We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation.\nWe study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive,\nsymmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence\nclass and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes\nis a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar\nfuzzy partition. Finally, we define an (a, b)-level set of a bipolar fuzzy relation and investigate some\nrelationships between bipolar fuzzy relations and their (a, b)-level sets.
Loading....