Fuzzy Set Field and Fuzzy Metric

Abstract

The notation of fuzzy set field is introduced. A fuzzy metric is redefined on fuzzy set field and on arbitrary fuzzy set in a field. The metric redefined is between fuzzy points and constitutes both fuzziness and crisp property of vector. In addition, a fuzzy magnitude of a fuzzy point in a field is defined.

Constructing the Fuzzy Hyperbola and Its Applications in Analytical Fuzzy Plane Geometry

“…(Fuzzy Set) Let X be a nonempty set and μ: X ⟶ [0, 1] be a mapping. A � (α, μ(α)): α ∈ X 􏼈 􏼉 is called a fuzzy set in X with membership function μ [5].…”

“…By the aid of the membership function, the hyperbolic segment FH 1,...,5 can be defned by μ (x, y)|FH 1,..., 5 􏼐 􏼑 � sup α: Where (x, y) lies on FH α that passes through five same points on…”

“…(i) Te support of fuzzy points H i∞ , (i � 1, 3,4,5) should be taken as a compact set (ii) Te cores of fuzzy points H i∞ , (i � 1, 3, 4, 5) should lie on the core hyperbola CH Fuzzy hyperbola segments FH i∞ , (i � 1, 3, 4, 5) can be formed by using fuzzy same points H i∞ , (i � 1, 3, 4, 5) on the hyperbola, and their fuzzy points extended to infnity. Te important detail here is connecting the same points of H 1 with H 1∞ , H 3 with H 3∞ , H 4 with H 4∞ , H 5 with H 5∞ segments with these connecting same points give us the fuzzy hyperbola.…”

“…Gebray and Reddy dealt with properties of fuzzy set, fuzzy metric, fuzzy feld, fuzzy set feld, and fuzzy magnitude [5].…”