Statistically pre-Cauchy Fuzzy real-valued sequences de fi ned by Orlicz function

In this article we have defined statistically pre-Cauchy sequence of fuzzy real numbers defined by Orlicz function. We have proved a necessary and sufficient condition for a sequence X = (Xk) of fuzzy real numbers to be statistically pre-Cauchy. We have also established some other results.


Introduction
Fast [8] and Schoenberg [13] independently extended the concept of convergence of a sequence to statistical convergence with the help of natural density function on the set of positive integers.For a set K ∈ N the density is defined by where |{k ∈ K : k ≤ n}| denote the numbers of element in the set.
The notion of statistically pre-Cauchy for real sequences was introduced by Connor, Fridy and Kline [2].A real-valued sequence x = (x k ) is said to be statistically convergent to a number L if for each ε > 0, lim A sequence x = (x k ) is said to be statistically pre-Cauchy if An Orlicz function is defined by the mapping M : [0, ∞) → [0, ∞), which is continuous, non-decreasing and convex with M (0) = 0, M(x) > 0 for x > 0 and M (x) → ∞ as x → ∞.If the convexity of M is replaced by the sub-additivity M (x + y) ≤ M (x) + M (y), then it is called the modulus function.An Orlicz function may be bounded or unbounded.
A fuzzy real number X is a fuzzy set on R, more precisely a mapping X : R → I(= [0, 1]), associating each real number t with its grade of membership X(t), which satisfy the following conditions: , is open in the usual topology of R, for all a ∈ I.
The class of all upper-semi-continuous, normal, convex fuzzy real numbers is denoted by R(I).
The set of real numbers R can be embedded into R(I), since each r ∈ R can be regarded as a fuzzy number r given by r(t) = ½ 1 for t = r, 0, for t 6 = r.
The additive identity and multiplicative identity of R(I) are denoted by 0 and 1 respectively.The α-level set of a fuzzy real number X is defined by Consider the mapping d :

Clearly d define a metric on R(I).
A fuzzy real-valued sequence (X k ) is a function from the set of natural numbers into R(I).It is denoted by (X k ), where X k ∈ R(I), for all k ∈ N .
The set E F of sequences taken from R(I) is closed under addition and scalar multiplication defined as follows: Let E F be the class of sequence of fuzzy real numbers, the linearity of E F can be understand as follows: For A sequence (X k ) of fuzzy real numbers is said to be convergent to the fuzzy real number X 0 , if for every ε > 0, there exists k 0 ∈ N such that d(X k , X 0 ) < ε, for all k ≥ k 0 .
A sequence (X k ) of fuzzy real numbers is said to be bounded if where R * (I) denotes the set of all positive fuzzy real numbers.
In this article we have defined statistically pre-Cauchy sequence of fuzzy real numbers and defined by Orlicz function.
A fuzzy real valued sequence X = (X k ) is said to be statistically pre-Cauchy if for each ε > 0,

Main Results
Theorem 3.1.Let M be an Orlicz function, then the sequence (X k ) of fuzzy real numbers is statistically pre-Cauchy if and only if Proof.First we suppose that for some ρ > 0.
We have Conversely suppose that (X k ) is statistically pre-Cauchy.For a given ε > 0, choose δ > 0 such that M (δ) < ε 2 and since M is continuous so there exists a positive integer K such that M (X) < K 2 , for all X.We have Since X is statistically pre-Cauchy, the R.H.S can be made less than ε.

Thus we have lim
This completes the proof.
Theorem 3.2.Let M be an Orlicz function, then the sequence (X k ) of fuzzy real numbers is statistically convergent to L if and only if lim Proof.First we suppose that lim We have Thus (X k ) is statistically convergent to L.
Converse part can be proved in a similar manner to the previous theorem.
Theorem 3.3.A statistically convergent fuzzy real valued sequence is statistically pre-Cauchy.
Proof.Let the fuzzy real-valued sequence (X k ) is statistically convergent.Let A ⊂ N be such that δ(A) = 1.Select B ⊂ A such that A\B is finite and B × B ⊂ {(j, k) : d(X k , X m ) < ε} for some ε > 0. Now for n ∈ N , we have This completes the proof.

n→∞ 1 n
|{k ≤ n : |x k − L| ≥ ε}| = 0, where the vertical bars indicate the number of elements in the enclosed set.