Applied Mathematics, 2013, 4, 1669-1672
Published Online December 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.412227
Open Access AM
Extension Error Set Based on Extension Set
Qixin Ye, Jinge Zhou
Guangdong University of Technology, School of Management, Guangzhou, China
Received July 31, 2013; revised August 31, 2013; accepted September 7, 2013
Copyright © 2013 Qixin Ye, Jinge Zhou. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper gives the concepts of extension error set and fuzzy extension error set, discusses diverse extension error set
and fuzzy extension error set based on extension set and error set, and puts forward the relevant propositions and opera-
tions. Finally, it provides proofs of the soundness and completeness for the propositions and operations.
Keywords: Extension Set; Error Set; Fuzzy Extension Error Set
1. Introduction
In the field of fuzzy mathematics, the research of set
mainly concentrates on the static form of fuzzy set and
its effective forms of reasoning and rule. However, the
dynamic changes of the fuzzy set are important parts of
set research. In this paper, firstly, we study extension er-
ror set and fuzzy extension error set’s dynamic concept
based on the theory of error eliminating and extenics.
Then, we research diverse extension error set and fuzzy
extension error set, and put forward the relevant proposi-
tions and operations. Finally, we provide proofs of the
soundness and completeness for the propositions and
operations. In one word, because of the study of exten-
sion error set, this paper has very important theoretical
and practical significance in different fields.
2. Basic Definitions
2.1. Matter-Element [1-6]
Definition 2.1.1 An ordered triple composed of the meas-
ure vm of Om about cm, with matter Om as object, and cm
as characteristic

,,
mmm
M
Ocv
As the fundamental element for matter description, it’s
referred to as 1-dimensional matter-element, and m,
vm are referred to as the three key elements of matter-
element M, within which, the two-tuples composed of cm
and is referred to as the characteristic-ele-
ment of matter .
O

,
mmm
vcv

m
For convenience, the whole matter-element is ex-
pressed as


£
M
, the whole matter is expressed as


£m O, and whole characteristic as . The domain
of measure of characteristic cm is expressed as

£m c


m
Vc ,
referred to as the domain of measure of cm.
A matter with multiple characteristics, similar to 1-
dimensional matter-element, can be defined as a multi-
dimensional matter-element:
Definition 2.1.2 The array composed of matter m,
n-names of characteristics of and the
corresponding measure of about
O
12
,,,
mm mn
cc c

1, 2,, nO
mi
vim


1, 2,,
mi
ci n

11
22 ,,
mm m
mm mmm
mn mn
Oc v
cv
M
OCV
cv








is referred to as n-dimensional matter-element, wherein
1
2
m
m
m
mn
c
c
C
c













, .
1
2
m
m
m
mn
v
v
V
v







2.2. Affair-Element
Interaction between matters is referred to as affair, de-
scribed by af fair-element.
Definition 2.2.1 The ordered triple composed of action
, action’s characteristic and the obtained measure
of about
a
O
a
va
c
a
Oa c
O


,,
aaa
A
Ocv

Q. X. YE, J.-G. ZHOU
1670
is used as the fundamental element for affair description,
referred to as 1-dimensional affair-element.
Basic characteristics of action include dominating ob-
ject, acting object, receiving object, time, location, de-
gree, mode, and tool, etc.
Definition 2.2.2 The array composed of action a,
n-characteristics 12 and the obtained meas-
ure , of about
O
,,,
aa an
cc c
an
va O
12
,,,
aa
vv12 ,,,
aa an
cc c

11
22 ,,
aa a
aa aaa
an an
Oc v
cv OCV A
cv









is referred to as n-dimensional affair-element, wherein
11
22
,
aa
aa
aa
an an
cv
cv
CV
cv
 
 
 

 
 
 

2.3. Relation-Element
In the boundless universe, there is a network of relations
among any matter, affair, person, information, knowl-
edge and other matter, affair, person, information and
knowledge. Because of interaction and interplay among
these relations, the matter-element, affair-element and
relation-element describing them also have various rela-
tions with other matter-elements, affair-elements and
relation-elements, and the changes of theses relations
will also be interacting and interplaying. Relation-ele-
ment is a formalized tool to describe this kind of phe-
nomena.
3. The Research of Extension Error Set
We research extension error set based on the theory of
Extenics, and explore classical extension error set, fuzzy
extension error set, multivariate extension error set.
Moreover, we put forward the relevant propositions and
operations. According to thses propositions and opera-
tions, we provide some proofs.
3.1. The Definition of Extension Error Set
Suppose is an object set, is a set of asso-
ciation rules, if

Ut

St
      


    


  

,,,,
,,,
,,
EUtMtAtRtxtfSut
Ut MtAt Rtut
fUtrxt fS ut



,
we call that “E” is an extension error set for association
rule in domain . In detail, is a domain,

St

Ut U


St (incidence - standard) is a set of association rules,
M refers to the matter-element, A representative affair-
element, R represents the relationship between relation-
element;






X
tfSut

represents the correlation
functions of extension error set, R is the real number field,
T refers to the time. In this paper we take extension error
set as a complex system, its’ elements as subsystems.
So,
 

 


,,UututxtExt

0,
  

 


,,UututxtExt

0,
 

 


,,UututxtExt
0,











 


,,
0,
UututxtExt
TfS ut
0,


 











 
 
,,
0,
UututxtExt
TfS ut
0,


 
 



 


,,UututxtETfSut
0,
are called extension error set’s extension of the domain,
negative extension field, extension, stable domain and
negative stable region, critical region respectively.




 
 
 
 
1) andcontradictious
2)completely can not push out
contain 3)some part can not push out
4)possible can not push out
St ut
ut St
St ut
St ut
St ut








In the definition




fS ut should be a general
situation






St
12
SS
,fut .
Propositi on 3 .1.1 In U, if , then
1
,f f
2
      



     

  

11
11
,,,,
,,, ,
´,
EUtMtAtRtxtfSut
Ut MtAt RtutUt
fUtrxtfut



 
      



     
 
  

22
22
,,,,
,,, ,
,
EUtMtAtRtxtfSut
Ut MtAt RtutUt
fUtrxtfS ut



have 12
EE

, vice versa.
Proof, when 121 ,SSf f
2

,














12
, ,UxtfSutfSutyt  ut
Open Access AM

Q. X. YE, J.-G. ZHOU 1671
so, when

,ut U
 
t xtE
  
2
have
 
,,t E

1
,uut y



x
tyt
e can kn, ow so, conversely, if , t
12
EE，
2
S. Also

12
EE
hen ,w
1
S if f1f2 Is true in U

ut U, have




12
f
Sut fSut , so, for
 
u
    
,,,t xtEutyt
     
,,,utxtEutyt
12
12
E
E
, have



x
tyt,
This contradiction with 1
EE2 . so, 12
f
f. 
The end.
3.2. The Class of Extension Error Set
-
 
We according to the features of the elements can be di
vided:
1) Classic extension error set

  

 


E
  


 
 

,
,,,
´0,1,
xtfSut
Ut MtAt RtutU
fUxt fSut



2) Fuzzy extension error set

,,,UtMtAtRt
  

 



    


   

,,,UtMtAtRt,
,,,
´0,1,
E xtfSut
Ut MtAt RtutU
fUxt fSut



3) Have critical point extension error set


  







,,,,UtMtAtRtxtf
 




 

,, ,
,,
E Sut
Ut MtAt RtutU
fUxtfS ut


 
4. The Research of Fuzzy Extension Error
Set
-
eration of extension error set.
This section mainly research the definition, relation, op
4.1. The Definition of Fuzzy Extension Error Set
Definition 4.1.1 Suppose U is object set, S is a set of
association rules in U, if
  

,, ,0,1EuxuUxfSufU
, we call
that E is a fuzzy extensio
n error set for S in U .
Error Set
Defi uppose
4.2. The Relation between Fuzzy Extension
s
4.2.1. Equation
nition 4. 2 . 1 S
  


,0,1fSufU,
1,Ux
,Euxu
  


2222
,, ,0,EuyuUyfSufU
1f , i
uU

, have


1
,uxE

, , make

2
,ux E
x
y

and 12 SS

, we 12 EE
ociation
1
4.2.2. Sub
call that  for ass
S or S2.
set
Definition 4.2.2 Suppose U, Uare subset in U, and
rule
1 2
  


f
1, ,0,1UxfSuU
, ,Euxu
  


1,
of E

2222
,, ,0,EuyuUyfSufU if
for association rule S, is the subset
, or

12
UU
so EE

1
E
 2,
12
21
EE


finition, there are clearlytablished the follow-
ing proposition:
.
By de es
Proposition 4.2.1 Suppose 1 E
, 2
E
, 3
E
 are subset
for association rule S:
3
, then
pose are fuzzy sets for
association rule S
1) 11 EE

,
2) if 12 EE, 1 E

osition 4.E 2.3 Supp 23
EE

.
, EPro 1

S1, E2

2:
Then
  


1,, ,0,1EuxuUx SufU
,

f
 


222
,0,1fSufU If
2
,,EuyuUy
uU

,


1
,ux E

,


2
,uxE, have
x
y

, then
12
EE


or 21 EE

for association rule in
1
, 2
E
, 3
E
 fuzzy
n rule S1, S2 ,S3 in U
U.
Proposition 3.1.2.4 Suppose are
subsets for associatio
4.3. The Operations of between Fuz Extension
Error Sets
If
E
1) 11
EE

,
2) if 12 EE, 13 EE

,then 23 EE
 .
zy
4.3.1. The Union of Fuzzy Extension Error Set


,, ,xySS
12 0f

,
nition 4.3.1.1
then the definition of Fuzzy ex-
tension error set’s union for association rule 12
,SS:
Defi Suppose 1 E
 and 2
E
 are fuzzy
sets for association rule S1, S2 in U, and
 


312
,, ,,max,uzuxEuyEy

, then
312
EEE ,Ez x

, means union. ets for
ule S1, S2, then

Proposition 3.2.2.1 Supposeubs
association r
2
E1
, 2
E
 are s
1) 111
EEE

;
2) 12 21 EE EE

;
 
f EE 3) i1


, thn e112 EEEE
2


  .
4.he InteFuzzy Extension Error Set
Definition 4.3.2.1 Suppose and are fuzzy sets
3.2. Trsection of
1
2 
for association rule S1, S2 in U and
E E
 


312
,,,,, in,EuzuxEuyEzxy, then m

Open Access AM

Q. X. YE, J.-G. ZHOU
Open Access AM
1672
n 4.3.2.1 Suppose , ar
sets for association
5. Conclusion
Extenics and error eliminating theory have increasingly
tion of academia and industry,
[1] W. Cai, C. Y. Yang and W. C. Lin, “Extension Engineer-
ing Methods,”003.
i, “Study on
. H. Wang, “The Re lated Mat-
X. Peng and W. Q. Ye, “The Principle of
Basic Matter-
[2] C. Y. Yang, G. H. Wang, Y. Li and W. Ca
312
EEE
 means intersection. 
Propositio 1 E
, 2
E
3
E
e sub-
rule S1, S2, S3then
E; 1) 111 EE

2) 12 21
EE EE
 
;
3)
 
123
EEE EEE

ter
Knowledge Reasoning Based on Extended Formulas,” In-
ternational Conference on AIAI, Springer, New York,
Vol. 9, 2005, pp. 797-805.
[3] Y. Q. Yu, Y. Huang and M
12
 
3
espe- Extension Detecting with Extension Sets,” International
Conference on Computer, Communication and Control
Technologies (CCCT), Florida, July 2003, pp. 113-118.
[5] Z. Chen and Y. Q. Yu, “To Find the Key Matter-Ele-
-Elements in Extension Detecting and Application,”
Proceedings of IEEE International Conference on Infor-
mation Technology and Applications (ICITA), July 2005,
pp. 411-418.
[4] Y. Q. Yu, H.
attracted the atten
cially in the fields of management and decision-making.
So we study the extension error set and fuzzy extension
error set. But, what we have done is not enough. It’s in
administrative before our theory is perfect. So, we call
for more scholars from all over the world to do research
about extenics and error eliminating theory. Only in this
way, can they have wider value of applications in more
fields.
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