Abstract
This paper introduces a new notion to define equivalence to the non-reflexive fuzzy relation equation. The most important condition for the fuzzy equivalence relations is reflexive, but the condition of reflexive is unsuccessful in many cases of fuzzy relation in real life problems that creates problem to form partition tree. Therefore, this paper defines the equivalence for the cases of fuzzy relation that satisfies the symmetry, and transitive. That is, it proves the reflexive to non-reflexive fuzzy relations through alpha-cut relations. Further, this paper defines tolerance to the non-reflexive fuzzy relation equation through alpha-cut relations and it proves the entire upper left, lower right and centre sub matrices of every weakly-similarity relation matrix are weaklysimilarity relation matrices.
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