Applied Mathematical Modelling 36 (2012) 4539–4541
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Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Erratum
Erratum to ‘‘Fuzzy rough set model on two different universes
and its application’’ [Appl. Math. Model. 35 (4) (2011) 1798–1809]
Bingzhen Sun a,b, Weimin Ma a,⇑
a
b
School of Economics and Management, Tongji University, Shanghai 200092, China
School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China
a r t i c l e
i n f o
Article history:
Received 14 October 2011
Received in revised form 1 March 2012
Accepted 10 November 2011
Available online 14 December 2011
Keywords:
Fuzzy sets
Rough sets
Fuzzy relation
Fuzzy compatible relation
Fuzzy rough sets
a b s t r a c t
The aim of this paper is to correct two mistakes in [Appl. Math. Model. 35 (4) (2011) 1798–
1809], which are: one of the properties of fuzzy rough set between two different universes
and the definition of the upper approximation with the property for degree fuzzy rough set
between two different universes.
For the modified properties and the definition of upper approximation for degree fuzzy
rough set between two different universes, we claim that the results obtained are correct
and the errors have not further implications.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
In this presentation we correct an aspect of the article by Sun and Ma [1] referent to the example presented in [1]. In Section 2 of cited work, it can be observed, there existing some mistakes in the property of the lower and upper approximation
operators for fuzzy rough set on two different universes. Also, the upper approximation of the degree fuzzy rough set on two
different universes is mistaken.
2. The correction
In [1], we present the following definition for fuzzy rough set over two universes.
e a be a fuzzy compatible relation of universe U and V. For any X(X # V), we define
Let U, V be non-empty finite universes, R
e a on U and V as follows, respectively [1]:
the lower and upper approximations of X about R
n
o
e a ðuÞ # X ;
apre ðXÞ ¼ u 2 Uj R
Ra
n
o
e a ðuÞ \ X – £ :
apre ðXÞ ¼ u 2 Uj R
Ra
If apre ðXÞ ¼ apre ðXÞ, then X is called definable set on U and V at the threshold value a. Otherwise, X is called fuzzy rough
Ra
Ra
set on U and V at the threshold value a.
DOI of original article: 10.1016/j.apm.2010.10.010
⇑ Corresponding author.
E-mail addresses: [email protected] (B. Sun), [email protected] (W. Ma).
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2011.11.072
4540
B. Sun, W. Ma / Applied Mathematical Modelling 36 (2012) 4539–4541
Based on the above definition, we present some properties for fuzzy rough set over two universes in [1].
e a be fuzzy compatible relation of U V. For any X, Y # V. Then
Theorem 2.1 [1]. Let U, V be two non-empty finite universes, R
lower approximation apre ðXÞ and upper approximation apre ðXÞ have following property:
Ra
Ra
(1) apre ðXÞ # X # apre ðXÞ:
Ra
Ra
We show why the relation apre ðXÞ # X # apre ðXÞ dose not hold.
Ra
Ra
Clearly, X # V and the universe U and V are two different universes. Moreover, it can be easily seen that
apre ðXÞ; apre ðXÞ # U by the definition. Therefore, apre ðXÞ X and X apre ðXÞ.
Ra
Ra
Ra
Ra
In [1], we give two types generalization models for fuzzy rough set on two different universes. One of the generalization
model is degree fuzzy rough set on two different universes. The definition of the degree fuzzy rough set on two different
universes as follows.
e be fuzzy relation of U and V. For any X # U, a 2 (0, 1] we define the lower
Let U, V be two non-empty finite universes, R
e a on U and V as followings, respectively:
and upper approximations of X according to the degree k about R
n
o
k
e a ðuÞj j R
e a ðuÞ \ Xj 6 k ;
apre
ðXÞ ¼ u 2 Uk R
Ra
n
o
k
e a ðuÞ \ Xj P k :
apre ðXÞ ¼ u 2 Uk R
Ra
where k is finite integer, jXj denotes the cardinality of set.
As a matter of fact, the degree fuzzy rough set on two different universes is a generalization of the generalized fuzzy rough
set [2] and the classical degree rough set on the same universe [3]. Then the definition of the lower and upper approximations are similar to the case of classical degree rough set on the same universe. It can be observed that the upper approximation is incorrect for degree fuzzy rough set on two different universes. In following, we present the correct forms of the
upper approximation.
e be fuzzy relation of U and V. For any X # U, a 2 (0, 1] we define
Definition 2.1. Let U, V be two non-empty finite universes, R
e a on U and V as followings, respectively.
the lower and upper approximations of X according to the degree k about R
n
o
k
e a ðuÞj j R
e a ðuÞ \ Xj 6 k ;
apre
ðXÞ ¼ u 2 Uk R
Ra
n
o
k
e a ðuÞ \ Xj > k :
apre
ðXÞ ¼ u 2 Uk R
Ra
where k is finite integer, jXj denotes the cardinality of sets.
Like the Theorem 3.1 in [1] and Theorem 3.20 in [3], we present the properties for the lower and upper approximations of
degree fuzzy rough set on two different universes in [1].
e a is fuzzy compatible relation of U V. For any X, Y # V, a 2
Theorem 2.2 ([1]). Let U, V be two non-empty finite universes, R
(0, 1],k is finite integer. Then the degree approximation operators satisfies the following relations:
(1) aprk ðXÞ # X # apr k ðXÞ; apr k ðXÞ # aprk ðXÞ.
eR a
eR a
eR a
eR a
(2) aprk ðX \ YÞ ¼ aprk ðXÞ \ aprk ðYÞ; aprk ðX [ YÞ ¼ apr k ðXÞ [ apr k ðYÞ.
eR a
eR a
eR a
eR a
eR a
eR a
Like the property in Theorem 2.1, it can be easily seen that the first relation apr k ðXÞ # X # aprk ðXÞ does not hold with the
eR a
eR a
same reason of Theorem 2.1.
The next, we use the example presented in [1] to illustrate the other relations also are not satisfy.
Example ([1]). Let U = {x1, x2, x3, x4, x5, x6} be finite universe of six sufferers. V = {y1, y2, y3, y4, y5, y6} be the finite of six
e a be fuzzy compatible relation between the universe U and V.
symptoms. R 2 F(U V), for any a 2 ð0; 1; R
According to the statistic datum, the degree value of every sufferer in universe U shows the symptoms in the universe V
are given as following, respectively.
R(xi, yj)
y1
y2
y3
y4
y5
y6
x1
x2
x3
x4
x5
x6
0.8
0.7
0.5
0.4
0.3
0.1
0.6
0.3
0.2
0.6
0.5
0.7
0.4
0.2
0.6
0.5
0.7
0.8
0.7
0.5
0.3
0.7
0.3
0.6
0.3
0.8
0.7
0.5
0.6
0.3
0.2
0.6
0.1
0.3
0.8
0.2
B. Sun, W. Ma / Applied Mathematical Modelling 36 (2012) 4539–4541
4541
Let X = {y1, y3, y5} # V. Taking a = 0.5 and k = 3.
By the Definition 3.3 in Ref. [1], we can obtain the classes of the fuzzy compatible for the elements in the universe U as
following, respectively:
e 0:5 ðx1 Þ ¼ fy ; y ; y g;
R
1
2
4
e 0:5 ðx2 Þ ¼ fy ; y ; y ; y g;
R
1
4
5
6
e 0:5 ðx4 Þ ¼ fy ; y ; y ; y g;
R
2
3
4
5
e 0:5 ðx3 Þ ¼ fy ; y ; y g;
R
1
3
5
e 0:5 ðx5 Þ ¼ fy ; y ; y ; y g;
R
2
3
5
6
e 0:5 ðx6 Þ ¼ fy ; y ; y g;
R
2
3
4
We can calculate the lower approximation and upper approximation of X(X # V) as following, respectively.
3
apre
ðXÞ ¼ fx1 ; x2 ; x3 ; x4 ; x5 ; x6 g;
R 0:5
3
apre
ðXÞ ¼ f£g:
R 0:5
It can be observed that apr 3 ðXÞ apr3 ðXÞ.
eR 0:5
eR 0:5
Then we take Y = {y1, y2, y6} # V. Similarly, we can calculate the lower approximation and upper approximation of
Y(Y # V) as following, respectively.
3
apre
ðYÞ ¼ fx1 ; x2 ; x3 ; x4 ; x5 ; x6 g;
R 0:5
3
apre
ðYÞ ¼ f£g:
R 0:5
Also, we calculate lower approximation of X [ Y and upper approximation of X \ Y as following, respectively.
3
apre
ðX \ YÞ ¼ fx1 ; x2 ; x3 ; x6 g;
R 0:5
3
apre
ðX [ YÞ ¼ fx5 g;
R 0:5
It can be easily verified that apr 3 ðX \ YÞ – apr3 ðXÞ \ apr3 ðYÞ and apr 3 ðX [ YÞ – apr3 ðXÞ [ apr3 ðYÞ.
eR 0:5
eR 0:5
eR 0:5
eR 0:5
eR 0:5
eR 0:5
In following, we present the corrected results in Theorem 2.3:
e a is the fuzzy compatible relation of U V. For any X, Y # V, a 2
Theorem 2.2. Let U, V be two non-empty finite universes, R
(0, 1],k is the finite integer. Then the degree approximation operators satisfies the following relations:
(1) aprk ðX \ YÞ # aprk ðXÞ \ apr k ðYÞ.
eR a
eR a
eR a
(2) aprk ðX [ YÞ aprk ðXÞ [ apr k ðYÞ.
eR a
eR a
eR a
Proof. It could be easily verified by the definition as similar as way of reference [3]. h
3. Conclusion
In this paper, we show that our previously paper published in [1] contain slight flaws and then provide the correct versions. It could further perfect the research on fuzzy rough set over two universes.
Acknowledgements
The author would like to express his sincere thanks to the Editor-in-Chief Professor M. Cross for the valuable comments
and recommendations. The work was partly supported by the National Natural Science Foundation of China (71161016,
71071113). A Ph.D. Programs Foundation of Ministry of Education of China (20100072110011), a Foundation for the Author
of National Excellent Doctoral Dissertation of PR China (200782), Shanghai Pujiang Program, and Shanghai Philosophical and
Social Science Program (2010BZH003), the Fundamental Research Funds for the Central Universities.
References
[1] B.Z. Sun, W.M. Ma, Fuzzy rough set model on two different universes and its application, Appl. Math. Model. 35 (4) (2011) 1798–1809.
[2] W.Z. Wu, W.X. Zhang, Generalized fuzzy rough sets, Inform. Sci. 15 (2003) 263–282.
[3] W.X. Zhang, W.Z. Wu, J.Y. Liang, D.Y. Li, Rough Set Theory and Methodology, Science Press, Beijing, 2001.