Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 147130, 8 pages
doi:10.1155/2011/147130
Research Article
On the Fuzzy Convergence
Abdul Hameed Q. A. Al-Tai
College of Qu’ranic Studies, University of Babylon, Babylon, Iraq
Correspondence should be addressed to Abdul Hameed Q. A. Al-Tai, ahbabil@yahoo.com
Received 9 July 2011; Accepted 19 September 2011
Academic Editor: Ch Tsitouras
Copyright q 2011 Abdul Hameed Q. A. Al-Tai. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to introduce and study the fuzzy neighborhood, the limit fuzzy number,
the convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence
on the base which is adopted by Abdul Hameed every real number r is replaced by a fuzzy
number r either triangular fuzzy number or singleton fuzzy set fuzzy point. And then, we will
consider that some results respect effect of the upper sequence on the convergent fuzzy sequence,
the bounded fuzzy sequence, and the Cauchy fuzzy sequence.
1. Introduction
Zadeh 1 introduced the concept of fuzzy set in 1965. Kramosil and Michálek 2 introduced
the concept of fuzzy metric space using continuous t-norms in 1975.
Matloka 3 introduced bounded and convergent sequences of fuzzy numbers and
studied some of their properties in 1986. Sequences of fuzzy numbers also were discussed by
Nanda 4, Kwon 5, Esi 6, and many others.
In 2010, Al-Tai defined the fuzzy metric space, the fuzzy sequence, and many other
concepts, on other base, that the family of fuzzy real numbers is R Z ∪ Q ∪ Q . Z is the
family of fuzzy integer numbers, where every r ∈ Z r ∈ Z is a singleton fuzzy set fuzzy
point see 7–12. Q and Q are the family of fuzzy rational numbers and the family of fuzzy
irrational numbers, respectively, where every r ∈ Q r ∈ Q or r ∈ Q r ∈ Q is a triangular
fuzzy number 13 and by using the representation Theorem the resolution principle 14.
In this research, we will consider the fuzzy neighborhood, the limit fuzzy number, the
convergent fuzzy sequence, the bounded fuzzy sequence, and the Cauchy fuzzy sequence,
and then we will introduce some results about properties of the above concepts on the base
which depended by Al-Tai to define the fuzzy metric space.
2
Journal of Applied Mathematics
Definition 1.1. Let X, d be a fuzzy metric space. If x ∈ X, then a neighborhood of x is a family
of fuzzy numbers N ε x consisting of all fuzzy numbers y ∈ X, such that dx, y < ε ε > 0.
The fuzzy number ε will be called the fuzzy radius of N ε x.
Definition 1.2. A fuzzy number x is a limit fuzzy number of the family of the fuzzy numbers
E, if every fuzzy neighborhood of x contains a fuzzy number y /
x, such that y ∈ E.
Definition 1.3 Al-Tai 13. If we have the family of the fuzzy numbers X and the family of
the fuzzy natural numbers N, the fuzzy sequence xn in X is a fuzzy function from N to X.
A fuzzy sequence xn consists of all ordered tuples sequences at degree α, for all
α ∈0, 1; xj,n,α xj,1,α , xj,2,α , . . . ∈ x1 α, x2 α, . . . x1 , x2 , . . .α is its α-cut, j ∈ N.
The fuzzy value of the fuzzy function at n ∈ N is denoted by xn .
Definition 1.4. A fuzzy sequence xn in a fuzzy metric space X, d is said to be convergent,
if there is a fuzzy number x ∈ X, such that for every ε > 0, there is a fuzzy natural number
N, with n ≥ N, which implies that
dxn , x < ε.
1.1
That is, for every εj,α ∈ ε α α-cut of ε, there is a natural number N, with n ≥ N, such
that
d xj,n,α , xj,α < εj,α ,
1.2
where xj,n,α ∈ xn α α-cut of xn , xj,α ∈ xα α-cut of x, N ∈ N, and n ∈ n, for all α ∈0, 1. x
will be called the fuzzy limit of xn , and we write
lim xn x.
n→∞
1.3
If xj,n,α is not convergent at any α ∈0, 1, then xn is said to be divergent.
Definition 1.5. The family of the fuzzy numbers E in the fuzzy metric space X, d is bounded,
if there is a fuzzy real number r > 0 and a fuzzy number x ∈ X, such that
d y, x < r
1.4
for all y ∈ X. That is, if rα ∈ rα α-cut of r and xα ∈ xα α-cut of x ∈ X, then
d xα , yα < rα ,
where yα ∈ yα α-cut of all y ∈ X, for all α ∈0, 1.
1.5
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3
Definition 1.6. The fuzzy sequence xn in a fuzzy metric space X, d is said to be bounded
if its fuzzy range is bounded.
Definition 1.7. A fuzzy sequence xn in a fuzzy metric space X, d is said to be a Cauchy
fuzzy sequence if for every ε > 0, there is a fuzzy natural number N, such that
1.6
dxn , xm < ε,
whenever n ≥ N and m ≥ N. In other words, every xj,n,α ∈ xn α α-cut of xn is a Cauchy
sequence; that is, for every εj,α ∈ εα α-cut of ε, there is a natural number N, such that
d xj,n,α , xj,m,α < εj,α
1.7
for every n ≥ N, m ≥ N, where xj,n,α ∈ xn α α-cut of xn , xj,m,α ∈ xm α α-cut of xm , n ∈ n,
m ∈ m, and N ∈ N, for all α ∈0, 1.
Examples 1.8. 1 1/n 1, 1/n, 1/n − 1 converges to 0, since for all xn,α ∈ n α/nn 1, n − α/nn − 1 α-cut of 1/n 1, 1/n, 1/n − 1, for all εα > 0, ∃N ∈ N ∈ N :
dα xn,α , 0 < εα , for all n > N, for all α ∈0, 1.
Therefore,
lim
n→∞
1
1
1
, ,
n1 n n−1
0.
1.8
2 1/en1 , 1/en , 1/en−1 converges to 0, since for all xn,α ∈ e − 1α 1/en1 , 1 −
eα e/en α-cut of 1/en1 , 1/en , 1/en−1 , for all εα > 0, ∃N ∈ N ∈ N : dα xn,α , 0 < εα , for
all n > N, for all α ∈0, 1.
Therefore,
lim
n→∞
1
e
,
n1
1
1
, n−1
n
e e
1.9
0.
√
√ √
√
3 sinn · x1 , x, x2 / n sinnx1 , sinnx, sinnx2 / n 1 , n, n2 converges to 0, where α-cut of sinnx1 , sinnx, sinnx2 is sinnx − sinnx1 α sinnx1 , sinnx − sinnx2 α sinnx2 a1 α, a2 α.
√ √
√
And α-cut of n1 , n, n2 is
√
n−
√ √ √ √
√ n 1 α
n 1, n −
n 2 α
n 2 b1 α, b2 α.
1.10
Since
a1 α a1 α a2 α a2 α
a1 α a1 α a2 α a2 α
∀xn,α ∈ min
,
,
,
, max
,
,
,
,
b2 α b1 α b2 α b1 α
b2 α b1 α b2 α b1 α
0∈
/ b1 α, b2 α,
1.11
4
Journal of Applied Mathematics
√
√ √
α-cut of sinnx1 , sinnx, sinnx2 / n1 , n, n2 , for all εα > 0 , ∃N ∈ N ∈ N :
dα xn,α , 0 < εα , for all n > N, for all α ∈0, 1.
Therefore,
lim
n→∞
sinnx1 , sinnx, sinnx2 0.
√ √ √ n 1 , n, n 2
1.12
4 1/2, 1/3, 1/4 converges to 1/2, 1/3, 1/4.
5 n is divergent and unbounded.
6 cosn · x1 , x, x2 cosnx1 , cosnx, cosnx2 is divergent, since for all xn,α ∈
cosnx − cosnx1 α cosnx1 , cosnx − cosnx2 α cosnx2 α-cut of cosn · x1 , x, x2 is divergent. And it is bounded.
Theorem 1.9. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is bounded for all α ∈0, 1, then xn is bounded.
Proof. Let xj,α , be a supremum of a sequence xj,n,α , j 1, 2, . . ., for all α ∈0, 1. We have
from the hypothesis, the upper sequence x1,n,α is bounded for all α ∈0, 1; that is, there is a
real number r1,α ∈ rα α-cut of r ∈ R, such that
dx1,i,α , 0 < r1,α
1.13
for all x1,i,α ∈ x1,n,α , i ∈ N, for all α ∈0, 1.
When j 2,
dx2,i,α , 0 ≤ dx2,i,α , x1,α dx1,α , 0
< r2,α
r1,α r2,α ,
1.14
where r2,α
max{dx2,i,α , x1,α , x2,i,α ∈ x2,n,α }. That is, there is a real number r2,α ∈ rα α-cut
of r ∈ R, such that
dx2,i,α , 0 < r2,α
1.15
for all x2,i,α ∈ x2,n,α , i ∈ N, for all α ∈0, 1.
When j 3,
dx3,i,α , 0 ≤ dx3,i,α , x2,α dx2,α , 0
< r3,α
r2,α r3,α ,
1.16
where r3,α
max{dx3,i,α , x2,α , x3,i,α ∈ x3,n,α }. That is, there is a real number r3,α ∈ rα
α-cut of r ∈ R, such that
dx3,i,α , 0 < r3,α
for all x3,i,α ∈ x3,n,α , i ∈ N, for all α ∈0, 1.
1.17
Journal of Applied Mathematics
5
And so on, when j m 1,
dxm1,i,α , 0 ≤ dxm1,i,α , xm,α dxm,α , 0
< rm1,α
rm,α rm1,α ,
1.18
where rm1,α
max{dxm1,i,α , xm,α , xm1,i,α ∈ xm1,n,α }. That is, there is a real number
rm1,α ∈ rα α-cut of r ∈ R, such that
dxm1,i,α , 0 < rm1,α
1.19
for all xm1,i,α ∈ xm1,n,α , i ∈ N, m ∈ N, for all α ∈0, 1. By Definitions 1.5 and 1.6, xn is
bounded.
Theorem 1.10. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is bounded for all α ∈0, 1, then xn is convergent.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is bounded for all α ∈
0, 1. By Theorem 1.9, xn is bounded; that is, every sequence xj,n,α , j 2, 3, . . ., of the
α-cut of xn is bounded for all α ∈0, 1, but every sequence xj,n,α , j 1, 2, . . ., of the αcut of xn is monotonic, for all α ∈0, 1; then, every sequence xj,n,α , j 1, 2, . . ., of the
α-cut of xn is convergent, for all α ∈0, 1 see 15, Theorem 3.14. By Definition 1.4, xn is convergent.
Theorem 1.11. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is convergent for all α ∈0, 1, then xn is convergent.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is convergent for all α ∈
0, 1, so it is bounded see 15, Theorem 3.2-c. By Theorem 1.10, xn is convergent.
Theorem 1.12. Every convergent fuzzy sequence xn in a fuzzy metric space X, d is bounded.
Proof. Let xn be a convergent fuzzy sequence in a fuzzy metric space X, d. By
Definition 1.4, every sequence xj,n,α , j 1, 2, . . ., of the α-cut of xn is convergent for all
α ∈0, 1; that is, the upper sequence x1,n,α of xn is bounded for all α ∈0, 1 see 15,
Theorem 3.2-c. By Theorem 1.9, xn is bounded.
Theorem 1.13. Every bounded fuzzy sequence xn in the fuzzy metric space R, d is convergent.
Proof. Let xn be a bounded fuzzy sequence in a R, d. By Definition 1.5, we get that
the upper sequence x1,n,α of xn is bounded for all α ∈0, 1. By Theorem 1.10, xn is
convergent.
Theorem 1.14. Let xn be a fuzzy sequence in a fuzzy metric space X, d; then xn converges to
x ∈ X if and only if every fuzzy neighborhood of x contains all but finitely many of the terms of xn .
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Journal of Applied Mathematics
Proof. Suppose that N ε x is a fuzzy neighborhood of x. For ε > 0, dy, x < ε, y ∈ X, imply
y ∈ N ε x. We have from the hypothesis xn converges to x, so for same ε > 0, there exists
N ∈ N, such that n ≥ N, implying dxn , x < ε. That is, xn ∈ N ε x.
Conversely, suppose that every fuzzy neighborhood of x contains all but finitely many
of xn . Fix ε > 0, and let N ε x be the set of all y ∈ X, such that dy, x < ε.
By assumption, there exists N ∈ N corresponding to N ε x, such that xn ∈ N ε x if
n ≥ N. Thus dxn , x < ε if n ≥ N. Hence xn converges to x.
Theorem 1.15. In any fuzzy metric space X, d, every convergent fuzzy sequence is a Cauchy fuzzy
sequence.
Proof. Let xn be a convergent fuzzy sequence in a fuzzy metric space X, d. By
Definition 1.4, every sequence xj,n,α , j 1, 2, . . ., of the α-cut of xn is convergent for
all α ∈0, 1; then every sequence xj,n,α , j 1, 2, . . ., of the α-cut of xn is a Cauchy
sequence for all α ∈0, 1 see 15, Theorem 3.11-a; by Definition 1.7, xn is a Cauchy fuzzy
sequence.
Theorem 1.16. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is bounded for all α ∈0, 1,then xn is a Cauchy fuzzy sequence.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is bounded for all
α ∈0, 1. By Theorem 1.10, xn is convergent. By Theorem 1.15, xn is a Cauchy fuzzy
sequence.
Theorem 1.17. In the fuzzy metric space R, d, every bounded fuzzy sequence is a Cauchy fuzzy
sequence.
Proof. Let xn be a bounded fuzzy sequence in R, d. By Theorem 1.13, xn is a convergent
fuzzy sequence. By Theorem 1.15, xn is a Cauchy fuzzy sequence.
Theorem 1.18. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is convergent for all α ∈0, 1, then xn is a bounded fuzzy sequence.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is convergent for all α ∈
0, 1. By Theorem 1.11, xn is a convergent fuzzy sequence. By Theorem 1.12, xn is a
bounded fuzzy sequence.
Theorem 1.19. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is convergent for all α ∈0, 1, then xn is a Cauchy fuzzy sequence.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is convergent for all α ∈
0, 1. By Theorem 1.11, xn is a convergent fuzzy sequence. By Theorem 1.15, xn is a
Cauchy fuzzy sequence.
n
Theorem 1.20. Every Cauchy fuzzy sequence in R , the n-dimensional fuzzy Euclidean space, is
convergent.
Journal of Applied Mathematics
7
n
Proof. Let xm be a Cauchy fuzzy sequence in R . By Definitions 1.7 and 1.3, xm consists of all
Cauchy sequences at degree α, xm,α x1,α , x2,α , . . . ∈ x1 , x2 , . . .α α-cut of xm for all
α ∈0, 1, where every element xi,α ∈ xm,α is xi,1,α , xi,2,α , . . . , xi,n,α ∈ xi,1 , xi,2 , . . . , xi,n α αcut of xi xi,1 , xi,2 , . . . , xi,n for all α ∈0, 1 Al-Tai 13. By Theorem 3.11-c see 15,
xm,α is a convergent sequence for all α ∈0, 1. By Definition 1.4, xm is a convergent fuzzy
sequence.
Theorem 1.21. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is a Cauchy sequence for all α ∈0, 1; then xn is convergent.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is a Cauchy sequence
for all α ∈0, 1, so it is convergent see 15, Theorem 3.11-c. By Theorem 1.11, xn is
convergent.
Theorem 1.22. Every Cauchy fuzzy sequence in the fuzzy metric space R, d is bounded.
Proof. Let xn be a Cauchy fuzzy sequence in R, d. By Theorem 1.20, xn is a convergent
fuzzy sequence. By Theorem 1.12, xn is bounded.
Theorem 1.23. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is a Cauchy sequence for all α ∈0, 1, then xn is bounded.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is a Cauchy sequence for
all α ∈0, 1. By Theorem 1.21, xn is convergent. By Theorem 1.12, xn is bounded.
Theorem 1.24. Let xn be a fuzzy sequence in the fuzzy metric space R, d. If the upper sequence
of the α-cut of xn is a Cauchy sequence for all α ∈0, 1, then xn is Cauchy.
Proof. Suppose that the upper sequence x1,n,α of the α-cut of xn is a Cauchy sequence
for all α ∈0, 1; then it is convergent see 15, Theorem 3.11-c. By Theorem 1.19, xn is
Cauchy.
Theorem 1.25. The limit fuzzy number of a convergent fuzzy sequence xn in a fuzzy metric space
X, d is unique.
Proof. Let xn be a fuzzy sequence in a fuzzy metric space X, d which converges to x ∈
X, d. By Definition 4 and Theorem 3.2-b 15, we have that every sequence xn,α of the
α-cut of xn has a unique limit point xα ∈ xα α-cut of x for all α ∈0, 1, and hence, xn converges to a unique limit fuzzy number x.
Theorem 1.26. If y n is a convergent fuzzy sequence in the fuzzy metric space R, d, then a fuzzy
sequence xn is convergent for all n ≥ N, as 0 ≤ xn ≤ y n .
Proof. Let y1,n,α be the upper sequence of the α-cut of yn for all α ∈0, 1. Since 0 ≤ xn ≤
y n , then 0 ≤ x1,n,α ≤ yn,α 16, where x1,n,α is an upper sequence of the α-cut of
xn for all α ∈0, 1. By Theorem 7.4.4 Comparison Test 17, x1,n,α is convergent for all
α ∈0, 1. By Theorem 1.11, xn is convergent.
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Journal of Applied Mathematics
2. Conclusion
Fuzzy neighborhood, limit fuzzy point, convergent fuzzy sequence, bounded fuzzy sequence,
and Cauchy fuzzy sequence can be defined on the dependent base by Al-Tai, these concepts
help us to show that the fuzzy sequence in R, d is convergent, if an upper sequence of the
α-cut of it is convergent, bounded, or Cauchy for all α ∈0, 1. The fuzzy sequence in R, d
is bounded, if an upper sequence of the α-cut of it is convergent, bounded, or Cauchy for all
α ∈0, 1. The fuzzy sequence in R, d is Cauchy, if an upper sequence of the α-cut of it is
convergent, bounded, or, Cauchy for all α ∈0, 1. The limit fuzzy number of a convergent
fuzzy sequence is unique. And other results about above concepts.
References
1 L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
2 I. Kramosil and J. Michálek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5,
pp. 336–344, 1975.
3 M. Matloka, “Sequences of fuzzy numbers,” BUSEFAL, vol. 28, pp. 28–37, 1986.
4 S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 33, no. 1, pp. 123–126, 1989.
5 J.-S. Kwon, “On statistical and p-Cesaro convergence of fuzzy numbers,” Journal of Computational and
Applied Mathematics, vol. 7, no. 1, pp. 757–764, 2003.
6 A. Esi, “On some new paranormed sequence spaces of fuzzy numbers defined by Orlicz functions
and statistical convergence,” Mathematical Modelling and Analysis, vol. 11, no. 4, pp. 379–388, 2006.
7 J. Zhan, Y. B. Jun, and W. A. Dudek, “On ∈, ∈ V q-fuzzy filters of pseudo-BL algebras,” Bulletin of the
Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 57–67, 2010.
8 M. N. Mukherjee and S. P. Sinha, “On some weaker forms of fuzzy continuous and fuzzy open
mappings on fuzzy topological spaces,” Fuzzy Sets and Systems, vol. 32, no. 1, pp. 103–114, 1989.
9 P. Dheena and S. Coumaressane, “Generalization of ∈, ∈ V q-fuzzy subnear-rings and ideals,” Iranian
Journal of Fuzzy Systems, vol. 5, no. 1, pp. 79–97, 2008.
10 P. M. Pu and Y. M. Liu, “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith
convergence,” Journal of Mathematical Analysis and Applications, vol. 76, no. 2, pp. 571–599, 1980.
11 S. Jafari and R. M. Latif, “Applications of fuzzy points via fuzzy θ−open sets and fuzzy θ−closure
operator,” Tech. Rep. number TR 392, 2008.
12 L. Zadeh, “A fuzzy set theoretic interpretation of linguistic hedges,” Memorandum number
ERLM335, University of California, Berkeley, Calif, USA, 1972.
13 A.-H. Q. A. Al-Tai, “On the fuzzy metric spaces,” European Journal of Scientific Research, vol. 47, no. 2,
pp. 214–229, 2010.
14 S. Chandra and C. Bector, Fuzzy Mathematical Programming and Fuzzy Matrix Games, vol. 1st, Springer,
New York, NY, USA, 2005.
15 W. Rudin, Principles of Mathematics Analysis, McGraw-Hill, New York, NY, USA, 3rd edition, 1953.
16 G. J. George and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall PTR, New Jersey, NJ, USA, 1st
edition, 1995, Theory and Applications.
17 J. E. Hutchinson, Introduction to Mathematical Analysis, 1994, Revised by R. J. Loy, 1995.
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