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A Note on Zadeh's Extension Principle

Received: 27 November 2014     Accepted: 2 December 2014     Published: 27 December 2014
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Abstract

For a mapping, fuzzy sets obtained by Zadeh's extension principle are images of other fuzzy sets on the domain of the mapping under the mapping. Some relationships between images of level sets of one or two fuzzy sets under a mapping and another fuzzy set obtained from the one or two fuzzy sets by Zadeh's extension principle are known. In the present paper, the known results are extended to more general ones, and some useful results for applications are derived by the extended ones.

Published in Applied and Computational Mathematics (Volume 4, Issue 1-2)

This article belongs to the Special Issue New Advances in Fuzzy Mathematics: Theory, Algorithms, and Applications

DOI 10.11648/j.acm.s.2015040102.13
Page(s) 10-14
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Zadeh’s Extension Principle, Fuzzy Inner Product, Fuzzy Distance

References
[1] D. Dubois, W. Ostasiewicz, and H. Prade, "Fuzzy sets: history and basic notions", in Fundamentals of Fuzzy Sets, D. Dubois and H. Prade, Eds. Boston: Kluwer, 2000, pp.21–124.
[2] S. K. Gupta and D. Dangar, "Duality for a class of fuzzy nonlinear optimization problem under generalized convexity", Fuzzy Optim. and Decis. Mak., 13 (2014) 131–150.
[3] M. Kurano, M. Yasuda, J. Nakagami, and Y. Yoshida, "Markov-type fuzzy decision processes with discounted reward on a closed interval", European J. Oper. Res., 92 (1996) 649–662.
[4] H. T. Nguyen, "A note on the extension principle for fuzzy sets", J. Math. Anal. Appl., 64 (1978) 369–380.
[5] H.-C. Wu, "Duality theory in fuzzy optimization problems formulated by the Wolfe's primal and dual pair", Fuzzy Optim. and Decis. Mak., 6 (2007) 179–198.
[6] H.-C. Wu, "The optimality conditions for optimization problems with fuzzy-valued objective functions", Optimization, 57 (2008) 473–489.
[7] H.-C. Wu, "The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions", Fuzzy Optim. and Decis. Mak., 8 (2009) 295–321.
[8] Y. Yoshida, "A time-average fuzzy reward criterion in fuzzy decision processes", Inform. Sci., 110 (1998) 103–112.
[9] L. A. Zadeh, "Fuzzy sets", Inform. and Control, 8 (1965) 338–353.
Cite This Article
  • APA Style

    Masamichi Kon. (2014). A Note on Zadeh's Extension Principle. Applied and Computational Mathematics, 4(1-2), 10-14. https://doi.org/10.11648/j.acm.s.2015040102.13

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    ACS Style

    Masamichi Kon. A Note on Zadeh's Extension Principle. Appl. Comput. Math. 2014, 4(1-2), 10-14. doi: 10.11648/j.acm.s.2015040102.13

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    AMA Style

    Masamichi Kon. A Note on Zadeh's Extension Principle. Appl Comput Math. 2014;4(1-2):10-14. doi: 10.11648/j.acm.s.2015040102.13

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  • @article{10.11648/j.acm.s.2015040102.13,
      author = {Masamichi Kon},
      title = {A Note on Zadeh's Extension Principle},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {1-2},
      pages = {10-14},
      doi = {10.11648/j.acm.s.2015040102.13},
      url = {https://doi.org/10.11648/j.acm.s.2015040102.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2015040102.13},
      abstract = {For a mapping, fuzzy sets obtained by Zadeh's extension principle are images of other fuzzy sets on the domain of the mapping under the mapping. Some relationships between images of level sets of one or two fuzzy sets under a mapping and another fuzzy set obtained from the one or two fuzzy sets by Zadeh's extension principle are known. In the present paper, the known results are extended to more general ones, and some useful results for applications are derived by the extended ones.},
     year = {2014}
    }
    

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    AB  - For a mapping, fuzzy sets obtained by Zadeh's extension principle are images of other fuzzy sets on the domain of the mapping under the mapping. Some relationships between images of level sets of one or two fuzzy sets under a mapping and another fuzzy set obtained from the one or two fuzzy sets by Zadeh's extension principle are known. In the present paper, the known results are extended to more general ones, and some useful results for applications are derived by the extended ones.
    VL  - 4
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Author Information
  • Graduate School of Science and Technology, Hirosaki University, Hirosaki, Japan

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