper set in (X, R–1).
Then for any 𝑥, 𝓎 ∈ X, AC (𝑥) Λ R –1 (𝑥, 𝓎) ≤ AC (𝓎). So AC (𝑥) Λ R (𝓎, 𝑥) ≤ AC (𝓎). Thus
PPT Slide
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So
PPT Slide
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Hence
PPT Slide
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.
Corollary 3.5. Let ( X, R ) be an IF approximation space and A ∈ IF( X ). If R is reflexive, then the following are equivalent:
PPT Slide
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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
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(2) R is symmetric
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(3) R is transitive
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Proof. (1) and (2) are obvious. (3) By Proposition 3.2,
PPT Slide
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Also,
PPT Slide
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Proposition 3.7. Let ( X, R ) be an IF approximation space. Then
R is symmetric
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Proof. By Remark 2.5,
PPT Slide
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, because R is symmetric. Similarly we have that
PPT Slide
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.
Theorem 3.8. Let R be an IF relation on X and let
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be an IF topology on X . If one of the following conditions is satisfied, then R is an IF preorder.
  • (1)is a closure operator of.
  • (2)is an interior operator of.
Proof. Suppose that
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satisfies (1). By Remark 2.5,
PPT Slide
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for each 𝑥 ∈ X . Since
PPT Slide
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is a closure operator of
PPT Slide
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, for each 𝑥 ∈ X ,
PPT Slide
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Thus R is reflexive. For any 𝑥, 𝓎, z X ,
PPT Slide
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. Then by Remark 2.5 and Proposition 2.4,
PPT Slide
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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
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Obviously, RA = ∅ iff
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for some ( a, b ) ∈ I I or A (𝑥) and A (𝓎) are non-comparable for all 𝑥, 𝓎 ∈ X .
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
PPT Slide
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Proof. (1) (⇒) Suppose that
PPT Slide
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. Note that for each 𝑥 ∈ X ,
PPT Slide
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Then A (𝓎) ∨ RC (𝑥, 𝓎) ≥ A (𝑥) for any 𝑥, 𝓎 ∈ X . Since A (𝑥) > A (𝓎) for each (𝑥, 𝓎) ∈ RA , we have
PPT Slide
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(⇐) Suppose that for each (𝑥, 𝓎) ∈ RA, RC (𝑥, 𝓎) ≥ A(𝑥) ∨ A (𝓎). Let z X .
(i) If A (z) > A (𝓎), then
PPT Slide
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(ii) If A ( z ) ≤ A (𝓎), then
A (𝓎) ∨ RC ( z , 𝓎) ≥ A (𝓎) ∨ ( A ( z ) ∨ A (𝓎)) ≥ A (𝓎) ≥ A ( z ).
Hence
PPT Slide
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for any z X . Thus
PPT Slide
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.
PPT Slide
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Then A (𝑥) Λ R (𝓎, 𝑥) ≤ A (𝓎) for any 𝑥, 𝓎 ∈ X . Since A (𝑥) > A (𝓎) for each (𝑥, 𝓎) ∈ RA , we have
PPT Slide
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(⇐) Suppose that for any (𝑥, 𝓎) ∈ RA, R (𝓎, 𝑥) ≤ A (𝑥) Λ A (𝓎). Let z X .
(i) If A (𝑥) > A ( z ), then
PPT Slide
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(ii) If A (𝑥) ≤ A ( z ), then
A (𝑥) Λ R ( z , 𝑥) ≤ A (𝑥) Λ ( A (𝑥) Λ A ( z )) ≤ A (𝑥) ≤ A ( z ).
Thus
PPT Slide
Lager Image
. Hence
PPT Slide
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.
Corollary 3.11. Let ( X, R ) be a reflexive IF approximation space. Then for each A ∈ cIF( X ) with RA ≠ ∅,
PPT Slide
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Proof. By the above proposition and the reflexivity of R , it can be easily proved.
Let R 1 and R 2 be two IF relations on X . We denote R 1 R 2 if R 1 (𝑥, 𝓎) ≤ R 2 (𝑥, 𝓎) for any 𝑥, 𝓎 ∈ X . And R 1 = R 2 if R 1 R 2 and R 2 R 1 .
Proposition 3.12. Let ( X, R 1 ) and ( X, R 2 ) be two IF approximation spaces. Then for each A ∈ IF( X ),
PPT Slide
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Proof. (1) For each 𝑥 ∈ X ,
PPT Slide
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Thus we have
PPT Slide
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. Dually,
PPT Slide
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PPT Slide
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Thus we have
PPT Slide
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. Moreover, since R 1 R 1 R 2 and R 2 R 1 R 2 , we have
PPT Slide
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and
PPT Slide
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Thus
PPT Slide
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Hence we have
PPT Slide
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. By Proposition 2.4,
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Proposition 3.13. Let ( X, R 1 ) and ( X, R 2 ) be two reflexive IF approximation spaces. Then for each A ∈ IF( X ),
PPT Slide
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Proof. (1) By Theorem 2.6,
PPT Slide
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and
PPT Slide
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. Thus we have
PPT Slide
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Similarly, we can prove that
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(2) The proof is similar to (1).
Proposition 3.14. Let ( X, R 1 ) and ( X, R 2 ) be two IF approximation spaces. If R 1 is reflexive, R 2 is transitive and R 1 R 2 , then
PPT Slide
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Proof. By Theorem 2.6,
PPT Slide
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For each 𝑥 ∈ X , by R 1 R 2 and the transitivity of R 2 , we have
PPT Slide
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Thus
PPT Slide
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. So
PPT Slide
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. By Proposition 2.4,
PPT Slide
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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.
BIO
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
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