Intuitionistic Fuzzy Rough Approximation Operators

International Journal of Fuzzy Logic and Intelligent Systems.
2015.
Sep,
15(3):
208-215

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 30, 2015

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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.
X
be a nonempty set. An
intuitionistic fuzzy set A
is an ordered pair
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
.
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
For any IF set
A
= (𝜇
_{A}
, 𝜈
_{A}
) of
X
, the value
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
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
of IF(
X
) is called an
intuitionistic fuzzy topology
on
X
in the sense of Lowen (
[18]
), if
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
. The interior and closure of
A
denoted by int(
A
) and cl(
A
) respectively for each
A
∈ IF(
X
) are defined as follows:
An IF topology
is called an
Alexandrov topology
[19]
if (2) in Definition 2.1 is replaced by
Definition 2.2.
(
[20]
) An IF set
R
on
X
×
X
is called an
intuitionistic fuzzy relation
on
X
. Moreover,
R
is called
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,
R^{C}
is called the
complement
of
R
if
R^{C}
(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}
≠
R^{C}
.
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
, is defined as follows:
Similarly, the
intuitionistic fuzzy upper approximation
of
A
∈ IF(
X
) with respect to (
X, R
), denoted by
, is defined as follows:
The pair
is called the
intuitionistic fuzzy rough set
of
A
with respect to (
X
,
R
).
and
are called the
intuitionistic fuzzy lower approximation operator
and the
intuitionistic fuzzy upper approximation operator
, respectively. In general, we refer to
and
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
), {
A_{j}
∣
j
∈
J
} ⊆ IF(
X
) and (
a, b
) ∈
I
⊗
I
,
Remark 2.5.
Let (
X, R
) be an IF approximation space. Then
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
(2)
R
is transitive
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
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:
Proof.
(1) ⇒ (2). Suppose that
. Since for each 𝑥 ∈
X
,
we have
Thus
A
is a lower set in (
X, R
).
Hence
.
Corollary 3.3.
Let (
X, R
) be an IF approximation space and
A
∈ IF(
X
). If
R
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 and
A
∈ 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
).
Then for any 𝑥, 𝓎 ∈
X, A^{C}
(𝑥) Λ
R
^{–1}
(𝑥, 𝓎) ≤
A^{C}
(𝓎). So
A^{C}
(𝑥) Λ
R
(𝓎, 𝑥) ≤
A^{C}
(𝓎). Thus
So
Hence
.
Corollary 3.5.
Let (
X, R
) be an IF approximation space and
A
∈ IF(
X
). If
R
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 transitive
Proof.
(1) and (2) are obvious. (3) By Proposition 3.2,
Also,
Proposition 3.7.
Let (
X, R
) be an IF approximation space. Then
R
is symmetric
Proof.
By Remark 2.5,
, because
R
is symmetric. Similarly we have that
.
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.
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. 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
Obviously,
R_{A}
= ∅ iff
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
R_{A}
≠ ∅. Then we have
Proof.
(1) (⇒) Suppose that
. Note that for each 𝑥 ∈
X
,
Then
A
(𝓎) ∨
R^{C}
(𝑥, 𝓎) ≥
A
(𝑥) for any 𝑥, 𝓎 ∈
X
. Since
A
(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). Let
z
∈
X
.
(i) If
A
(z) >
A
(𝓎), then
(ii) If
A
(
z
) ≤
A
(𝓎), then
A
(𝓎) ∨
R^{C}
(
z
, 𝓎) ≥
A
(𝓎) ∨ (
A
(
z
) ∨
A
(𝓎)) ≥
A
(𝓎) ≥
A
(
z
).
Hence
for any
z
∈
X
. Thus
.
Then
A
(𝑥) Λ
R
(𝓎, 𝑥) ≤
A
(𝓎) for any 𝑥, 𝓎 ∈
X
. Since
A
(𝑥) >
A
(𝓎) for each (𝑥, 𝓎) ∈
R_{A}
, we have
(⇐) Suppose that for any (𝑥, 𝓎) ∈
R_{A}, R
(𝓎, 𝑥) ≤
A
(𝑥) Λ
A
(𝓎). Let
z
∈
X
.
(i) If
A
(𝑥) >
A
(
z
), then
(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
. Hence
.
Corollary 3.11.
Let (
X, R
) be a reflexive IF approximation space. Then for each
A
∈ cIF(
X
) with
R_{A}
≠ ∅,
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
),
Proof.
(1) For each 𝑥 ∈
X
,
Thus we have
. Dually,
Thus we have
. Moreover, since
R
_{1}
⊆
R
_{1}
∪
R
_{2}
and
R
_{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 each
A
∈ IF(
X
),
Proof.
(1) By Theorem 2.6,
and
. Thus we have
Similarly, 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. If
R
_{1}
is reflexive,
R
_{2}
is transitive and
R
_{1}
⊆
R
_{2}
, then
Proof.
By Theorem 2.6,
For each 𝑥 ∈
X
, by
R
_{1}
⊆
R
_{2}
and the transitivity of
R
_{2}
, we have
Thus
. So
. By Proposition 2.4,
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

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
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- (1) for each (a, b) ∈I⊗I,,
- (2)A, B∈impliesAᑎB∈,
- (3) {Aj∣j∈J} ⊆implies ∪j∈JAj∈.

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- (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,

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3. IF Rough Approximation Operator

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- (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

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- (2) ⇒ (3). This is obvious.
- (3) ⇒ (1). Suppose thatACis an upper set in (X, R–1).

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- (1)is a closure operator of.
- (2)is an interior operator of.

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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

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1968
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Coker D.
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** DOI : 10.1016/S0165-0114(96)00076-0**

Atanassov K. T.
1986
“Intuitionistic fuzzy sets,”
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20
87 -
96
** DOI : 10.1016/S0165-0114(86)80034-3**

Coker D.
,
Demirci M.
1996
“An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense,”
BUSEFAL
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67 -
76

Lee S. J.
,
Lee E. P.
2000
“The category of intuitionistic fuzzy topological spaces,”
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Citing 'Intuitionistic Fuzzy Rough Approximation Operators
'

@article{ E1FLA5_2015_v15n3_208}
,title={Intuitionistic Fuzzy Rough Approximation Operators}
,volume={3}
, url={http://dx.doi.org/10.5391/IJFIS.2015.15.3.208}, DOI={10.5391/IJFIS.2015.15.3.208}
, number= {3}
, journal={International Journal of Fuzzy Logic and Intelligent Systems}
, publisher={Korean Institute of Intelligent Systems}
, author={Yun, Sang Min
and
Lee, Seok Jong}
, year={2015}
, month={Sep}