International Journal of Management and Fuzzy Systems
Volume 2, Issue 2, April 2016, Pages: 15-21

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

(S. Saha)
(B. Nath)
(S. Roy)

*Corresponding author

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

Contents

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

1. R. P. Agnew, On summability of multiple sequences, American Journal of Mathematics; 1(4), 62-68, (1934).
2. M. A. Beigi, T. Hajjari, E. G. Khani, A New Index for Fuzzy Distance Measure,Appl. Math. Inf. Sci.; 9(6),3017-3025, (2015).
3. M. A. Beigi, M. A. Beigi, T. Hajjari, A Theoretical Development on Fuzzy Distance Measure, Journal of Mathematical Extension (JME); 9(3),15-38, (2015).
4. J. S. Connor, The statistical and strong p-Cesáro convergence of sequences, Analysis; 8, 47-63, (1988).
5. J. S. Connor, J. Fridy, J. Kline, Statistically pre-Cauchy sequences, Analysis; 14, 311-317, (1994).
6. P.Das,P. Kostyrko, W. Wilczyńskiand P. Malik, I and I*convergence of double Sequences, Math. Slovaca; 58(5), 605-620, (2008).
7. P. Das, E. Savas, On I-statistically pre-Cauchy Sequences, Taiwanese Journal of Mathematics,18 (1), 115-126, (2014).
8. A. J. Dutta, A. Esi, B. C. Tripathy, Statistically convergence triple sequence spaces defined by Orlicz function,Journal of Mathematical Analysis; 4(2), 16-22,(2013).
9. A. J. Dutta, B. C. Tripathy, Statistically pre-Cauchy Fuzzy real-valued sequences defined by Orlicz function,Proyecciones Journal of Mathematics; 33(3), 235-243, (2014).
10.  H. Dutta, A Characterization of the Class of statistically pre-Cauchy Double Sequences of Fuzzy Numbers, Appl. Math. Inf. Sci.; 7(4), 1437-1440, (2013).
11. A. Esi, Statistical convergence of triple sequences in topological groups, Annals of the Universityof Craiova, Mathematics and Computer Science Series; 40(1), 29-33, (2013).
12. A. Esi, -Statistical convergence of triple sequences on probabilistic normed space, GlobalJournal of Mathematical Analysis; 1(2), 29-36, (2013).
13. H. Fast, Surla convergence statistique, Colloq. Math.;2, 241-244, (1951).
14. J. A. Fridy, On statistical convergence, Analysis, 5, 301-313, (1985).
15. P. Kostyrko, T. Šalát, W. Wilczyński,I-convergence, Real Anal.Exchange; 26, 669686, (2000- 2001).
16. V. A. Khan, Q. M. Danish Lohani, Statistically pre-Cauchy sequences and Orlicz functions, Southeast Asian Bull. Math.; 31, 1107-1112, (2007).
17. P. Kumar, V. Kumar, S. S. Bhatia,Multiple sequence of Fuzzy numbers and theirstatistical convergence, Mathematical Sciences, Springer, 6(2), 1-7, (2012).
18. J. S. Kwon, On statistical and p-Cesáro convergence of fuzzy numbers, Korean J. Comput. Appl. Math.; 7, 195–203, (2000).
19. I. J. Maddox, A tauberian condition for statistical convergence, Math. Proc. Camb. PhilSoc.; 106, 272-280, (1989).
20. M. Matloka, Sequences of fuzzy numbers, BUSEFAL; 28, 28-37, (1986).
21. F. Moričz, Statistical convergence of multiple sequences. Arch. Math.; 81, 82–89, (2003).
22. S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets and Systems; 33,123-126, (1989).
23. M. Nath, S. Roy, Some new classes ofidealconvergentdifference multiple sequences of fuzzy real numbers,Journal of Intelligent and Fuzzy systems; 31(3), 1579-1584, (2016).
24. F, Nuray, E. Savas, Statistical convergence of sequences of fuzzy numbers. Math. Slovaca; 45, 269–273, (1995).
25. A. Şahiner, M. Gürdal, F. K. Düden, Triple sequences and their statistical convergence, Seluk J.Appl. Math; 8(2), 49-55, (2007).
26. A. Sahiner, B. C. Tripathy, Some I -related Properties of Triple Sequences, Selcuk J. Appl.Math.; 9(2), 9-18, (2008).
27. T.Šalát, On statistically convergent sequences of real numbers, Math. Slovaca; 30, 139-150, (1980).
28.  T. Šalát, B.C. Tripathy, M. Ziman, On some properties of I-convergence, Tatra Mt. Math. Publ.; 28, 279286, (2004).
29. E. Savas, On statistically convergent sequences of fuzzy numbers, Inform. Sci.; 137(1-4), 277-282,(2001).
30. E. Savas, On strongly-summable sequences of fuzzy numbers, Information Sciences, 125, 181-186, (2000).
31. E. Savas, P. Das, A generalized statistical convergence via ideals, Applied Mathematics Letters;24, 826-830, (2011).
32. E. Savas, A. Esi, Statistical convergence of triple sequences on probabilistic normed space, Annals of the University of Craiova, Mathematics and Computer Science Series; 39(2), 226 -236, (2012).
33. M Sen, S. Roy, Some I-convergent double classes of sequences of Fuzzy numbers defined by Orlicz functions, Thai Journal of Mathematics; 11(1), 111–120, (2013),
34. B. C. Tripathy, A. J. Dutta, Statistically convergence triple sequence spaces defined by Orliczfunction,Journal of Mathematical Analysis; 4(2), 16-22, (2013).
35. B. C. Tripathy,R. Goswami, On triple difference sequences of real numbers in probabilistic normed spaces, Proyecciones Journal of Mathematics; 33(2), 157-174, (2014).
36. B. C. Tripathy, B. Sarma,Statistically convergent double sequence spaces defined by Orlicz function, Soochow Journal of Mathematics, 32(2), 211-221, (2006).
37. B. C. Tripathy, M. Sen, On generalized statistically convergent sequences, Indian Jour. Pure Appl. Math.; 32(11), 1689-1694, (2001).
38. B. K. Tripathy, B. C. Tripathy,On I-convergence of double sequences, Soochow Journal of Mathematics; 31(4), 549560, (2005).
39. L. A. Zadeh, Fuzzy sets, Information and Control; 8, 338-353, (1965).

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