Statistically Convergent Fuzzy Sequence Spaces by Fuzzy Metric

Paritosh Chandra Das
2014 Kyungpook Mathematical Journal  
In this article we study different properties of the statistically convergent and statistically null sequence classes of fuzzy real numbers with fuzzy metric, like completeness, solidness, sequence algebra, symmetricity and convergence free. Definition 2.1. A fuzzy real number X is a fuzzy set on R, i.e. a mapping X : R → I(= [0, 1]) associating each real number t with its grade of membership X(t). Definition 2.3. If there exists t 0 ∈ R such that X(t 0 ) = 1, then the fuzzy real number X is
more » ... led normal. Definition 2.4. A fuzzy real number X is said to be upper-semi continuous if, for each ε > 0, X −1 ([0, a + ε) ), is open in the usual topology of R for all a ∈ I. The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by R(I). Throughout the article, by a fuzzy real number we mean that the number belongs to R(I). Definition 2.5. The α-level set [X] α of the fuzzy real number X, for 0 < α ≤ 1, is defined by [X] α = {t ∈ R : X(t) ≥ α}. If α = 0, then it is the closure of the strong 0-cut. (The strong α -cut of the fuzzy real number X, for 0 ≤ α ≤ 1 is the set {t ∈ R : X(t) > α}). Let X, Y ∈ R(I) and α-level sets be [X] α = [a α
doi:10.5666/kmj.2014.54.3.413 fatcat:5ouqkgumkbbevggiw4oyingx3e