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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 ,, 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,, nO mi vim 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 vv12 ,,, 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 f1f2 Is true in U ut U, have 12 f Sut fSut , so, for u ,,,t xtEutyt ,,,utxtEutyt 12 12 E E , have x tyt, This contradiction with 1 EE2 . 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. 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, “The Element Research of Extension Detecting,” International Conference on Computer, Communication and Control Technologies (CCCT), Florida, July 2003. Science Press, Beijing, 2 |