Homeomorphism Problems of Fuzzy Real Number Space and The Space of Bounded Functions with Same Monotonicity on [-1,1]
Keywords:
fuzzy numbers, fuzzy structured element, standard monotonic bounded functions, fuzzy functional, homeomorphismAbstract
In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space ε1 and the space B[−1, 1], which defined as a set of standard monotonic bounded functions with monotonicity on interval [−1, 1]. Furthermore, a new approach based upon the monotonic bounded functions has been proposed to create fuzzy numbers and represent them by suing fuzzy structured element. In order to make two different metrics based space in B[−1, 1], Hausdorff metric and Lp metric, which both are classical functional metrics, are adopted and their topological properties are discussed. In addition, by the means of introducing fuzzy functional to space B[−1, 1], we present two new fuzzy number’s metrics. Finally, according to the proof of homeomorphism between fuzzy number space ε1 and the space B[−1, 1], it’s argued that not only does it give a new way to study the fuzzy analysis theory, but also makes the study of fuzzy number space easier.
References
M.-S. Yang, C.-H. Ko (1996), On a class of fuzzy c-numbers clustering procedures for fuzzy data, Fuzzy sets and systems,84(1):49-60. http://dx.doi.org/10.1016/0165-0114(95)00308-8
U. Reuter(2011), A fuzzy approach for modeling non-stochastic heterogeneous data in engi- neering based on cluster analysis, Integrated Computer-Aided Engineering,18(3):281-289.
A. Li, Y. Shi, J. He, Y. Zhang (2011), A fuzzy linear programming-based classification method, International Journal of Information Technology & Decision Making, 10(06):1161- 1174. http://dx.doi.org/10.1142/S0219622011004750
K. Lin, P. Pai, Y. Lu, P. Chang (2013), Revenue forecasting using a least-squares support vector regression model in a fuzzy environment, Information Sciences, 220:196-209. http://dx.doi.org/10.1016/j.ins.2011.09.003
H. Wang, S. Guo, L. Yue (2014), An approach to fuzzy multiple linear regression model based on the structural element theory, Systems Engineering - Theory & Practice, 34(10): 26-28.
P. Diamond, P. Kloeden (1989), Characterization of compact subsets of fuzzy sets, Fuzzy Sets and Systems, 29(3):341-348. http://dx.doi.org/10.1016/0165-0114(89)90045-6
P. Diamond, P. Kloeden (1990), Metric spaces of fuzzy sets, Fuzzy sets and systems, 35(2): 241-249. http://dx.doi.org/10.1016/0165-0114(90)90197-E
P. Diamond, P. E. Kloeden, P. E. Kloeden (1994), A. Mathematician, P. E. Kloeden, Metric spaces of fuzzy sets: theory and applications, World Scientific. http://dx.doi.org/10.1142/2326
R. Goetschel, W. Voxman (1983), Topological properties of fuzzy numbers, Fuzzy Sets and Systems, 10(1):87-99. http://dx.doi.org/10.1016/S0165-0114(83)80107-9
L. Gerg (1992), Generalisation of the goetschel-voxman embedding, Fuzzy sets and systems, 47(1): 105-108. http://dx.doi.org/10.1016/0165-0114(92)90066-D
M. L. Puri, D. A. Ralescu (1983), Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications, 91(2):552-558. http://dx.doi.org/10.1016/0022-247X(83)90169-5
C.X. Wu, M. Ma (1991), Embedding problem of fuzzy number space: Part i, Fuzzy Sets and Systems, 44(1):33-38. http://dx.doi.org/10.1016/0165-0114(91)90030-T
C.X. Wu, M. Ma(1992), Embedding problem of fuzzy number space: Part ii, Fuzzy Sets and Systems, 45(2):189-202. http://dx.doi.org/10.1016/0165-0114(92)90118-N
S. Guo(2002), Method of structuring element in fuzzy analysis, Journal of Liaoning Technical University, 21(5):670-673.
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