Bipolar Fuzzy Relations

• Main Authors: Jeong-Gon Lee, Kul Hur
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Mathematics
• Subjects: bipolar fuzzy relation; bipolar fuzzy reflexive (resp., symmetric and transitive) relation; bipolar fuzzy equivalence relation; bipolar fuzzy partition; (a,b)-level set
• Online Access: https://www.mdpi.com/2227-7390/7/11/1044

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-level sets.