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 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 Email: 411682485@qq.com 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 ,,mmmMOcv 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,mmmvcvmFor convenience, the whole matter-element is ex- pressed as £M, the whole matter is expressed as £mO, and whole characteristic as . The domain of measure of characteristic cm is expressed as £mcmVc , 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 O12,,,mm mncc c1, 2,, nOmivim1, 2,,mici n 1122 ,,mm mmm mmmmn mnOc vcvMOCVcv is referred to as n-dimensional matter-element, wherein 12mmmmnccCc, . 12mmmmnvvVv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 aOavacaOacO,,aaaAOcv 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 ancc canvaO12,,,aavv12,,,aa ancc c1122 ,,aa aaa aaaan anOc vcv OCV Acv is referred to as n-dimensional affair-element, wherein 1122, aaaaaaan ancvcvCVcv       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      ,,,,,,,,,EUtMtAtRtxtfSutUt MtAt RtutfUtrxt 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; XtfSut 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,UututxtExtTfS ut0,  ,,0,UututxtExtTfS 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) andcontradictious2)completely can not push out contain 3)some part can not push out 4)possible can not push out St utut StSt utSt utSt ut In the definition fS ut should be a general situation St12SS,fut . Propositi on 3 .1.1 In U, if , then 1,f f2       1111,,,, ,,, ,´, EUtMtAtRtxtfSutUt MtAt RtutUtfUtrxtfut        2222,,,,,,, ,, EUtMtAtRtxtfSutUt MtAt RtutUtfUtrxtfS ut have 12EE, vice versa. Proof, when 121,SSf f2, 12, ,UxtfSutfSutyt  utOpen Access AM Q. X. YE, J.-G. ZHOU 1671so, when ,ut U t xtE2 have ,,t E1,uut yxtyte can kn, ow so, conversely, if , t12EE，2S. Also12EEhen ,w1S if f1f2 Is true in Uut U, have 12fSut fSut , so, for u ,,,t xtEutyt  ,,,utxtEutyt1212EE , have xtyt, This contradiction with 1EE2. so, 12ff. The end. 3.2. The Class of Extension Error Set - We according to the features of the elements can be divided: 1) Classic extension error set    E  ,,,, ´0,1,xtfSutUt MtAt RtutUfUxt fSut 2) Fuzzy extension error set ,,,UtMtAtRt      ,,,UtMtAtRt, ,,,´0,1,E xtfSutUt MtAt RtutUfUxt fSut 3) Have critical point extension error set   ,,,,UtMtAtRtxtf  ,,,,, E SutUt MtAt RtutUfUxtfS ut  4. The Research of Fuzzy Extension Error Set - eration of extension error set. This section mainly research the definition, relation, op4.1. 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 SetDefi uppose 4.2. The Relation between Fuzzy Extensions 4.2.1. Equation nition 4. 2 . 1 S ,0,1fSufU, 1,Ux,Euxu 2222,, ,0,EuyuUyfSufU1f , iuU, have 1 ,uxE, , make2 ,ux E xy and 12SS, we 12EEociation 14.2.2. Subcall that  for assS 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 12UU so EE1E 2, 1221EEfinition, there are clearlytablished the follow- ing proposition: . By de esProposition 4.2.1 Suppose 1E, 2E, 3E are subset for association rule S: 3, then pose are fuzzy sets for association rule S 1) 11EE, 2) if 12EE, 1Eosition 4.E2.3 Supp 23EE. , EPro 1S1, E2 2:Then  1,, ,0,1EuxuUx SufU, f222,0,1fSufU If 2,,EuyuUyuU, 1,ux E, 2,uxE, have xy, then 12EE or 21EE for association rule in 1, 2E, 3E fuzzy n rule S1, S2 ,S3 in U U. Proposition 3.1.2.4 Suppose aresubsets for associatio4.3. The Operations of between Fuz Extension Error Sets If E1) 11EE, 2) if 12EE, 13EE,then 23EE . zy4.3.1. The Union of Fuzzy Extension Error Set ,, ,xySS12 0f, nition 4.3.1.1then the definition of Fuzzy ex- tension error set’s union for association rule 12,SS: Defi Suppose 1E and 2E are fuzzy sets for association rule S1, S2 in U, and  312,, ,,max,uzuxEuyEy, then 312EEE ,Ez x, means union. ets for ule S1, S2, then Proposition 3.2.2.1 Supposeubsassociation r2 E1, 2E are s1) 111EEE; 2) 12 21EE EE;  f EE3) i1, thn e112EEEE2  . 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 , arsets for association 5. Conclusion Extenics and error eliminating theory have increasinglytion 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. Ca312EEE means intersection. Propositio 1E, 2E3Ee sub- rule S1, S2, S3then E; 1) 111EE2) 12 21EE EE ; 3) 123EEE EEE terKnowledge 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 M12  3espe- 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 attencially 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. REFERENCES ment Research of Extension Detecting,” International Conference on Computer, Communication and Control Technologies (CCCT), Florida, July 2003. [6] Y. Huang, B. Zeng and M. H. Wang, “TheElement Research of Extension Detecting,” International Conference on Computer, Communication and Control Technologies (CCCT), Florida, July 2003. Science Press, Beijing, 2