Some new classes of Ideal convergent difference
multiple sequence spaces of Fuzzy real numbers
Dr. Munmun Nath
Department of Mathematics S.S. College, Hailakandi , Assam, India
Dr. Santanu Roy
Department of Mathematics
National Institute of Technology Silchar, Silchar, Assam, India
Abstract- In 1981 H. Kizmaz defined the difference sequence spaces for crisp set and in 2007 Sahiner et al.
was first introduced the idea of triple sequence spaces. In this article, we introduce the classes of fuzzy
real-valued multiple sequences
(
)(
,
),
(
)(
,
),
(
)
(
,)
) ( 3 )
( 0 3 )
(
3
c
p
c
p
c
mp
R F I m
F I m
F
I
and
) ( )
( 0 ( ) ,
3 c m p
R F
I
where
p
p
nlk is a triple sequence of bounded strictly positive numbers. We studydifferent topological properties of these spaces like completeness, solid, monotone, symmetricity convergence free etc. We prove some inclusion results also.
Keywords – Ideal convergence (I-convergence), Fuzzy real-valued triple sequence, multiple sequences, solid, monotone, symmetric, convergence free, sequence algebra etc.
I. Introduction
In 1965 Zadeh [32] introduced the concepts of fuzzy sets and fuzzy set operations and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets. Fuzzy set theory, compared to other mathematical theories, is perhaps the most easily adaptable theory to practice. 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. This representation suits well the uncertainties encountered in practical life, which make fuzzy sets a valuable mathematical tool. In fact the fuzzy set theory has become an area of active area of research in science and engineering for the last 40 years. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It extends the scope and results of classical mathematical analysis by applying fuzzy logic to conventional mathematical objects, such as functions, sequences and series etc. The ideas of fuzzy set theory have been used widely not only in many engineering applications, such as, computer programming [11], population dynamics [3], quantum physics [20], control of chaos [10], bifurcation of non-linear dynamical system [13], approximation theory [2] etc., but also in various branches of mathematics, such as, theory of metric and topological spaces [8], theory of linear systems [22], studies of convergence of sequences of functions [4,14]. While studying fuzzy topological spaces, we face many situations where we need to deal with convergence of fuzzy numbers.
II. Definitions and background
Throughout the article N and R denote the sets of natural, and real numbers respectively and
w
,
c
,
c
0,
denotethe spaces of all, convergent, null and bounded sequences respectively.
A fuzzy real number X is a fuzzy set on
R
, i.e. a mappingX
:R
→L
(= [0,1] ) associating each real numbert with its grade of membership
X
(t). Every real number r can be expressed as a fuzzy real numberr
as follows:r
(t) =
otherwise
r
t
if
0
1
The -level set of a fuzzy real number
X
,0 < ≤ 1 denoted by
]
[
X
is defined as}.
)
(
:
{
]
[
X
t
R
X
t
A fuzzy real number X is called convex if
X
(
t
)
X
(
s
)
X
(
r
)
min(
X
(
s
),
X
(
r
)),
wheres
t
r
.
Ifthere exists
t
0
R
such thatX
(
t
0)
1
,
then the fuzzy real number X is called normal. A fuzzy realnumber X is said to be upper semi-continuous if for each
0
,
[
0
,
)),
1
a
X
for alla
L
is open in the usual topology of R. The set of all upper semi continuous, normal, convex fuzzy number is denoted by R(L).Let D be the set of all closed bounded intervals
R L
X
X
X
,
on the real line R. ThenX
Y
if and only ifL L
Y
X
andX
R
Y
R.
Also let
(
,
)
max
(
|
-
|
,
|
-
Y
|
).
R L R L
Y
X
X
Y
X
d
Then(
D
,
d
)
is a complete metric space.Let
d
:
R
(
L
)
R
(
L
)
R
be defined by.
)
(
,
for
,
)
]
[
,
]
([
sup
)
,
(
1 0
L
R
Y
X
Y
X
d
Y
X
d
Then
d
defines a metric onR
(
L
)
and
R
(
L
),
d
is a complete metric space.In order to generalize the notion of convergence of real sequences, Kostyrko, Šalát and Wilczyński [18] introduced the idea of Idealconvergence for single sequences in 2000-2001. Later on it was further developed by Šalát et. al. [26], Das et. al. [6], Tripathy and Tripathy [30], Tripathy and Sen [29], Kumar and Kumar [19], Sen and Roy [27] and many others.
Let X be a non empty set. A non-void class
I
2
X (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)
A
,
B
I
A
B
I
and (ii) AIand BABI. A non-empty family of setsX
F
2
is said to be a filter on X if(i) F
(ii) A, B F A B F
For any ideal I, there is a filter F(I) given by
}.
\
:
{
)
(
I
K
N
N
K
I
F
An ideal
I
2
X is said to be non-trivial if I and X I.A subset E of N is said to have density
(
E
)
if)
(
1
lim
)
(
1
k
n
E
n
k E n
exists.
The details about the ideals of
2
NNare introduced and investigated by Tripathy and Tripathy [29]. Throughout the article, the ideals of2
NNN will be denoted byI
3.
.Example 1. Let
N N N
I
3(
)
2
i.e. the class of all subsets of
N
N
N
of zero natural density. Then)
(
3
I
is an ideal of
2
NNN.Throughout
R F R F F F
F F
c
c
c
c
w
),
(
),
(
),
(
),
(
)
,
(
)
(
3 3 3 0 3 3 03
denote the spaces of all, bounded,convergent in Pringsheim’s sense, null in Pringsheim’s sense, regularly convergent and regularly null fuzzy real valued triple sequences respectively.
A triple sequence can be defined as a function
x
:
N
N
N
R
(
C
).
A fuzzy real valued triple sequence
X
X
nlk is a triple infinite array of fuzzy real numbersX
nlk for allN
k
l
n
,
,
and is denoted byX
nkl whereX
nlk
R
(
L
).
A fuzzy real-valued triple sequence
X
X
nlk is said to beI
3-convergent to the fuzzy numberX
0,
if for all,
0
the set{(
n
,
l
,
k
)
N
N
N
:
d
(
X
nlk,
X
0)
}
I
3.
We write
I
3
lim
X
nlk
X
0.
A fuzzy real-valued triple sequence
X
X
nlk is said to beI
3-bounded if there exists a real number
suchthat the set
{(
n
,
l
,
k
)
N
N
N
:
d
(
X
nlk,
0
)
}
I
3.
A fuzzy real-valued triple sequence space
E
F is said to be solid ifF
nlk E
Y
whenever Ynlk Xnlk for all
N
k
l
n
,
,
andX
nlk
E
F.
.
A fuzzy real-valued triple sequence space
E
F is said to be monotone ifE
F contains the canonical pre-image of all its step spaces.A fuzzy real-valued triple sequence
E
Fis said to be symmetric if S(X) E
F, for all XE
F, where S(X)denotes the set of all permutations of the elements of
X
X
nlkA fuzzy real-valued triple sequence space
E
F is said to be sequence algebra if,
F nlk
nlk
Y
E
X
whenever
.
, nlk F
nlk Y E
A fuzzy real-valued triple sequence space
E
F is said to be convergence free if F nlk E Y whenever . F nlk E X and
X
nlk
0
impliesY
nlk
0
.
In 1981, H. Kizmaz [16 ] the difference sequence spaces for crisp set and studied the spaces
,
c
,
c
0l
. Later on Seva [ ] introduced the notion of difference sequences for fuzzy real numbers a studied different properties of the spaces. In 2006, this concept was further generalised by Tripathy and Esi [ 31] as
Z
m
x
k
w
:
mx
k
Z
wherem
0
be an integer andZ
c
,
c
0,
l
andm k k k
m
x
x
x
for all
k
N
. Thereafter Tripathy and B. A. Baruah introduced the difference sequences of fuzzy real numbers L. we introduced the following for triple sequence:m k m l m n m k l m n k m l m n k l m n m k m l n m k l n k m l n nlk nlk
m
X
X
X
X
X
x
X
X
X
, , , , , , , , , , , , , ,Let
p
p
nkl be a triple sequence of bounded strictly positive numbers. We introduce the following triplesequence spaces:
)},
(
some
for
,
0
)]
,
(
[
lim
:
)
(
{
)
,
)(
(
( ) 3 3 0 03
c
p
X
X
w
I
d
X
X
X
R
L
nlk p nlk m F nlk m F
I
:
lim
[
(
,
0
)]
0
},
{
)
,
)(
(
3 3) ( 0
3
nlk p nlk m F nlk m F I
X
d
I
w
X
X
p
c
:
there
exists
a
real
number
0
such that
the
set
{
)
,
)(
(
3 ) (3
F nlk m F I
w
X
X
p
}.
}
)]
0
,
(
[
:
)
,
,
{(
n
l
k
N
N
N
d
X
I
3nlk
p nlk
m
)
,
(
)
(
( )3
c
p
X
mR F I
nlk
if and only if(
)(
,
)
) (
3
c
p
X
mF I
nlk
and the following limits exist:),
(
some
for
,
each
for
,
0
)]
,
(
[
lim
3
d
X
I
k
N
I
R
L
I
m nlk k pnlk
k
)
(
some
for
,
each
for
,
0
)]
,
(
[
lim
I
3
d
mX
nlkJ
l pnLk
l
N
J
l
R
L
).
(
some
for
,
each
for
,
0
)]
,
(
[
lim
and
I
3
d
mX
nlkK
n pnlk
n
N
K
n
R
L
)
,
(
)
(
0 ( )3
c
p
X
nlk
I F R
mif and only if
(
)(
,
)
) ( 0
3
c
p
X
nlk
I F
mand the following limits exist:
N
k
X
d
I
pnlknlk
m
lim
[
(
,
0
)]
0
,
for
each
3
N
l
X
d
I
pnlknlk
m
lim
[
(
,
0
)]
0
,
for
each
3
.
each
for
,
0
)]
0
,
(
[
lim
and
I
3d
X
n
N
nlk
p nlk
m
Let}.
)]
0
,
(
[
sup
:
)
(
{
)
,
)(
(
, , 3 ) (3
nlk p nlk m k l n F nlk m F
X
d
w
X
X
p
Then
(
)(
.
)
) (
3 m
p
F
M pnlk m nlk m k
l n
nlk
Y
X
d
Y
X
,
sup
,
,
where
.
sup
),
,
1
(
max
, ,
nlkk l n
p
H
H
M
Also, we define the following sequence spaces :
),
,
)(
(
)
,
)(
(
)
,
(
)
(
( ) 3 ( ) 3 ( )3
c
p
c
p
mp
F m
F I m
BP F
I
),
,
)(
(
)
,
)(
(
)
,
(
)
(
0( ) 3 0( ) 3 ( )3
c
p
c
p
mp
F m
F I m
BP F
I
),
,
)(
(
)
,
(
)
(
)
,
(
)
(
( ) 3 ( ) 3 ( )3
c
p
c
p
mp
F m
R F I m
BR F
I
).
,
)(
(
)
,
(
)
(
)
,
(
)
(
0( ) 3 0( ) 3 ( )3
c
p
c
p
mp
F m
R F I m
BR F
I
For particular values of p and ideal I, these sequence spaces reduce to many well known sequence spaces.
Let
X
nlk andY
nlk be two fuzzy real valued triple sequences. Then we say thatX
nlk
Y
nlk for almost alln, l andk relative to
I
3(in short a.a. n, l & k rI
3) if the set.
}
:
)
,
,
{(
n
l
k
N
N
N
X
nlk
Y
nlk
I
3Note. Let
p
p
nkl be a triple sequence of bounded strictly positive numbers and
p
nlkH
sup
. Then for sequences
a
nkl andb
nkl of complex numbers, we have the following inequality:),
(
nkl nklnkl p
nkl p
nkl p
nkl
nkl
b
D
a
b
a
where
max(
1
,
2
).
1
HD
We procure the following existing result.
Lemma 1. If a sequence space
E
F is solid, then it is monotone.For the crisp set case, one may refer to Kamthan and Gupta [15], p.53.
III. MAIN Result
Theorem 1. The following statements are equivalent:
(i)
(
)(
,
).
) (
3
c
p
X
mF I
nlk
(ii) There exists a sequence
Y
3(
c
)(
m,
p
)
Fnlk
such that
mX
nlk
mY
nlk for a.a. n, l & k rI
3.(iii) There exists a subset M {(ni,lj,km)NNN:i, j,mN} of NNN such that
)
(
I
3F
M
and
X
3(
c
)(
m,
p
).
Fm k l ni j
Proof. (i) (ii). Let
(
)(
,
).
)(
3
c
p
X
I F mnlk
Then there existsX
0
R
(
L
)
such that
(
,
)
0
.
lim
03
nlk
p nlk
m
X
X
d
I
So for any
0
,
we have the set
(
n
,
l
,
k
)
N
N
N
:
lim
d
(
X
,
X
0)
I
3nlk
p nlk
m
.
Let us consider the increasing sequences
(
T
j),
(
M
j)
and(
N
j)
of natural numbers such that ifp
T
j.
jM
q
and
r
N
j.
Then the set
( , )
1 . and; ; : )
, ,
( 0 I3
j X
X d r
k q l p n N N N k l
n pn lk
nlk
m
We define the sequence
Y
nlk as follows:nlk nlk
X
Y
if
n
T
1 orl
M
1 ork
N
1Also for all (n,l,k ) with
T
j
n
T
j1 orM
j
l
M
j1 orN
j
k
N
j1,
let
Y
nlk
X
nlkif
(
,
0)
1
,
j
X
X
d
m nlk pnlk
otherwise
Y
nlk
X
0.
We show that
Y
3(
c
)(
m,
p
).
Fnlk
Let
0
.
We choose j such that. 1
j
We see that for
n
T
jl
M
j andk
N
j,
(
,
0)
.
nlk
p nlk m
Y
X
d
Hence
Y
3(
c
)(
m,
p
).
Fnlk
Next let
T
j
n
T
j1,
M
j
l
M
j1 andN
j
k
N
j1,
then the set
n
l
k
N
N
N
mX
nlk mY
nlk
A
(
,
,
)
:
(
,
)
1
.
:
)
,
,
(
0I
3j
X
X
d
N
N
N
k
l
n
m nlk pn lk
This implies
A
I
3 and so
mX
nlk
mY
nlk for a.a. n, l & k r.I
3.
(ii) (iii). Suppose there exists a sequence
Y
3(
c
)(
m,
p
)
Fnlk
such thatX
nlk
Y
nlk for a.a. n, l &k r.
I
3.
LetM
(
n
,
l
,
k
)
N
N
N
:
mX
nlk
mY
nlk
. ThenM
F
(
I
3).
We can enumerate
M
asM
{(
n
i,
l
j,
k
m)
N
N
N
:
i
,
j
,
m
N
},
on neglecting the rows andcolumns those contain finite number of elements. Then
).
,
)(
(
3
c
p
X
F mm k l
(iii) (i) From (iii), (i) follows immediately.
Theorem 2. If
,
sup
, ,
nlkk l n
p
H
then the classes of sequences
(
)
(
,
),
)(
3
c
mp
BP F
I
)
,
(
)
(
0( )3
c
mp
BP F
I
)
,
(
)
(
( )3
c
mp
BR F
I
and
(
)
(
,
)
) ( 0
3
c
mp
BR F
I
are complete metric spaces with respect to the metric defined by
,
)]
,
(
[
sup
)
,
(
, ,
M p
nlk m nlk m k l n
nlk
Y
X
d
Y
X
where
M
max
(
1
,
H
).
Proof. Let us consider the space
(
)
(
,
).
) (
3
c
mp
BP F
I
Let
) (i
X
be a Cauchy sequence in
(
)
(
,
)
(
)(
,
),
)( 3 )
(
3
c
p
mp
F m
BP F
I
where () nlk(i).
.
iX
X
Since
(
)(
,
)
) (
3 m
p
F
is complete, so there exists(
( ))(
,
)
3
p
X
F
m
such that,
lim
X
(i)X
i
whereX
X
nlk.
We claim that
(
)
(
).
) (
3
c
p
X
I F BPSince
) (i
X
is Cauchy, so for a given
0
1
,
there existsn
0
N
such that
,
3
,
( ))
(
i j
X
X
for all
i
,
j
n
0.
(
,
)
3
,
) ( m )
(
Mp j nlk i
nlk m
nlk
X
X
d
for all
i
,
j
n
0.
,
3
)
,
(
() m ( ) pnlk
M j
nlk i
nlk
m
X
X
d
for all
i
,
j
n
0.
Again since
,
(
)
(
,
),
) ( 3 ) ( ) (
p
c
X
X
i j
I F BP
mso there exists fuzzy numbers
Y
i andY
j such that the sets
(
)
3
)
,
(
:
)
,
,
(
n
l
k
N
N
N
d
X
()Y
F
I
3A
M p
i i nlk m
n lk
And
(
)
3
)
,
(
:
)
,
,
(
n
l
k
N
N
N
d
X
( )Y
F
I
3B
M p
j j nlk m
n lk
Then
A
B
F
(
I
3).
Let(
n
,
l
,
k
)
A
B
.
Then
(
,
)
(
,
)
(
,
)
(
,
)
) ( )
( m ) ( )
(
m j
j nlk m j
nlk i
nlk m i
nlk i
j
i
Y
d
Y
X
d
X
X
d
X
Y
Y
,
for alli
,
j
n
0.
Hence
Y
I is a Cauchy sequence fuzzy real numbers. So there exists a fuzzy real number Y such that.
lim
Y
iY
i
Let
0
be given. Since,
) (X
X
i
so there existst
N
such that
3
, M
1 )
(
X t X(1) The number t can be chosen in such a way that together with (1) we get
.
3
)
,
(
Y
tY
pnLk
d
Since ) (t nlk
X
is I-convergent to
Y
t so we have
(
).
3
)
,
(
:
)
,
,
(
n
l
k
N
N
N
d
X
()Y
F
I
3C
m nlkt i pn lk
So for each
(
n
,
l
,
k
)
C
,
we have
(
,
)
(
,
m ())
2
(
(),
)
(
,
)
2 nlk nlk nlk
nlk p
t p
t t nlk m p
t nlk nlk
m p
nlk
m
X
Y
D
d
X
X
D
d
X
Y
D
d
Y
Y
d
3 3
3
2 2
D D
D
(say),
where
max
(
1
,
2
),
1
HD
sup
,.
nkk n
p
H
Hence
mX
nlk is I-convergent to Y.This implies that
(
)
(
,
)
) (
3
c
mp
BP F
I
is complete.
Using similar technique the other cases can be established.
Theorem3.The classes of sequences
(
)(
,
),
(
)(
,
),
(
)
(
,
),
) ( 3 )
( 0 3 )
(
3
c
p
c
p
c
mp
R F I m
F I m
F
I
)
,
(
)
(
),
,
(
)
(
),
,
(
)
(
( ) 3 0 ( ) 3 ( )3
c
p
c
p
c
mp
BR F I m
BP F I m
BP F
I
are closed under addition and scalar multiplication
Proof. – We consider the space
(
)(
,
)
)( 0
3
c
mp
F
I
. Let
X
nlk,
Y
nlk
(
)(
,
)
)( 0
3
c
mp
F
I
and
0
1
.
Let
,
be two scalars. Then there existsX
0andY
0
R
L
be such that
,
0
lim
03
nlk
p nlk
m
X
X
d
I
and 3
lim
,
0
0
nlk
p nlk m
Y
Y
d
I
nlk
nlk
pnlknlk m p
nlk m p
nlk m nlk
m
X
Y
X
Y
D
d
X
X
D
d
Y
Y
d
,
0
0
,
0
,
0Where
1
2
,
1
max
HD
and H =sup
p
nlk
.
Taking I- limits on both sides of the above inequality, we have
nlk
nlk nlkp nlk m p
nlk m
p nlk
m nlk m
Y
Y
d
D
I
X
X
d
D
I
Y
X
Y
X
d
I
0 3
0 3
0 0 3
,
lim
,
lim
,
lim
,
0
.
lim
0 03
nlk
p nlk
m nlk
m
X
Y
X
Y
d
I
)
,
)(
(
( )0
3
c
p
Y
X
I F mnlk m nlk
m
.
the space(
)(
,
)
)( 0
3
c
mp
F
I
is closed under addition. Similarly we can show that
(
)(
,
)
)( 0
3
c
mp
F
I
is closed under scalar multiplication.
Theorem 4.The classes of sequences
(
)(
,
),
(
)
(
,
),
)
( 0 3 )
( 0
3
c
p
c
mp
R F I m
F
I
)
,
(
)
(
3
c
0I(F) BP
mp
and
)
,
(
)
(
0( )3
c
mp
BR F
I
are solid as well as monotone.
Proof. We consider the space
(
)(
,
).
) ( 0
3
c
mp
F
I
Let
(
)(
,
)
) ( 0
3
c
p
X
mF I
nlk
andY
nlk be such thatd
(
mY
nlk,
0
)
d
(
mX
nlk,
0
),
for alln
,
l
,
k
N
.
Let
0
be given. Then the solidness of(
)(
,
).
) ( 0
3
c
mp
F
I
follows from the following inclusion relation:
(
,
0
)
}
{(
,
,
)
:
(
,
0
)
}.
:
)
,
,
{(
nlk
pnlk
nlk m p
nlk
m
X
n
l
k
N
N
N
d
Y
d
N
N
N
k
l
n
Also by Lemma 1. it follows that the classes of sequences under consideration are monotone Using similar technique the other cases can be established.. ■
Theorem 5. The classes of sequences
(
)(
,
),
(
)
(
,
),
)( 3 )
(
3
c
p
c
mp
R F I m
F
I
)
,
(
)
(
( )3
c
mp
BP F
I
and
)
,
(
)
(
( )3
c
mp
BR F
I
are neither solid nor monotone in general
Proof. The result follows from the following example.
Example 1. Let
A
I
3,p
nlk
1
,
for all
n
,
l
,
k
N
.
We consider the sequence
X
nlk defined by:
X
nlk(
t
)
otherwise
,
0
3
1
0
for
,
3
1
0
3
1
for
,
3
1
nlk
t
nlkt
t
nlk
nlkt
otherwise
X
nlk
0
.
Let m = 3, then we have
3X
nlk(
t
)
otherwise
,
0
3
3
3
3
1
3
1
0
for
,
27
9
3
3
3
3
3
1
0
3
3
3
3
1
3
1
for
,
27
9
3
3
3
3
3
1
k
l
n
nlk
t
t
k
l
n
nk
lk
nl
k
l
n
nlk
t
k
l
n
nlk
t
k
l
n
nk
lk
nl
k
l
n
nlk
Then
X
nlk
Z
,
for
.
)
,
(
)
(
and
)
,
(
)
(
),
,
(
)
(
),
,
)(
(
( ) 3 3 ( ) 3 3 ( ) 3 3 ( ) 33
c
p
c
p
c
p
c
p
Z
I F
I F R
I F BP
I F BR
Let
K
{
i
,
j
,
k
:
i
j
k
3
q
:
q
N
}
.We consider the sequence
Y
nlk defined by:
otherwise
,
0
,
)
,
,
(
if
,
X
n
l
k
K
Y
nlk nlkThen
Y
nlk belongs to the canonical pre-image of K step space of Z,for
(
)(
,
),
(
)
(
,
),
(
)
(
,
)
and
(
)
(
3,
)
.
) ( 3 3
) ( 3 3 ) ( 3 3 ) (
3
c
p
c
p
c
p
c
p
Z
I F
I F R
I F BP
I F BR
But
Y
nlk
Z
,
for
.
)
,
(
)
(
and
)
,
(
)
(
),
,
(
)
(
),
,
)(
(
( ) 3 3 ( ) 3 3 ( ) 3 3 ( ) 33
c
p
c
p
c
p
c
p
Z
I F
I F R
I F BP
I F BR
Hence the class of sequences under consideration are not monotone.
Therefore by Lemma 1. the class of sequences are not solid. ■
Theorem 6. The classes of sequences
(
)
(
,
)
(
)
(
,
)
) ( 0 3 )
( 0
3
c
p
and
c
mp
BR F I m
R F
I
are symmetric for
)
(
3 3
I
P
I
andI
I
(
f
),
the class of finite subsets of N, otherwise they are not symmetric.Proof. Let
I
3
I
3(
P
)
and
I
I
(
f
).
Then)
,
(
)
(
)
,
(
)
(
0( ) 3 03
c
p
c
mp
R F m
R F
I
and
(
)
(
,
)
3(
0)
(
,
).
)( 0
3
c
p
c
mp
BR F m
BR F
I
Now for
(
)
(
,
)
(
3(
0)
(
,
))
) ( 0
3
c
p
c
p
X
mR F m
BR F I
nlk
and given
0
,
we have
(
,
)
}
:
)
,
,
{(
n
l
k
N
N
N
d
mX
nlkX
0 pnlk
is a finite subset ofN
N
N
.
Hence the classes of sequences
(
)
(
,
)
(
)
(
,
)
) ( 0 3 )
( 0
3
c
p
and
c
mp
BR F I m
R F
I
are symmetric.
Theorem 7. The classes of sequences
(
)(
,
),
(
)(
,
),
(
)
(
,
),
)( 3 )
( 0 3 )
(
3
c
p
c
p
c
mp
R F I m
F I m
F
I
),
,
(
)
(
( )3
c
mp
BP F
I
)
,
(
)
(
0( )3
c
mp
BP F
I
)
,
(
)
(
( )3
c
p
and
I F BR
mare not symmetric in general.
Proof. The proof follows from the following example.
Example 2. Let
I
3
I
3(
)
and
otherwise
3,
,
all
and
even
for
2,
n
l
k
N
p
nlk
We consider the sequence
X
nlk defined by:For
,
3
i
n
i
N
and for alll
,
k
N
.
)
(
t
X
nlkotherwise
,
0
3
0
for
,
3
1
0
3
for
,
3
1
3 3
3 3
n
t
n
t
t
n
n
t
otherwise
X
nlk
0
.
Let m = 3, then we have
3X
nlk(
t
)
otherwise
,
0
3
3
3
0
for
,
3
3
3
1
0
3
3
3
for
,
3
3
3
1
3 3 3
3
3 3 3
3
n
n
t
n
n
t
t
n
n
n
n
t
Then
X
nlk
Z
,
for
),
,
(
)
(
),
,
(
)
(
),
,
)(
(
),
,
)(
(
( ) 33 3 ) ( 3 3 ) ( 0 3 3 ) (
3
c
p
c
p
c
p
c
p
Z
I F
I F
I F R
I F BP
).
,
(
)
(
),
,
(
)
(
,
3c
0I(F) BP
3p
3c
I(F) BR
3p
We consider the rearrangement
