On Statistical Convergent Sequence Spaces Of Intuitionistic Fuzzy Numbers

In the present paper we introduce the classes of sequence stc , stc IFN 0 and stl ∞ of statistically convergent, statistically null and statistically bounded sequences of intuitionistic fuzzy number based on the newly defined metric on the space of all intuitionistic fuzzy numbers (IFNs). We study some algebraic and topological properties of these spaces and prove some inclusion relations too.


Introduction
Intuitionistic fuzzy set (IFS) is one of the generalizations of fuzzy sets theory [1].Out of several higher-order fuzzy sets, IFS was first introduced by Atanassov, have been found to be compatible to deal with vagueness.The conception of IFS can be viewed as an appropriate and alternative approach in case where available information is not sufficient to define the impreciseness by the conventional fuzzy set theory.In fuzzy sets the degree of acceptance is considered only but IFS is characterized by a membership function and a non-membership function such that the sum of both values is less than one.Presently IFSs are being studied and used in different fields of science.Let X be universe of discourse defined by X = {x 1 , x 2 , . . .x n }.The grade of membership of an element x i ∈ X in a fuzzy set is represented by real value in [0, 1].It does indicate the evidence for x i ∈ X, but does not indicate the evidence against x i ∈ X. Atanassov presented the concept of IFS, an IFS A in X is characterized by a membership function A (x) and a nonmembership function ν A (x).Here A (x) and ν A (x) are associated with each point in X, a real number in [0, 1] with the values of A (x) and ν A (x) at X representing the grade of membership and non-membership of x in A. When A is an ordinary (crisp) set, its membership function can take only two values zero and one.An IFS becomes a fuzzy set A when ν A (x) = 0 but A (x) ∈ [0, 1] for all S. Debnath, V. N. Mishra and J. Debnath x ∈ A. Burillo et.al. [4] proposed definition of intuitionistic fuzzy number (IFN).The notion of statistical convergence was introduced by Fast [11] and Schoenberg [20] independently.This concept has been generalized and developed by Fridy [12], ˇSalat [18], Connor [6], Connor et.al. [7], Et and Nuray [10] and many others.Nuray and Savas [17], extended the idea to apply to sequences of fuzzy numbers.Later on, Aytar and Pehlivan [2], Bilgin [3], Colak et.al. [5], Kwon [14], Tripathy and Baruah [21], Savas [18], Debnath et.al. [9] and many others extended the idea of statistical convergence to the sequences of fuzzy numbers.The existing literature on statistical convergence appears to have been restricted to sequences of real numbers, complex numbers or fuzzy numbers.As intuitionistic fuzzy numbers are generalization of fuzzy numbers, it is reasonable to think about the existing sequence spaces of real numbers, interval numbers and fuzzy numbers in terms of intuitionistic fuzzy numbers.Recently the authors have introduced convergent, null and bounded sequence spaces of intuitionistic fuzzy numbers.In the current paper we have introduced and studied the properties of the sequence spaces st c IF N , st c IF N 0 and st ℓ IF N ∞ of statistically convergent, null and bounded sequences of intuitionistic fuzzy numbers with the help of a newly defined metric.

Preliminaries
Definition 2.1.A fuzzy number X is a fuzzy subset of the real line R, i.e, a mapping X : R → I = [0, 1] associating each real number t with its grade of membership X(t).A fuzzy number X is normal if there exists t 0 ∈ R such that X(t 0 ) = 1.A fuzzy number X is upper semi continuous if for each ε > 0,

[15]
A sequence space E is said to be monotone if E contains the canonical pre image of all its step spaces.Definition 2.5.[15] A sequence space E is said to be sequence algebra

[15]
A sequence space E is normal implies that it is monotone.Definition 2.6.Let X be a given non-empty set.An intuitionistic fuzzy set(IFS) [1] in X is an object A given by an intuitionistic fuzzy subset of the real line.
ii) normal, i.e., there is any , and µ A (x) + ν A (x) ≤ 1 is called an intuitionistic fuzzy number.We denote by A = (µ A , ν A ), an intuitionistic fuzzy number and by R 2 (I), the set of all IFN.It is obvious that any fuzzy number B = {< x, µ A (x) >: x ∈ X} can be represented as an intuitionistic fuzzy number by Let D 2 be the set of all closed and bounded intervals of the form ([ ] α are closed and bounded intervals of the following form: with respect to the α−cuts of the fuzzy number 1 − ν X , the following equality is immediate: The additive identity and multiplicative identity of R 2 (I) are 0 and 1 respectively.
Let X, Y ∈ R 2 (I) and the α− level sets are A sequence X = (X k ) of intuitionistic fuzzy numbers is a function X from the set N of all positive integers into R 2 (I).Thus, a sequence of intuitionistic fuzzy numbers X is a correspondence from the set of positive integers to a set of intuitionistic fuzzy numbers, i.e., to each positive integer k there corresponds an intuitionistic fuzzy number X (k ).It is more common to write X k rather than X (k ) and to denote the sequence by (X k ) rather than X .The intuitionistic fuzzy number X k is called the k−th term of the sequence.Definition 2.8.A sequence X = (X k ) of IFN is said to be convergent to an intuitionistic fuzzy number X 0 , if there exists a positive integer n 0 such that d(X k , X 0 ) < ε for all k > n 0 .We write lim X k = X 0 .Definition 2.9.A sequence X = (X k ) of IFN is said to be Cauchy sequence if there exists a positive integer n 0 such that d(X k , X m ) < ε for all n, m ≥ n 0 .Definition 2.10.A sequence X = (X k ) of IFN is said to be bounded if the sets

Main Result
Let us define a mapping d2 : R ) is a complete metric space.Proof: Let {x k } be any Cauchy sequence in R 2 (I).Then there is a n 0 ∈ N such that d2 (x k , x m ) < ε for all k, m > n 0 .⇒sup α∈[0,1] d(x k(α) , x m(α) ) < ε.