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Separation theorems and minimax theorems for fuzzy sets

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Abstract

In this paper, we prove some theorems on fuzzy sets. We first show that, in order to demonstrate that the equality of shadows ofA andB implies the equality ofA andB, it is necessary to assume thatA andB are closed and thatS H (A)=S H (B) for any closed hyperplane hyperplaneH. We also obtain a separation theorem for two convex fuzzy sets in a Hilbert space. Finally, we investigate results relating to minimax theorems for fuzzy sets. We obtain a necessary and sufficient condition for compactness.

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Authors and Affiliations

  1. Yokosuka Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, Take Yokosuka-Shi, Kanagawa, Japan

    M. Takahashi (Engineer)

  2. Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan

    W. Takahashi (Associate Professor)

Authors
  1. M. Takahashi
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  2. W. Takahashi
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Additional information

Communicated by G. Leitmann

The authors wish to express their sincere thanks to Professor Hisaharu Umegaki for his invaluable suggestions and advice.

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Takahashi, M., Takahashi, W. Separation theorems and minimax theorems for fuzzy sets. J Optim Theory Appl 31, 177–194 (1980). https://doi.org/10.1007/BF00934109

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