{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T05:14:23Z","timestamp":1787030063861,"version":"build-2736575974"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2017,8,31]],"date-time":"2017-08-31T00:00:00Z","timestamp":1504137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors are perpendicular. We present the basic properties of orthogonality of complex fuzzy sets and various results on orthogonality with respect to complex fuzzy complement, complex fuzzy union, complex fuzzy intersection, and complex fuzzy inference methods. Finally, an example application of signal detection demonstrates the utility of the orthogonality of complex fuzzy sets.<\/jats:p>","DOI":"10.3390\/sym9090175","type":"journal-article","created":{"date-parts":[[2017,8,31]],"date-time":"2017-08-31T10:54:44Z","timestamp":1504176884000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":49,"title":["The Orthogonality between Complex Fuzzy Sets and Its Application to Signal Detection"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8591-2313","authenticated-orcid":false,"given":"Bo","family":"Hu","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang 550025, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lvqing","family":"Bi","sequence":"additional","affiliation":[{"name":"School of Electronics and Communication Engineering, Yulin Normal University, Yulin 537000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2283-9459","authenticated-orcid":false,"given":"Songsong","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Information Science and Engineering, Xiamen University, Xiamen 361005, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,8,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1109\/91.995119","article-title":"Complex fuzzy sets","volume":"10","author":"Ramot","year":"2002","journal-title":"IEEE Trans. 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