I-Statistically Pre-Cauchy Triple Sequences of Fuzzy Real Numbers
Sangita Saha, Bijan Nath, Santanu Roy^{*}
Department of Mathematics, National Institute of Technology Silchar, Assam, India
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To cite this article:
Sangita Saha, Bijan Nath, Santanu Roy. I-statistically Pre-Cauchy Triple Sequences of Fuzzy Real Numbers. International Journal of Management and Fuzzy Systems. Vol. 2, No. 2, 2016, pp. 15-21. doi: 10.11648/j.ijmfs.20160202.12
Received: August 18, 2016; Accepted: August 29, 2016; Published: October 11, 2016
Abstract: In this article, the notion of I-statistically pre-Cauchy sequence of fuzzy real numbers having multiplicity greater than two is introduced. We establish the criterion for any arbitrary triple sequence of fuzzy numbers to be I-statistically pre-Cauchy. It is shown that an I-statistically convergent sequence of fuzzy numbers is I-statistically pre-Cauchy. Moreover a necessary and sufficient condition for a bounded triple sequence of fuzzy real numbers to be I-pre-Cauchy is established.
Keywords: Ideal, Filter, Statistical Convergence, Ideal Convergence, I-Statistical Convergence, Triple Sequence of Fuzzy Numbers, I-statistical Pre-Cauchy, Orlicz Function
1. Introduction
Fuzzy set theory, compared to other mathematical theories, is perhaps the most easily adaptable theory to practice. The main reason is that a fuzzy set has the property of relativity, variability, and inexactness in the definition of its elements. Instead of defining an entity in calculus by assuming that its role is exactly known, we can use fuzzy sets to define the same entity by allowing possible deviations and inexactness in its role. Fuzzy set theory has become an area of active area of research in science and engineering for the last 46 years. While studying fuzzy topological spaces, many situations are faced, where we need to deal with convergence of fuzzy numbers. The concept of fuzzy set is first introduced by Zadeh [39] in 1965. Later on, the sequence of fuzzy numbers is discussed by several mathematicians such as Matloka [20], Nanda [22], Savas [30] and many others.
The notion of statistical convergence was first introduced by Fast [13]. After then it was studied by many researchers like Šalát [27], Fridy [14], Connor [4], Maddox [19], Kwon [18], Savas [29]. Different classes of statistically convergent sequences were introduced and investigated by Tripathy and Sen [37], Tripathy and Sarma [36] etc. Móricz [21] extended statistical convergence from single to multiple real sequences. Nuray and Savas [24] first defined the concepts of statistical convergence and statistically Cauchy for sequences of fuzzy numbers. The notion of statistically pre-Cauchy for real sequences was introduced by Connor, Fridy and Kline [5]. More works on statistically pre-Cauchy sequences are found in Khan and Lohani [16], Dutta [10], Dutta and Tripathy [9], Das et al. [7] etc.
Agnew [1] studied the summability theory of multiple sequences and obtained certain theorems which have already been proved for double sequences by the author himself. The different types of notions of triple sequences was introduced and investigated at the initial stage by Sahiner et al. [25], Sahiner and Tripathy [26]. Recently Savas and Esi [32] have introduced statistical convergence of triple sequences on probabilistic normed space. Later on, Esi [11] have introduced statistical convergence of triple sequences in topological groups. Some more works on triple sequences are found on Kumar et al. [17], Esi [12], Dutta et al. [8], Tripathy and Dutta [34], Tripathy and Goswami [35], Nath and Roy [23] etc.
The idea of statistical convergence is extended to I-convergence in case of real’s by using the notion of ideals of N. Kostyrko, Šalát and Wilczyński [15] introduced the concept of ideal convergence for single sequences in 2000-2001. Later on it was further developed by Šalát et al. [28], Das et al. [6], Sen and Roy [33] and many others. Savas and Das [31] used ideals to introduce the concept of I-statistical convergence for real’s which have extended the notion of statistical convergence. The notion of I-statistically pre-Cauchy sequences are also introduced by them. In the present article, we have extended these results to introduce the concept of I-statistically pre-Cauchy triple sequence of fuzzy real numbers.
A fuzzy real number on is a mapping associating each real number with its grade of membership (t). Every real number r can be expressed as a fuzzy real number as follows:
(t) =
The a-level set of a fuzzy real number , denoted by is defined as
A fuzzy real number X is called convex if min where If there exists such that then the fuzzy real number X is called normal. A fuzzy real number X is said to be upper semi-continuous if for each for all is open in the usual topology of The set of all upper semi continuous, normal, convex fuzzy number is denoted by The additive identity and multiplicative identity in are denoted by and respectively.
Let D be the set of all closed bounded intervals on the real line R. Then
if and only if and Also let
Then is a complete metric space.
Let be defined by
Then defines a metric on
Recent Development on Fuzzy Distance Measure are found in Beigi et al. ([2], [3]).
2. Preliminaries and Background
In this section, some notations and basic definitions which will be used in this article are recalled.
A triple sequence can be defined as a function where N, R and C denote the sets of natural numbers, real numbers and complex numbers respectively.
The notion of statistical convergence for triple sequences depends on the density of the subsets of A subset E of is said to have density or asymptotic density if
exists, where is the characteristic function of E.
The notion of Ideal convergence depends on the structure of the ideal I of the subset of the set of natural numbers N.
Let X be a non empty set. A non-void class (power set of X) is said to be an ideal if I is additive and hereditary, i.e. if I satisfies the following conditions:
(i). and
(ii).
A non-empty family of sets is said to be a filter on X if
(i). Φ F
(ii). A, B F A B F
(iii). A F and A B B F.
For any ideal I, there is a filter F(I) given by
An ideal is said to be non-trivial if I and X I.
The details about the ideals of are introduced and investigated by Tripathy and Tripathy [38].
Throughout the article, the ideals of will be denoted by
Example 2.1. Let i.e. the class of all subsets of of zero natural density.
Then is an ideal of .
Example 2.2. Let be the class of all subsets of such that implies
that there exists such that
Then is an ideal of
A fuzzy real-valued triple sequence is a triple infinite array of fuzzy real numbers for all and is denoted by where
A fuzzy real-valued triple sequence is said to be statistically convergent to the fuzzy real number if for all where denote the cardinality of the enclosed set and we write
A fuzzy real-valued triple sequence is said to be I-convergent to the fuzzy real number if for each the set
A fuzzy real-valued triple sequence is said to be bounded if
A fuzzy real-valued triple sequence is said to be I-statistically convergent to the fuzzy real number if for each and
A fuzzy real-valued triple sequence is said to be I-statistically pre-Cauchy if, for each and
3. Main Results
Theorem 3.1. If a triple sequence of fuzzy numbers is I-statistically convergent, then is
I-statistically pre-Cauchy but an I-statistically pre-Cauchy triple sequence of fuzzy numbers is not necessarily I-statistically convergent.
Proof. Let be I-statistically convergent to L. Then for each let,
Then for (complement of A), we have
Let Then (1a)
So for all
using (1a).
For any given let be chosen such that
Now for all
Hence is I-statistically pre-Cauchy.
But an I-statistically pre-Cauchy triple sequence of fuzzy numbers is not necessarily I-statistically convergent, as it can be seen from the following example.
Example 3.1. Consider the fuzzy triple sequence defined as follows:
where
Then for each the α-cut of is given by, Now it can be easily proved thatis I-statistically pre-Cauchy, but not I-statistically convergent. ■
Theorem 3.2. Let be a bounded triple sequence of fuzzy numbers. Then is I-statistically pre-Cauchy if and only if
Proof. Let (1b)
For each and
Therefore for any and using (1b)
Hence X is I-statistically pre-Cauchy.
Conversely let be I-statistically pre-Cauchy. Since is bounded, so there exists such that for all
The for each and
Since X is I-statistically pre-Cauchy, for any
Then for all
Let be chosen such that
Then for all we have
Hence ■
Theorem 3.3. Let be a bounded triple sequence of fuzzy numbers. Then is I-statistically convergent to L, if and only if
Proof. Let (1c)
For each and we have
Therefore for each and and using (1c),
Hence X is I-statistically convergent.
We can prove the converse part in a similar manner to the Theorem 3.2.
Theorem 3.4. Let a triple sequence of fuzzy number be I-statistically pre-Cauchy. If X has a subsequence which converges to L, and
then X is I -statistically convergent to L.
Proof. Let Since converges to L, there exists such that for all
Let and
Now
Since X is I-statistically pre-Cauchy, so for
Thus for all
Again, since, then the set
and so for all
Therefore for all
be chosen such that Now, for all
Hence is I-statistically convergent to L.
4. Conclusion
Convergence theory is used as a basic tool in, measure spaces, sequences of random variables, information theory etc. We have introduced and studied the notion of I-statistically pre-Cauchy sequence of fuzzy real numbers having multiplicity greater than two. The criterion for any arbitrary triple sequence of fuzzy numbers to be I-statistically pre-Cauchy is derived. Also a necessary and sufficient condition for a bounded triple sequence of fuzzy real numbers to be I-pre-Cauchy is established.
Acknowledgment
The authors are grateful to the anonymous reviewer and the Editor-in-chief for careful reading of the paper and for helpful comments and suggestions of addition of the list of two papers in the reference part that improved the presentation of the paper.
References