EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 574-585
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Intuitionistic Fuzzy Zweier I-convergent Double
Sequence Spaces Defined by Orlicz Function

Vakeel A. Khan1, Yasmeen1, Hira Fatima1, Ayaz Ahmad2,∗

1 Department of Mathematics, Aligarh Muslim University,Aligarh, India
2 Department of Mathematics,National Institute of Technology, Patna, India

Abstract. The purpose of this paper is to introduce the intuitionistic fuzzy Zweier I-convergent
double sequence spaces 2ZI(µ,ν)(M) and 2ZI0(µ,ν)(M) defined by Orlicz function and study the fuzzy
topology on the said spaces.

Key Words and Phrases: Ideal, filter, double I-convergence, intuitionistic fuzzy normed spaces.

1. Introduction and Preliminaries

After the pioneering work of Zadeh [29], a huge number of research papers have been
appeared on fuzzy theory and its applications as well as fuzzy analogues of the classical the-
ories. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in
various problems arising in field of science and engineering. It has a wide range of applica-
tions in various fields: population dynamics [3], chaos control [5], computer programming
[6], nonlinear dynamical system [7], etc. Fuzzy topology is one of the most important
and useful tools and it proves to be very useful for dealing with such situations where the
use of classical theories breaks down. The concept of intuitionistic fuzzy normed space
[25] and of intuitionistic fuzzy 2-normed space [21] are the latest developments in fuzzy
topology. Recently V. A. Khan and Yasmeen([12], [13]) studied the intuitionistic fuzzy
Zweier I-convergent sequence spaces defined by modulus function and Orlicz function.

The notion of statistical convergence is a very useful functional tool for studying the
convergence problems of numerical problems/matrices(double sequences) through the con-
cept of density. The notion of I-convergence, which is a generalization of statistical con-
vergence [4], was introduced by Kostyrko, Salat and Wilczynski [14] by using the idea of I
of subsets of the set of natural numbers N and further studied in [22]. Recently, the notion
of statistical convergence of double sequences x = (xij) has been defined and studied by
Mursaleen and Edely [20]; and for fuzzy numbers by Savaş and Mursaleen [26]. Quite

∗Corresponding author.

Email addresses: vakhanmaths@gmail.com (V. A. Khan), yasmeen9828@gmail.com (Yasmeen),
hirafatima2014@gmail.com (H. Fatima), ayaz1970@gmail.com (A. Ahmad)

http://www.ejpam.com 574 c© 2017 EJPAM All rights reserved.



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 575

recently, Das et al. [15] studied the notion of I and I∗- convergence of double sequences
in R.

We recall some notations and basic definitions used in this paper.

Definition 1. Let I ⊂ 2N be a non-trivial ideal in N. Then a sequence x = (xk) is said
to be I-convergent to a number L if, for every ε > 0, the set {k ∈ N :| xk − L |≥ ε} ∈ I.

Definition 2. Let I ⊂ 2N be a non-trivial ideal in N. Then a sequence x = (xk) is
said to be I-Cauchy if, for each ε > 0,there exists a number N = N(ε) such that the set
{k ∈ N :| xk − xN |≥ ε} ∈ I.

Definition 3. The five-tuple (X,µ, ν, ∗, �) is said to be an intuitionistic fuzzy normed
space(for short, IFNS) if X is a vector space, ∗ is a continuous t-norm, � is a continuous
t-conorm and µ, ν are fuzzy sets on X × (0,∞) satisfying the following conditions for
every x, y ∈ X and s, t > 0 :

(a) µ(x, t) + ν(x, t) ≤ 1,
(b) µ(x, t) > 0,
(c) µ(x, t) = 1 if and only if x = 0,
(d) µ(αx, t) = µ(x, t

|α|) for each α 6= 0,

(e) µ(x, t) ∗ µ(y, s) ≤ µ(x+ y, t+ s),
(f) µ(x, .) : (0,∞)→ [0, 1] is continuous,
(g) lim

t→∞
µ(x, t) = 1 and lim

t→0
µ(x, t) = 0,

(h) ν(x, t) < 1,
(i) ν(x, t) = 0 if and only if x = 0,
(j) ν(αx, t) = ν(x, t

|α|) for each α 6= 0,

(k) ν(x, t) � ν(y, s) ≥ ν(x+ y, t+ s),
(l) ν(x, .) : (0,∞)→ [0, 1] is continuous,
(m) lim

t→∞
ν(x, t) = 0 and lim

t→0
ν(x, t) = 1.

In this case (µ, ν) is called an intuitionistic fuzzy norm.

Definition 4. Let (X,µ, ν, ∗, �) be an IFNS. Then a sequence x = (xk) is said to be
convergent to L ∈ X with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε > 0
and t > 0, there exists k0 ∈ N such that µ(xk − L, t) > 1 − ε and ν(xk − L, t) < ε for all
k ≥ k0. In this case we write (µ, ν)− limx = L.

Definition 5. Let (X,µ, ν, ∗, �) be an IFNS. Then a sequence x = (xk) is said to be a
Cauchy sequence with respect to the intuitionistic fuzzy norm (µ, ν) if, for every ε > 0 and
t > 0, there exists k0 ∈ N such that µ(xk − xl, t) < ε and ν(xk − xl, t) < ε for all k, l ≥ k0.

Definition 6. Let K be the subset of natural numbers N. Then the asymptotic density of
K, denoted by δ(K), is defined as δ(K) = lim

n

1
n |{k ≤ n : k ∈ K}|, where the vertical bars

denotes the cardinality of the enclosed set.



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 576

A number sequence x = (xk) is said to be statistically convergent to a number ` if, for
each ε > 0, the set K(ε) = {k ≤ n :| xk − ` |> ε} has asymptotic density zero, i.e.

lim
n

1

n
|{k ≤ n :| xk − ` |> ε}| = 0.

In this case we write st− limx = `.

Definition 7. A number sequence x = (xk) is said to be statistically Cauchy sequence if,
for every ε > 0, there exists a number N = N(ε) such that

lim
n

1

n
|{j ≤ n :| xj − xN |≥ ε}| = 0.

The concepts of statistical convergence and statistical Cauchy for double sequences in
intuitionistic fuzzy normed spaces have been studied by Mursaleen and Mohiuddine[14].

Definition 8. Let I ⊂ 2N be a non trivial ideal and (X,µ, ν, ∗, �) be an IFNS. A se-
quence x = (xk) of elements of X is said to be I-convergent to L ∈ X with respect to the
intuitionistic fuzzy norm (µ, ν) if for every ε > 0 and t > 0 , the set

{k ∈ N : µ(xk − L, t) ≥ 1− ε or ν(xk − L, t) ≤ ε} ∈ I.

In this case L is called the I-limit of the sequence (xk) with respect to the intuitionistic
fuzzy norm (µ, ν) and we write I(µ,ν) − limxk = L.

2. I2-Convergence in an IFNS

Definition 9. Let (X,µ, ν, ∗, �) be an IFNS. Then, a double sequence x = (xij) is said to
be statistically convergent to L ∈ X with respect to the intuitionistic fuzzy norm (µ, ν) if,
for every ε > 0 and t > 0,

δ({(i, j) ∈ N× N : µ(xij − L, t) ≤ 1− ε or ν(xij − L, t) ≥ ε}) = 0.

or equivalently

lim
mn

1

mn
|{i ≤ m, j ≤ n, : µ(xij − L, t) ≤ 1− ε or ν(xij − L, t) ≥ ε}| = 0.

In this case we write st2(µ,ν) − limx = L.

Definition 9.2 Let (X,µ, ν, ∗, �) be an IFNS. Then, a double sequence x = (xij) is said
to be statistically Cauchy with respect to the intuitionistic fuzzy norm (µ, ν) if, for every



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 577

ε > 0 and t > 0, there exist N = N(ε) and M = M(ε) such that for all i, p ≥ N and
j, q ≥M,

δ({(i, j) ∈ N× N : µ(xij − xpq, t) ≤ 1− ε or ν(xij − xpq, t) ≥ ε}) = 0.

Definition 10. Let I2 be a non trivial ideal of N×N and (X,µ, ν, ∗, �) be an intuitionistic
fuzzy normed space. A double sequence x = (xij) of elements of X is said to be I2
convergent to L ∈ X with respect to the intuitionistic fuzzy norm (µ, ν) if, for each ε > 0
and t > 0,

{(i, j) ∈ N× N : µ(xij − L, t) ≤ 1− ε or ν(xij − L, t) ≥ ε} ∈ I2.

In this case we write I
(µ,ν)
2 − limx = L.

The approach of constructing new sequence spaces by means of the matrix domain of
a particular limitation method have been recently employed by Altay, Başar, Mursaleen
[1], Malkowsky [19] Ng and Lee [23], and Wang [28]. Şengönül [27] defined the sequence
y = (yi) which is frequently used as the Zp transformation of the sequence x = (xi) i.e,

yi = pxi + (1− p)xi−1
where x−1 = 0, p 6= 1, 1 < p <∞ and Zp denotes the matrix Zp = (zik) defined by

zik =

{ p, if (i = k),
1− p, (i− 1 = k); (i, k ∈ N)
0, otherwise .

Analogous to Başar and Altay [2], Şengönül [27] introduced the Zweier sequence spaces
Z and Z0 as follows

Z = {x = (xk) ∈ ω : Zpx ∈ c};

Z0 = {x = (xk) ∈ ω : Zpx ∈ c0}.

Khan, Ebadullah and Yasmeen [8] introduced the following classes of sequences:

ZI = {(xk) ∈ ω : ∃L ∈ C such that for a given ε > 0, {k ∈ N :| x/k − L |≥ ε} ∈ I};

ZI0 = {(xk) ∈ ω : for a given ε > 0; {k ∈ N :| x/k |≥ ε} ∈ I}

,



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 578

where (x
/
k) = (Zpx).

Recently V.A. Khan and Yasmeen [13] introduced the following sequence spaces:

ZI(µ,ν)(M) =
{

(xk) ∈ ω :
{
k ∈ N : M

(
µ(x

/
k−L,t)
ρ

)
≤ 1− ε or M

(
ν(x

/
k−L,t)
ρ

)
≥ ε
}
∈ I
}
,

ZI0(µ,ν)(M) =
{

(xk) ∈ ω :
{
k ∈ N : M

(
µ(x

/
k,t)

ρ

)
≤ 1− ε or M

(
ν(x

/
k,t)

ρ

)
≥ ε
}
∈ I
}
.

In this article we introduce the intuitionistic Zweier I-convergent dou-
ble sequence spaces defined by Orlicz function as follows:

2ZI(µ,ν)(M) =
{

(xij) ∈ 2ω :
{

(i, j) ∈ N× N : M
(
µ(x

//
ij −L,t)
ρ

)
≤ 1− ε or M

(
ν(x

//
ij −L,t)
ρ

)
≥ ε
}
∈ I2

}
;

2ZI0(µ,ν)(M) =
{

(xij) ∈ 2ω :
{

(i, j) ∈ N× N : M
(
µ(x

//
ij ,t)

ρ

)
≤ 1− ε or M

(
ν(x

//
ij ,t)

ρ

)
≥ ε
}
∈ I2

}
.

3. Main results

Theorem 1. The spaces 2ZI(µ,ν)(M) and 2ZI0(µ,ν)(M) are linear spaces.

Proof. We prove the result for 2ZI(µ,ν)(M). Similarly the result can be proved for

2ZI0(µ,ν)(M). Let (x
//
ij ), (y

//
ij ) ∈ 2ZI(µ,ν)(M) and let α, β be scalars. Then for a given

ε > 0, we have

A1 =

{
(i, j) ∈ N× N : M

(
µ
(
x
//
ij −L1,

t
2|α|

)
ρ1

)
≤ 1− ε or

(
ν
(
x
//
ij −L1,

t
2|α|

)
ρ1

)
≥ ε
}
∈ I2;

A2 =

{
(i, j) ∈ N× N : M

(
µ
(
y
//
ij −L2,

t
2|β|

)
ρ2

)
≤ 1− ε or M

(
ν
(
y
//
ij −L2,

t
2|β|

)
ρ2

)
≥ ε
}
∈ I2. Thus

Ac1 =

{
(i, j) ∈ N× N : M

(
µ
(
x
//
ij −L1,

t
2|α|

)
ρ1

)
> 1− ε or M

(
ν
(
x
//
ij −L1,

t
2|α|

)
ρ1

)
< ε
}
∈ F(I2);

Ac2 =

{
(i, j) ∈ N× N : M

(
µ
(
y
//
ij −L2,

t
2|β|

)
ρ2

)
> 1− ε or M

(
ν
(
y
//
ij −L2,

t
2|β|

)
ρ2

)
< ε
}
∈ F(I2).

Define the set A3 = A1 ∪ A2, so that A3 ∈ I2. It follows that Ac3 is a non-empty set in
F(I2).

We shall show that for ρ3 = max{2 | α | ρ1, 2 | β | ρ2} and for each (x
//
ij ), (y

//
ij ) ∈ 2ZI(µ,ν)(M),

Ac3 ⊂
{

(i, j) ∈ N× N : M
(
µ
(
(αx

//
ij +βy

//
ij )−(αL1+βL2),t

)
ρ3

)
> 1− ε or

M
(ν((αx//ij + βy

//
ij )− (αL1 + βL2), t

)
ρ3

)
< ε
}
.

Let (m,n) ∈ AI3. In this case



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 579

M
(µ(x//mn − L1,

t
2|α|
)

ρ3

)
> 1− ε or M

(ν(x//mn − L1,
t

2|α|
)

ρ3

)
< ε

and

M
(µ(y//mn − L2,

t
2|β|
)

ρ3

)
> 1− ε or M

(ν(y//mn − L2,
t

2|β|
)

ρ3

)
< ε.

we have

M
(
µ
(
(αx

//
mn+βy

//
mn)−(αL1+βL2),t

)
ρ3

)
≥M

(
µ
(
αx

//
mn−αL1,

t
2

)
ρ3

)
∗M

(
µ
(
βy
//
mn−βL2,

t
2

)
ρ3

)
= M

(
µ
(
x
//
mn−L1,

t
2|α|

)
ρ3

)
∗M

(
µ
(
y
//
mn−L2,

t
2|β|

)
ρ3

)
> (1− ε) ∗ (1− ε) = (1− ε)
and

M
(
ν
(
(αx

//
mn+βy

//
mn)−(αL1+βL2),t

)
ρ3

)
≤M

(
ν
(
αx

//
mn−αL1,

t
2

)
ρ3

)
�M

(
ν
(
βy
//
mn−βL2,

t
2

)
ρ3

)
= M

(
ν
(
x
//
mn−L1,

t
2|α|

)
ρ3

)
�M

(
ν
(
y
//
mn−L2,

t
2|β|

)
ρ3

)
> ε � ε = ε.
This implies that

Ac3 ⊂
{

(i, j) ∈ N× N : M
(
µ
(
(αx

//
ij +βy

//
ij )−(αL1+βL2),t

)
ρ3

)
> 1− ε or

M
(ν((αx//ij + βy

//
ij )− (αL1 + βL2), t

)
ρ3

)
< ε
}
.

Hence 2ZI(µ,ν)(M) is a linear space.

Theorem 2. Every open ball 2Bx//(r, t)(M) is an open set in 2ZI(µ,ν)(M).

Proof. Let 2Bx//(r, t)(M) be an open ball with centre x// and radius r with respect
to t.
That is

2Bx//(r, t)(M) = {(i, j) ∈ N× N : M
(µ(x

//
ij − L, t)
ρ

)
≤ 1− r or M

(ν(x
//
ij − L, t)
ρ

)
≥ r} ∈ I2.

Let y// ∈ 2B
c
x//

(r, t)(M). Then

M
(µ(x// − y//, t)

ρ

)
> 1− randM

(ν(x// − y//, t)
ρ

)
< r



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 580

.
Since M

(
µ(x//−y//,t)

ρ

)
> 1− r, there exists t0 ∈ (0, 1) such that

M
(µ(x// − y//, t0)

ρ

)
> 1− r and M

(ν(x// − y//, t0)
ρ

)
< r.

Putting r0 = M
(
µ(x//−y//,t0)

ρ

)
. We have r0 > 1 − r, there exists s ∈ (0, 1) such that

r0 > 1− s > 1− r.
For r0 > 1−s , we have r1, r2 ∈ (0, 1) such that r0∗r1 > 1−s and (1−r0)�(1−r2) ≤ s.
Putting r3 = max{r1, r2}, consider the ball 2B

c
y//

(1− r3, t− t0)(M). We prove that

2B
c
y//

(1− r3, t− t0)(M) ⊂ 2B
c
x//

(r, t)(M).

Let z// ∈ 2B
c
y//

(1− r3, t− t0)(M).

M
(
µ(y//−z//,t−t0)

ρ

)
> r3 and M

(
ν(y//−z//,t−t0)

ρ

)
< r3.

Therefore,

M
(µ(x// − z//, t)

ρ

)
≥M

(µ(x// − y//, t0)
ρ

)
∗M

(µ(y// − z//, t− t0)
ρ

)
≥ (r0 ∗ r3) ≥ (r0 ∗ r1) ≥ (1− s) > (1− r).

and

M
(ν(x// − z//, t)

ρ

)
≤M

(ν(x// − y//, t0)
ρ

)
�M

(ν(y// − z//, t− t0)
ρ

)
≤ (1− r0) � (1− r3) ≤ (1− r0) > (1− r2) < s < r.

Thus z// ∈ 2B
c
x//

(r, t)(M) and hence 2B
c
y//

(1− r3, t− t0)(M) ⊂ 2B
c
x//

(r, t)(M).

Remark 1. 2ZI(µ,ν)(M) is IFNS.

Define

2τ(µ,ν)(M) = {A ⊂ 2ZI(µ,ν)(M) : for each x ∈ A there exists t > 0 and r ∈ (0, 1) s. t. 2B
c
x//

(r, t)(M) ⊂ A}.
Then 2τ(µ,ν)(M) is a topology on 2ZI(µ,ν)(M).

Theorem 3. The topology 2τ(µ,ν)(M) on 2ZI(µ,ν)(M) is first countable.

Proof.
{
2Bx//

(
1
n,

1
n

)
(M) : n = 1, 2, 3, ......................

}
is a local base at x// the topology

2τ(µ,ν)(M) on 2ZI(µ,ν)(M) is first countable.

Theorem 4. 2ZI(µ,ν)(M) and 2ZI0(µ,ν)(M) are Housdorff spaces.



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 581

Proof. We prove the result for 2ZI(µ,ν)(M). Similarly the result can be proved for

2ZI0(µ,ν)(M).

Let x//, y// ∈ 2ZI(µ,ν)(M) such that x// 6= y//.

Then 0 < M
(
µ(x//−y//,t)

ρ

)
< 1

and 0 < M
(
ν(x//−y//,t)

ρ

)
< 1.

Putting r1 = M
(
µ(x//−y//,t)

ρ

)
and r2 = M

(
ν(x//−y//,t)

ρ

)
and r = max{r1, 1− r2}.

For each r0 ∈ (r, 1), there exists r3 and r4 such that

r3 ∗ r4 ≥ r0 and (1− r3) � (1− r4) ≤ (1− r0).

Putting r5 = max{r3, 1− r4} and consider the open balls 2Bx//(1−r5, t2)(M) and 2By//(1−
r5,

t
2)(M).

Then clearly 2Bx//(1− r5, t2) ∩ 2By//(1− r5, t2)(M) = φ.

For if there exists z// ∈ 2Bx//(1− r5, t2) ∩ 2By//(1− r5, t2)(M), then

r1 = M
(µ(x// − y//, t)

ρ

)
≥M

(µ(x// − z//, t2)

ρ

)
∗M

(µ(z// − y//, t2)

ρ

)
≥ r5 ∗ r5 ≥ r3 ∗ r3 ≥ r0 > r1.

and

r2 = M
(ν(x// − y//, t)

ρ

)
≤M

(µ(x// − z//, t2)

ρ

)
�M

(ν(z// − y//, t2)

ρ

)
≤ (1− r5) � (1− r5) ≤ (1− r4) � (1− r4) ≤ (1− r0) < r,

which is a contradiction.
Hence 2ZI(µ,ν)(M) is Housdorff.

Theorem 5. 2ZI(µ,ν)(m) is an IFNS. 2τ(µ,ν)(M) is a topology on 2ZI(µ,ν)(M) . Then a

sequence (x
//
ij ) ∈ 2ZI(µ,ν)(M), x

//
ij → x// if and if µ(x

//
ij − x//, t)(M) and

M
(ν(x

//
ij − x//, t)
ρ

)
→ 0

as i→∞, j →∞.
Proof. Fix t0 > 0. Suppose x

//
ij → x//. Then for r ∈ (0, 1), there exists n0 ∈ N such

that x
//
ij ∈ 2Bx//(r, t)(M) for all i ≥ n0, j ≥ n0.

2Bx//(r, t)(M) = {(i, j) ∈ N× N : M
(µ(x

//
ij − x//, t)
ρ

)
≤ 1− r or M

(ν(x
//
ij − x//, t)
ρ

)
≥ r} ∈ I2



V. A. Khan, Yasmeen, H. Fatima, A. Ahmad / Eur. J. Pure Appl. Math, 10 (3) (2017), 574-585 582

such that 2B
c
x//

(r, t)(M) ∈ F(I2).
Then

1−M
(µ(x

//
ij − x//, t)
ρ

)
< r

and M
(
ν(x

//
ij −x

//,t)

ρ

)
→ 0 as i→∞, j →∞

Conversely, if for each t > 0,M
(
µ(x

//
ij −x

//,t)

ρ

)
→ 1 and

M
(ν(x

//
ij − x//, t)
ρ

)
→ 0

as i→∞, j →∞,thenforr ∈ (0, 1), there exists n0 ∈ N such that 1−M
(
µ(x

//
ij −x

//,t)

ρ

)
< r

for all i ≥ n0, j ≥ n0.
Thus x

//
ij ∈ 2B

c
xij (r, t)(M) for all i ≥ n0, j0 ≥ n0 and hence x

//
ij → x//.

Theorem 6. A double sequence x = (x
//
ij ) ∈ 2ZI(µ,ν)(M). I-converges if and only if for

every ε > 0 and t > 0 there exists a number M = M(x, ε, t), N = N(x, ε, t) such that

{(M,N) ∈ N× N : M
(
µ(x

//
MN−L,

t
2
)

ρ

)
> 1− ε or M

(
ν(x

//
MN−L,

t
2
)

ρ

)
< ε} ∈ F(I2).

Proof. Suppose that 2I(µ,ν) − limx = L and let ε > 0 and t > 0. For a given ε > 0

choose, s > 0 such that (1−ε)∗(1−ε) > 1−s and ε�ε < s. Then for each x ∈ 2ZI(µ,ν)(M),

Ax(ε, t)(M) = {(i, j) ∈ N× N : M
(
µ(x

//
ij −L,

t
2
)

ρ

)
≤ 1− ε or M

(
ν(x

//
ij −L,

t
2
)

ρ

)
≥ ε} ∈ I2

which implies that

Acx(ε, t)(M) = {(i, j) ∈ N× N : M
(
µ(x

//
ij −L,

t
2
)

ρ

)
> 1− ε or M

(
ν(x

//
ij −L,

t
2
)

ρ

)
< ε} ∈ F(I2).

Conversely let us choose N ∈ Acx(ε, t)(M). Then

M
(µ(x

//
N − L,

t
2)

ρ

)
> 1− ε or M

(ν(x
//
N − L,

t
2)

ρ

)
< ε.

Now we want to show that there exists a number N = N(x, ε, t) such that

{(i, j) ∈ N× N : M
(µ(x

//
ij − x

//
N , t)

ρ

)
≤ 1− s or M

(ν(x
//
ij − x

//
N , t)

ρ

)
≥ s} ∈ I2.

For this, define for each x ∈ 2ZI(µ,ν)(M).

2Bx(ε, t)(M) = {(i, j) ∈ N× N : M
(µ(x

//
ij − x

//
N , t)

ρ

)
≤ 1− s or M

(ν(x
//
ij − x

//
N , t)

ρ

)
≥ s} ∈ I2.

Now we show that 2Bx(ε, t)(M) ⊂ 2Ax(ε, t)(M).



Suppose that 2Bx(ε, t)(M) ⊂ 2Ax(ε, t)(M).
Then there exists (m,n) ∈ 2Bx(ε, t)(M) and (m,n) /∈ 2Ax(ε, t)(M). Therefore we have

M
(µ(x

//
ij − x

//
N , t)

ρ

)
< 1− s

and M
(
µ(x

//
ij −L,

t
2
)

ρ

)
> 1− ε.

In particular M
(
µ(x

//
N −L,

t
2
)

ρ

)
> 1− ε.

Therefore we have

1− s ≥M
(µ(x

//
mn − x//N , t)

ρ

)
≥M

(µ(x
//
mn − L, t2)

ρ

)
∗M

(µ(x
//
N − L,

t
2)

ρ

)
≥ (1− ε) ∗ (1− ε) > 1− s,

which is not possible.

On the other hand M
(
ν(x

//
ij −x

//
N ,t)

ρ

)
≥ s and M

(
ν(x

//
ij −L,

t
2
)

ρ

)
< ε.

In particular M
(
ν(x

//
N −L,

t
2
)

ρ

)
< ε.

Therefore we have

s ≤M
(ν(x

//
mn − x//N , t)

ρ

)
≤M

(ν(x
//
mn − L, t2)

ρ

)
�M

(ν(x
//
N − L,

t
2)

ρ

)
≤ ε � ε < s,

which is not possible.
Hence 2Bx(ε, t)(M) ⊂ 2Ax(ε, t)(M), 2Ax(ε, t)(M) ∈ I2.
This implies that 2Bx(ε, t)(M) ∈ I2. Hence proved.

Acknowledgements

The authors would like to record their gratitude to the reviewer for his careful reading
and making some useful corrections which improved the presentation of the paper.

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