Abstract
In this paper, based on the fuzzy structured element, we prove that there is a bijection function between the fuzzy number space \(\varepsilon ^1\) 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 \(L_p\) metric, which both are classical functional metrics, is adopted and their topological properties is 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 \(\varepsilon ^1\) and the space \(B[-1, 1]\), it’s argued that not only it gives a new way to study the fuzzy analysis theory, but also make the study of fuzzy number space easier.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cong-Xin, W., Ming, M.: Embedding problem of fuzzy number space: part I. Fuzzy Sets Syst. 44(1), 33–38 (1991)
Congxin, W., Ming, M.: Embedding problem of fuzzy number space: part II. Fuzzy Sets Syst. 45(2), 189–202 (1992)
Diamond, P., Kloeden, P.: Characterization of compact subsets of fuzzy sets. Fuzzy Sets Syst. 29(3), 341–348 (1989)
Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. Fuzzy Sets Syst. 35(2), 241–249 (1990)
Diamond, P., Kloeden, P.E., Kloeden, P.E., Mathematician, A., Kloeden, P.E.: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, Singapore (1994)
Gergó, L.: Generalisation of the Goetschel-Voxman embedding. Fuzzy Sets Syst. 47(1), 105–108 (1992)
Goetschel, R., Voxman, W.: Topological properties of fuzzy numbers. Fuzzy Sets Syst. 10(1), 87–99 (1983)
Guo, S., Su, Z., Wang, L.: Method of structured element in fuzzy analysis and calculation. Fuzzy Syst. Math. 3, 011 (2004)
Guo, S.: Method of structuring element in fuzzy analysis. J. Liaoning Tech. Univ. 21(5), 670–673 (2002)
Li, A., Shi, Y., He, J., Zhang, Y.: A fuzzy linear programming-based classification method. Int. J. Inf. Tech. Decis. Making 10(06), 1161–1174 (2011)
Lin, K., Pai, P., Lu, Y., Chang, P.: Revenue forecasting using a least-squares support vector regression model in a fuzzy environment. Inf. Sci. 220, 196–209 (2013)
Puri, M.L., Ralescu, D.A.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91(2), 552–558 (1983)
Reuter, U.: A fuzzy approach for modelling non-stochastic heterogeneous data in engineering based on cluster analysis. Integr. Comput. Aided Eng. 18(3), 281–289 (2011)
Wang, H., Guo, S., Yue, L.: An approach to fuzzy multiple linear regression model based on the structural element theory. Syst. Eng. Theor. Pract. 34(10), 2628 (2014)
Yang, M.S., Ko, C.H.: On a class of fuzzy c-numbers clustering procedures for fuzzy data. Fuzzy Sets Syst. 84(1), 49–60 (1996)
Acknowledgements
This work has been partially supported by grants from National Natural Science Foundation of China (Grant NO.71331005, NO.71110107026.)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Wang, H., Guo, S., Shi, Y. (2015). Homeomorphism Between Fuzzy Number Space and the Space of Bounded Functions with Same Monotonicity on \([-1,1]\) . In: Zhang, C., et al. Data Science. ICDS 2015. Lecture Notes in Computer Science(), vol 9208. Springer, Cham. https://doi.org/10.1007/978-3-319-24474-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-24474-7_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24473-0
Online ISBN: 978-3-319-24474-7
eBook Packages: Computer ScienceComputer Science (R0)