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Existence and uniqueness for the construction of fuzzy sets from a solidly nested family

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Abstract

Given a pre-determined subset \(S\) of \((0,1]\) and a solidly nested family \({\mathcal {M}}\) of subsets of a universal set \(U\), in this paper, we propose a methodology to construct a fuzzy subset of \(U\) such that its range is \(S\) and the family consisting of all its \(\alpha \)-level sets is permutably identical to \({\mathcal {M}}\).

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Authors and Affiliations

  1. Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, 802, Taiwan

    Hsien-Chung Wu

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  1. Hsien-Chung Wu
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Correspondence to Hsien-Chung Wu.

Appendix

Appendix

The membership functions in Examples 4.1 and 4.2 are shown in Fig. 1 in which \(\kappa ^{\circ }\) is taken as the identity function.

Fig. 1
figure 1

The membership functions

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Wu, HC. Existence and uniqueness for the construction of fuzzy sets from a solidly nested family. Fuzzy Optim Decis Making 14, 1–41 (2015). https://doi.org/10.1007/s10700-014-9190-4

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  • DOI: https://doi.org/10.1007/s10700-014-9190-4

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