Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets
The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy set...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-12-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/7/1/11 |
| _version_ | 1811256127355092992 |
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| author | Hsien-Chung Wu |
| author_facet | Hsien-Chung Wu |
| author_sort | Hsien-Chung Wu |
| collection | DOAJ |
| description | The conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets. |
| first_indexed | 2024-04-12T17:35:24Z |
| format | Article |
| id | doaj.art-02c750aca7214043a33b1cd3c5692e8f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-12T17:35:24Z |
| publishDate | 2018-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-02c750aca7214043a33b1cd3c5692e8f2022-12-22T03:22:59ZengMDPI AGMathematics2227-73902018-12-01711110.3390/math7010011math7010011Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy SetsHsien-Chung Wu0Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, TaiwanThe conventional concept of α-level sets of fuzzy sets will be treated as the upper α-level sets. In this paper, the concept of lower α-level sets of fuzzy sets will be introduced, which can also be regarded as a dual concept of upper α-level sets of fuzzy sets. We shall also introduce the concept of dual fuzzy sets. Under these settings, we can establish the so-called dual decomposition theorem. We shall also study the dual arithmetics of fuzzy sets in R and establish some interesting results based on the upper and lower α-level sets.http://www.mdpi.com/2227-7390/7/1/11dual decomposition theoremdual fuzzy setsdual arithmeticslower α-level setsupper α-level sets |
| spellingShingle | Hsien-Chung Wu Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets Mathematics dual decomposition theorem dual fuzzy sets dual arithmetics lower α-level sets upper α-level sets |
| title | Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets |
| title_full | Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets |
| title_fullStr | Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets |
| title_full_unstemmed | Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets |
| title_short | Duality in Fuzzy Sets and Dual Arithmetics of Fuzzy Sets |
| title_sort | duality in fuzzy sets and dual arithmetics of fuzzy sets |
| topic | dual decomposition theorem dual fuzzy sets dual arithmetics lower α-level sets upper α-level sets |
| url | http://www.mdpi.com/2227-7390/7/1/11 |
| work_keys_str_mv | AT hsienchungwu dualityinfuzzysetsanddualarithmeticsoffuzzysets |