Bipolar Fuzzy Relations
Division of Applied Mathematics, Nanoscale Science and Technology Institute, Wonkwang University, Iksan 54538, Korea
Department of Applied Mathematics, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Korea
Author to whom correspondence should be addressed.
Mathematics 2019, 7(11), 1044; https://doi.org/10.3390/math7111044
Received: 22 September 2019 / Revised: 24 October 2019 / Accepted: 31 October 2019 / Published: 3 November 2019
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications)
We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an
-level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their -level sets.
Keywords: bipolar fuzzy relation; bipolar fuzzy reflexive (resp., symmetric and transitive) relation; bipolar fuzzy equivalence relation; bipolar fuzzy partition; (a, b)-level set
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MDPI and ACS Style
Lee, J.-G.; Hur, K. Bipolar Fuzzy Relations. Mathematics 2019, 7, 1044.
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Lee J-G, Hur K. Bipolar Fuzzy Relations. Mathematics. 2019; 7(11):1044.Chicago/Turabian Style
Lee, Jeong-Gon; Hur, Kul. 2019. "Bipolar Fuzzy Relations." Mathematics 7, no. 11: 1044.
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