Available online throug
ISSN 2229 – 5046
International Journal of Mathematical Archive- 9(3), March-2018 168
SOME RESULTS ON INTERVAL VALUED
INTUITIONISTIC FUZZY n- NORMED LINEAR SPACE
SANDEEP KUMAR
Assistant professor of mathematics, A. I. J. H. M. College, Rohtak, India.
(Received On: 21-01-18; Revised & Accepted On: 23-02-18)
ABSTRACT
I
n this paper we first define concept of interval-valued-intuitionistic –fuzzy n-normed space and then we obtain some results in this newly defined space.Mathematics Subject Classification: 46B20, 46B99, 46A19, 46S40, 47H10.
Key words and phrases: Normed space, interval-valued-intuitionstic-fuzzy n-normed space, fuzzy subset.
1. INTRODUCTION
The theory of Fuzzy sets was introduced by L.Zadeh [13] in 1965. Subsequently a progressive development was made in the theory of interval-valued fuzzy subsets in [14]. J.H.Park[9] studied the notion of intuitionistic fuzzy metric spaces. Many authors introduced the definition of fuzzy inner product space and fuzzy normed linear space in [4, 5, 6]. The origin and development of intuitionstic fuzzy set theory can be found in [7, 10, 11, 12]. The motive of this paper is to introduce the notion of interval- valued – intuitionistic –fuzzy n-normed space as generalization of intuitionstic-fuzzy n-normed space [20] and we obtain some results in this space. Further we introduce the generalized Cartesian product of the interval-valued-intuitionistic –fuzzy n-normed space and prove some of its properties.
2. PRELIMINARIES
In this section we provide the essential definitions and results necessary for the development of our theory.
Definition 2.1 [14]: An interval number on [0, 1], say
a
, is a closed sub interval of [0, 1] of the forma
= [a−, a+], where 0 ≤ a− ≤ a+ ≤ 1. Let D [0, 1] denote the family of all closed sub-intervals of [0, 1], that is,D [0, 1] = {
a
= [a−, a+]: a− ≤ a+ and a−, a+∈
[0, 1] }Definition 2.2 [14]: Let
a
i = [a
i−,
a
i+]∈
D[0, 1] for all i∈
Ω, Ω an index set. Define(a) inf i {
a
i : i∈
Ω } = [ + Ω ∈ − Ω∈ i i i
i
a
,
inf
a
inf
](b)supi {
a
i : i∈
Ω } = [+
Ω ∈ −
Ω
∈ i i i
i
a
a
,
sup
sup
]In particular, whenever
a
= [a−, a+],b
= [b−, b+], in D [0, 1], we define (i)a
≤b
if and only if a− ≤ b− and a+ ≤ b+(ii)
a
=b
if and only if a− = b− and a+ = b+ (iii)a
<b
if and only if a− < b− and a+ < b+.(iv) mini{
a
,b
}= [min {a−, b−}, min {a+, b+} ] (v) max i {a
,b
}= [max{a−, b−}, max{a+, b+}].Corresponding Author: Sandeep Kumar
© 2018, IJMA. All Rights Reserved 169
Definition 2.3 [14]: Let X be a set. A mapping
A
: X → D[0, 1] is called an Interval-valued fuzzy subset (briefly, an i-v fuzzy subset) of X, whereA
(x) = [A−(x), A+(x)], and A− and A+ are fuzzy subsets in X such thatA−(x) ≤ A+(x) for all x
∈
X.Definition 2.4 [14]: Let
A
be an interval-valued fuzzy subset of X and [t
1,
t
2]∈
D [0, 1]. Then the set∪
(A
; [t
1,
t
2]) = {x∈
X:A
(x) ≥ [t
1,
t
2]}is called an upper level subset of A. Note that∪
(A
; [t
1,
t
2]) = U (A−,t
1) ∩ U (A+;t
2) whereU (A−;
t
1) = {x∈
X : A−(x) ≥t
1}andU (A+;
t
2) = {x∈
X : A+(x) ≥t
2}.Definition 2.5 [2]: Let V denote a vector space of dimension n over a field F. A fuzzy subspace is a fuzzy subset μ of V
such that
μ(αx + βy) ≥ μ(x)
Λ
μ(y), x, y∈
V , α, β∈
F(field), whereΛ
stands for intersection.Definition 2.6: Let V denote a vector space of dimension n over a field F. A intuitionstic fuzzy subspace is a fuzzy subset μ,
η
of V such thatμ(αx + βy) ≥ μ(x)
Λ
μ(y)and
η
(αx + βy)≤
η
(x)∨
η
(y), x, y∈
V , α, β∈
F(field), whereΛ
and∨
stands for intersection and union respectively.Remark 2.7: A subset {
x
1,
x
2,...,
x
m} of V is intuitionstic-fuzzy linearly independent if it is linearly independentand μ(
m
i m
i i i
x
1 1
)
= =
Λ
=
∑
α
μ(x
i) ,η
(m
i m
i i i
x
1 1
)
= =
∨
=
∑
α
μ(x
i) for allα
i∈
F
,
α
i≠
0
,
i = 1,..., m,Λ
,∨
stands for intersection and union respectively.Definition 2.8 [20]: An intuitionistic fuzzy n-normed linear space (or) in short i-f-n-NLS is an object of the form A = {(X, N(
x
1,
x
2,...,
x
n, t), M(x
1,
x
2,...,
x
n, t) : (x
1,
x
2,...,
x
n)∈
Xn }where X is a linear space over a field F and N, M are fuzzy sets on Xn
×
(0,1), N denotes the degree of membership and M denotes the degree of non-membership of (x
1,
x
2,...,
x
n)∈
Xn×
(0,1) satisfying the following conditions: (i) N(x
1,
x
2,...,
x
n, t) +M(x
1,
x
2,...,
x
n, t)≤
1;(ii) N(
x
1,
x
2,...,
x
n, t) > 0;(iii) N(
x
1,
x
2,...,
x
n,t) = 1 if and only ifx
1,
x
2,...,
x
n are linearly dependent; (iv) N(x
1,
x
2,...,
x
n,t) is invariant under any permutation ofx
1,
x
2,...,
x
n. (v) N(x
1,
x
2,...,
cx
n, t) = N(x
1,
x
2,...,
x
n,c
t ), if c
≠
0, c∈
F(field).(vi)
N x x
( ,
1 2,...,
x
n+
x
n',
s
+ ≥
t
)
min
{
N x x
( ,
1 2,... , ),
x s N x x
n( ,
1 2,....,
x t
n', )
}
(vii) N(
x
1,
x
2,...,
x
n, t) : (0;1)→
[0; 1] is continuous in t; (viii) M(x
1,
x
2,...,
x
n, t) > 0;(ix) M(
x
1,
x
2,...,
x
n, t) = 0 if and only ifx
1,
x
2,...,
x
nare linearly dependent; (x) M(x
1,
x
2,...,
x
n,t) is invariant under any permutation ofx
1,
x
2,...,
x
n. (xi) M(x x
1,
2,...,
cx
n, t) = M(x
1,
x
2,...,
x
n,c
t ), if c
≠
0, c∈
F(field);(xii)
M x x
( ,
1 2,...,
x
n+
x
n',
s
+ ≤
t
)
max
{
M x x
( ,
1 2,... , ),
x s M x x
n( ,
1 2,....,
x t
n', )
}
© 2018, IJMA. All Rights Reserved 170
Remark 2.9:
(i) Let V be a vector space over a field F and let
U
1,
U
2,...,
U
n be the subspaces of V . V is said to be direct sum ofU
1,
U
2,...,
U
n if every element v∈
V can be written in one and only one wayv =
u
1+
u
2+
...
+
u
n whereu
i∈
U
i and denoted as V =U
1⊕
U
2⊕
...
⊕
U
n.(ii) Let V and W be vector spaces over a field F. Then the tensor product of two vectors is denoted by V
⊗
W and given t∈
V⊗
W, t can be uniquely written as , t =∑
⊗
j i
j i ij
x
y
t
,
where the sum is taken over all i, j for which
0
≠
ijt
.3. INTERVAL-VALUED-INTUITIONSTIC- FUZZY-n-NORMED LINEAR SUBSPACES
As a generalization of definition 2.8 we have the following notion of interval- valued-intuitionstic-fuzzy n-normed linear subspaces.
Definition 3.1: Let X be a linear space over a field F. An interval-valued fuzzy subsets
N
,
M
ofX
n×
R
is called an interval-valued- intituitionstic fuzzy n-norm if and only if1.
N
(
x
1,
x
2,...,
x
n,
t
)
+
M
(
x
1,
x
2,...,
x
n,
t
)
≤
1
2. For all t
∈
R with t≤
0,N
(
x
1,
x
2,...,
x
n,
t
)
=
0
.3. For all t
∈
R with t > 0,N
(
x
1,
x
2,...,
x
n,
t
)
=
1
iffx
1,
x
2,...,
x
n are linearly dependent.4.
N
(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n.5. For all t
∈
R with t > 0N
(
x
1,
x
2,...,
cx
n,
t
)
=
(
1,
2,...,
,
)
c
t
x
x
x
N
n , if c≠
0,c∈
F(field).6. For all s, t
∈
RN x x
( ,
1 2,...,
x
n+
x
n',
s
+ ≥
t
)
min
i{
N x x
( ,
1 2,... , ),
x s
nN x x
( ,
1 2,....,
x t
n', )
}
7.
N
(
x
1,
x
2,...,
x
n,
t
)
is a non decreasing function of t∈
R andlim
( ,
1 2,...,
n, )
1
t→∞
N x x
x t
=
.
8. For all t
∈
R with t≤
0,M
(
x
1,
x
2,...,
x
n,
t
)
>
0
.9. For all t
∈
R with t > 0,M
(
x
1,
x
2,...,
x
n,
t
)
=
0
iffx
1,
x
2,...,
x
n are linearly dependent. 10.M
(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n.11. For all t
∈
R with t > 0M
(
x
1,
x
2,...,
cx
n,
t
)
=
(
1,
2,...,
,
)
c
t
x
x
x
M
n , if c≠
0,c∈
F(field).12. For all s, t
∈
RM x x
( ,
1 2,...,
x
n+
x
n',
s
+ ≤
t
)
max
i{
M x x
( ,
1 2,... , ),
x s
nM x x
( ,
1 2,....,
x t
n', )
}
13.
M
(
x
1,
x
2,...,
x
n,
t
)
is a non increasing function of t∈
R andlim
t→∞M x x
( ,
1 2,...,
x t
n, )
=
0
Then (X,
N
,
M
) is called an interval- valued Intuitionstic fuzzy n-normed linear space in short i-v-i-f-n NLS.The following example agrees with our notion of i-v-i-f-n-NLS
Example 3.2: Let (X,
•
,
•
,...,
•
) bean n-normed space . Define)
,
,...,
,
(
x
1x
2x
t
N
n =
>
≤
n nx
x
x
t
when
x
x
x
t
when
,...,
,
1
,...
,
0
2 1 2 1and
M
(
x
1,
x
2,...,
x
n,
t
)
=
>
≤
n nx
x
x
t
when
x
x
x
t
when
,...,
,
0
,...
,
1
2 1 2 1Then (X,
N
,
M
) is an i-v-i-f-n-NLS where0
=[0,0] and1=[1,1].
© 2018, IJMA. All Rights Reserved 171
Theorem 3.3: Let (X,
N
1,
M
1) and (X,N
2,
M
2) be two i-v-i-f-n-NLS. DefineN
(x
1,
x
2,...,
x
n,t ) = (N
1
N
2 )(x
1,
x
2,...,
x
n,t )=
min
i{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
.
andM
(x
1,
x
2,...,
x
n,t ) = (M
1
M
2 )(x
1,
x
2,...,
x
n,t )=
max
i{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
.
for all (
x
1,
x
2,...,
x
n,t)∈
X
n×
R
. Then (X,N
1
N
2 ,M
1
M
2) or (X,N
,M
) is an i-v-i-f-n-NLS.Proof: (1) Since
N
1(
x
1,
x
2,...,
x
n,
t
)
+
M
1(
x
1,
x
2,...,
x
n,
t
)
≤
1
.
and
N
2(
x
1,
x
2,...,
x
n,
t
)
+
M
2(
x
1,
x
2,...,
x
n,
t
)
≤
1
.
It follows that
.
1
)
,
,...,
,
(
1 21
x
x
x
t
≤
M
n -N
1(
x
1,
x
2,...,
x
n,
t
)
and
M
2(
x
1,
x
2,...,
x
n,
t
)
≤
1
.
-N
2(
x
1,
x
2,...,
x
n,
t
)
By definition
max
{
1
N
1(
x
1,
x
2,...,
x
n,
t
),
1
N
2(
x
1,
x
2,...,
x
n,
t
)
}
.
i
−
−
≥
max
i{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
.
⇒
min
i{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
.
=
1
-(max
i{
1
−
N
1(
x
1,
x
2,...,
x
n,
t
),
1
−
N
2(
x
1,
x
2,...,
x
n,
t
)
}
.
)≤
1
−
max
i{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
.
⇒
min
i{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
+
max
i{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
≤
1
⇒
N
(x
1,
x
2,...,
x
n,t ) +M
(x
1,
x
2,...,
x
n,t )=1
(2) For all t
∈
R with t≤
0, sinceN
1(
x
1,
x
2,...,
x
n,
t
)
=
0
andN
2(
x
1,
x
2,...,
x
n,
t
)
=
0
⇒
min
{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
=
0
i
⇒
N
(
x
1,
x
2,...,
x
n,
t
)
=
0
.(3) Since for all t
∈
R with t > 0,N
1(
x
1,
x
2,...,
x
n,
t
)
=
1
andN
2(
x
1,
x
2,...,
x
n,
t
)
=
1
iffx
1,
x
2,...,
x
n are linearly dependent.⇒
min
{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
=
1
iiff
x
1,
x
2,...,
x
n are linearly dependent.⇒
N
(
x
1,
x
2,...,
x
n,
t
)
=
1
iffx
1,
x
2,...,
x
n are linearly dependent.(4) Since
N
1(
x
1,
x
2,...,
x
n,
t
)
andN
2(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n.⇒
min
i{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
is also invariant under any permutation ofx
1,
x
2,...,
x
n.⇒
N
(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n.(5) Since for all t
∈
R with t > 0,N
1(
x
1,
x
2,...,
cx
n,
t
)
=
1( ,
1 2,...,
n,
)
t
N x x
x
c
and 2( ,
1 2,...,
n,
)
t
N x x
x
© 2018, IJMA. All Rights Reserved 172
⇒
min
{
N
1(
x
1,
x
2,...,
cx
n,
t
),
N
2(
x
1,
x
2,...,
cx
n,
t
)
}
i = ) , ,..., , ( ), , ,..., , (min 1 1 2 2 1 2
c t x x x N c t x x x
N n n
i
⇒
N
(
x
1,
x
2,...,
cx
n,
t
)
=
( , ,..., , )2 1 c t x x x N n .
(6) For all s, t
∈
R(
,
,...,
,
)
min
{
(
1,
2,...
,
),
(
1,
2,....,
',
)
}
' 2
1
x
x
x
s
t
N
x
x
x
s
N
x
x
x
t
x
N
n ni n
n
+
+
≥
, since{
(
,
,...
,
),
(
,
,....,
,
)
}
min
)
,
,...,
,
(
x
1x
2x
x
's
t
N
1x
1x
2x
x
's
t
N
2x
1x
2x
x
't
s
N
n+
n+
=
i n+
n+
n+
n+
and{
(
,
,...
,
),
(
,
,....,
,
)
}
min
iN
1x
1x
2x
n+
x
n's
+
t
N
2x
1x
2x
n+
x
n't
+
s
{
min
{
(
,
,...
,
),
(
,
,....,
,
)},
min
{
(
,
...,
,
),
(
,
,...,
,
)
}
min
i iN
1x
1x
2x
ns
N
1x
1x
2x
n't
iN
2x
1x
2x
ns
N
2x
1x
2x
n't
≥
{
min
{
(
,
,...
,
),
(
,
,....,
,
)}
,
min
{
(
,
,....,
,
),
(
,
,...,
,
)}
}
min
i iN
1x
1x
2x
ns
N
2x
1x
2x
s
iN
1x
1x
2x
n't
N
2x
1x
2x
n't
≥
{
(
,
,...
,
),
(
,
,....,
,
)
}
min
iN
x
1x
2x
ns
N
x
1x
2x
n't
≥
(7)
N
1(
x
1,
x
2,...,
x
n,
t
)
,N
2(
x
1,
x
2,...,
x
n,
t
)
is a non decreasing function of t∈
R and1 1 2
lim
( ,
,...,
n, )
1
t→∞N x x
x t
=
,
lim
2( ,
1 2,...,
n, )
1
t→∞N x x
x t
=
⇒
N
(
x
1,
x
2,...,
x
n,
t
)
is a non decreasing function of t∈
R andlim
( ,
1 2,...,
n, )
1
t→∞N x x
x t
=
(8) For all t
∈
R with t≤
0,M
1(
x
1,
x
2,...,
x
n,
t
)
>
0
,M
2(
x
1,
x
2,...,
x
n,
t
)
>
0
andM
(x
1,
x
2,...,
x
n, t)= (M
1
M
2 ) (x
1,
x
2,...,
x
n,t )=
max
i{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
.
⇒
M
(
x
1,
x
2,...,
x
n,
t
)
>
0
.(9) For all t
∈
R with t > 0,M
1(
x
1,
x
2,...,
x
n,
t
)
=
0
,M
2(
x
1,
x
2,...,
x
n,
t
)
=
0
iffx
1,
x
2,...,
x
n are linearly dependent. So,M
(x
1,
x
2,...,
x
n, t) = (M
1
M
2 ) (x
1,
x
2,...,
x
n,t )=
max
i{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
.
=0 iffx
1,
x
2,...,
x
n are linearly dependent.(10) By definition of
M
(
x
1,
x
2,...,
x
n,
t
)
, and using the fact thatM
1(
x
1,
x
2,...,
x
n,
t
)
,M
2(
x
1,
x
2,...,
x
n,
t
)
isinvariant under any permutation of
x
1,
x
2,...,
x
n.It is clear thatM
(
x
1,
x
2,...,
x
n,
t
)
is invariant under any permutation ofx
1,
x
2,...,
x
n.(11) For all t
∈
R with t > 0M
1(
x
1,
x
2,...,
cx
n,
t
)
=
M x x
1( ,
1 2,...,
x
n,
t
)
c
, andM
2(
x
1,
x
2,...,
cx
n,
t
)
=
)
,
,...,
,
(
1 22
c
t
x
x
x
M
n , if c≠
0,c∈
F(field).⇒
M
(x
1,
x
2,...,
cx
n,t ) = (M
1
M
2 )(x
1,
x
2,...,
cx
n,t )=
max
{
M
1(
x
1,
x
2,...,
cx
n,
t
),
M
2(
x
1,
x
2,...,
cx
n,
t
)
}
.
i=
max
iM x x
1( ,
1 2,...,
x
n,
t
t M
),
2( ,
x x
1 2,...,
x
n,
t
) .
c
c
=
M
(x
1,
x
2,...,
x
n,© 2018, IJMA. All Rights Reserved 173 (12) For all s, t
∈
R(
,
,...,
,
)
max
{
1(
1,
2,...
,
),
1(
1,
2,....,
',
)
}
' 2
1
1
x
x
x
x
s
t
M
x
x
x
s
M
x
x
x
t
M
n ni n
n
+
+
≤
and
M
2(
x
1,
x
2,...,
x
n+
x
n',
s
+
t
)
≤
max
i{
M
2(
x
1,
x
2,...
x
n,
s
),
M
2(
x
1,
x
2,....,
x
n',
t
)
}
⇒
M
(
x
1,
x
2,...,
x
n+
x
n',
s
+
t
)
= (M
1
M
2 )(x
1,
x
2,...,
x
n+
x
n',s+t )=
max
i{
M x x
1( ,
1 2,...,
x
n+
x s t M
n',
+
),
2( ,
x x
1 2,...,
x
n+
x s t
n',
+
) .
}
{
max
{
(
,
,...
,
),
(
,
,....,
,
)},
max
{
(
,
...,
,
),
(
,
,...,
,
)
}
max
2 1 2 2 1 2 '' 2 1 1 2
1
1
x
x
x
s
M
x
x
x
t
M
x
x
x
s
M
x
x
x
t
M
n ni n
n i
i
≤
{
max
{
(
,
,...
,
),
(
,
,....,
,
)}
,
max
{
(
,
,....,
,
),
(
,
,...,
,
)}
}
max
i iM
1x
1x
2x
ns
M
2x
1x
2x
s
iM
1x
1x
2x
n't
M
2x
1x
2x
n't
≤
{
(
,
,...
,
),
(
,
,....,
,
)
}
max
M
x
1x
2x
ns
M
x
1x
2x
n't
i≤
.Hence (X,
N
1
N
2 ,M
1
M
2) or (X,N
,M
) is an i-v-i-f-n-NLS.Remark 3.4: Let (X,
N
1,
M
1) and (X,N
2,
M
2) be two i-v-i-f-n-NLS. DefineN
(x
1,
x
2,...,
x
n,t ) = (N
1∪
N
2 )(x
1,
x
2,...,
x
n,t )=
max
i{
N
1(
x
1,
x
2,...,
x
n,
t
),
N
2(
x
1,
x
2,...,
x
n,
t
)
}
.
andM
(x
1,
x
2,...,
x
n,t ) = (M
1∪
M
2 )(x
1,
x
2,...,
x
n,t )=
min
{
M
1(
x
1,
x
2,...,
x
n,
t
),
M
2(
x
1,
x
2,...,
x
n,
t
)
}
.
ifor all (
x
1,
x
2,...,
x
n,t)∈
X
n×
R
. Then (X,
N
1∪
N
2 ,M
1∪
M
2 ) or (X,N
,M
) is an not i-v-i-f-n-NLS.4. GENERALIZED CARTESIAN PRODUCT OF THE INTERVAL-VALUED-INTUITIONISTIC
FUZZY-n-NORMED LINEAR SPACES
We now proceed to our new notion of generalized cartesian product of the Interval-valued-intuitionistic fuzzy n -normed linear spaces in the following theorem.
Theorem 4.1: Let
S = {(X, (
A
1(x
1,
x
2,...,
x
n, t);B
1(x
1,
x
2,...,
x
n, t)); (x
1,
x
2,...,
x
n)∈
Xn}and
T = {(Y,
A
2(y
1,
y
2,...,
y
n, t);B
2(y
1,
y
2,...,
y
n, t)); (y
1,
y
2,...,
y
n)∈
Yn}be two interval-valued-intuitionistic fuzzy n-normed linear spaces. Then
S
×
∗,◊ T ={(X×
Y,A
(z
1,
z
2,...,
z
n, t),B
(z
1,
z
2,...,
z
n,t)); (z
1,
z
2,...,
z
n)∈
(X×
Y )n} is an i-v-i-f-n-NLS withA
(z
1,
z
2,...,
z
n , t) =A
1 (x
1,
x
2,...,
x
n ,t)∗
A
2(y
1,
y
2,...,
y
n, t)and
B
(z
1,
z
2,...,
z
n, t) =B
1(x
1,
x
2,...,
x
n, t)◊
B
2(y
1,
y
2,...,
y
n, t);where
z
i=
(
x
i,
y
i)
, i = 1, 2,……, n.Proof: As
A
1(
x
1,
x
2,...,
x
n,
t
)
+
B
1(
x
1,
x
2,...,
x
n,
t
)
≤
1
.
(1) and it follows that)
,
,...,
,
(
1
)
,
,...,
,
(
1 2 1 1 21
x
x
x
t
A
x
x
x
t
B
n≤
−
nand
B
2(
x
1,
x
2,...,
x
n,
t
)
≤
1
−
A
2(
x
1,
x
2,...,
x
n,
t
)
So,≥
−
◊
−
(
,
,...,
,
)
)
(
1
(
,
,...,
,
))
1
(
A
1x
1x
2x
nt
A
2x
1x
2x
nt
B
1(
x
1,
x
2,...,
x
n,
t
)
◊
B
2(
x
1,
x
2,...,
x
n,
t
)
Then,
(
,
,...,
,
)
2(
1,
2,...,
,
)
^
2 1
1
x
x
x
t
A
x
x
x
t
© 2018, IJMA. All Rights Reserved 174 = 1- (
1
−
A
1(
x
1,
x
2,...,
x
n,
t
)
◊
1
−
A
2(
x
1,
x
2,...,
x
n,
t
)
)≤
1-(B
1(
x
1,
x
2,...,
x
n,
t
)
◊
B
2(
x
1,
x
2,...,
x
n,
t
)
) (2) Thus,)
,
,...,
,
(
)
,
,...,
,
(
2 1 2^
2 1
1
x
x
x
t
A
x
x
x
t
A
n◊
n≤
1-(B
1(
x
1,
x
2,...,
x
n,
t
)
◊
B
2(
x
1,
x
2,...,
x
n,
t
)
)where a
^
◊
b= 1 -((1 - a)◊
(1 - b)) is defined as the dual t-norm with respect to◊
So, if≤
∗
(
,
,...,
,
)
)
,
,...,
,
(
1 2 2 1 21
x
x
x
t
A
y
y
y
t
A
n n(
,
,...,
,
)
2(
1,
2,...,
,
)
^
2 1
1
x
x
x
t
A
y
y
y
t
A
n◊
nthen by (2), we have,
≤
∗
(
,
,...,
,
)
)
,
,...,
,
(
1 2 2 1 21
x
x
x
t
A
y
y
y
t
A
n n 1-(B
1(
x
1,
x
2,...,
x
n,
t
)
◊
B
2(
y
1,
y
2,...,
y
n,
t
)
)(
A
1(
x
1,
x
2,...,
x
n,
t
)
∗
A
2(
y
1,
y
2,...,
y
n,
t
)
)
+(B
1(
x
1,
x
2,...,
x
n,
t
)
◊
B
2(
y
1,
y
2,...,
y
n,
t
)
)≤
1A
(z
1,
z
2,...,
z
n,
t
) +B
(z
1,
z
2,...,
z
n,
t
)≤
1.Similarly we can verify the other conditions. Thus, S
×
∗,◊ T is an i-f-n-NLS.Theorem 4.2: The generalized Cartesian product of the interval-valued-intuitionistic-fuzzy n-normed linear spaces is
commutative. In other words if S and T are two interval-valued-intuitionistic-fuzzy n-normed linear spaces. Then S= T
⇒
S×
∗,◊ T = T×
∗,◊ S.Proof: Assume S=T, (
x
1,
x
2,...,
x
n), (y
1,
y
2,...,
y
n)∈
Xn. Then,)
,
,...,
,
(
)
,
,...,
,
(
1 2 2 1 21
x
x
x
t
A
y
y
y
t
A
n∗
n =A
2(
x
1,
x
2,...,
x
n,
t
)
∗
A
1(
y
1,
y
2,...,
y
n,
t
)
and
B
1(
x
1,
x
2,...,
x
n,
t
)
◊
B
2(
y
1,
y
2,...,
y
n,
t
)
=B
2(
x
1,
x
2,...,
x
n,
t
)
◊
B
1(
x
1,
x
2,...,
x
n,
t
)
Thus, S×
∗,◊ T = T×
∗,◊ S. However the converse is not true. For example, LetS= {(X, (
A
1(x
1,
x
2,...,
x
n, t),B
1(x
1,
x
2,...,
x
n, t));A
1(x
1,
x
2,...,
x
n,t)=α
,1
B
(x
1,
x
2,...,
x
n,t) =β
, (x
1,
x
2,...,
x
n)∈
Xn}T= {(Y,
A
2(x
1,
x
2,...,
x
n, t),B
2(x
1,
x
2,...,
x
n, t));A
2(x
1,
x
2,...,
x
n)=γ
,2
B
(x
1,
x
2,...,
x
n) =δ
, (x
1,
x
2,...,
x
n)∈
Xn} Then)
,
,...,
,
(
)
,
,...,
,
(
1 2 2 1 21
x
x
x
t
A
x
x
x
t
A
n∗
n =α
∗
γ
=γ
∗
α
=A
2(
x
1,
x
2,...,
x
n,
t
)
∗
A
1(
y
1,
y
2,...,
y
n,
t
)
. and)
,
,...,
,
(
)
,
,...,
,
(
1 2 2 1 21
x
x
x
t
B
x
x
x
t
B
n◊
n =β
◊
δ
=δ
◊
β
=B
2(
x
1,
x
2,...,
x
n,
t
)
◊
B
1(
x
1,
x
2,...,
x
n,
t
)
.So, we obtain S
×
∗,◊ T = T×
∗,◊ S, but S≠
T,if α ≠γ
orβ ≠ δ
.Theorem 4.3: The generalized Cartesian product of the interval-valued-intuitionistic-fuzzy -n-normed linear spaces is distributive with respect to union and intersections. In other words if
S = {(X, (
A
1(x
1,
x
2,...,
x
n, t);B
1(x
1,
x
2,...,
x
n, t)); (x
1,
x
2,...,
x
n)∈
Xn}and T = {(Y,
A
2(y
1,
y
2,...,
y
n, t);B
2(y
1,
y
2,...,
y
n, t)); (y
1,
y
2,...,
y
n)∈
Yn}and U= {(Y,
A
3 (y
1,
y
2,...,
y
n, t);B
3(y
1,
y
2,...,
y
n, t)); (y
1,
y
2,...,
y
n)∈
Yn}are the interval-valued-intuitionistic-fuzzy n-normed linear spaces, then S
×
∗,◊ (T
U) = (S×
∗,◊ T)
(S×
∗,◊ U)and S
×
∗,◊ (T
U) =(S×
∗,◊ T)
(S×
∗,◊ U)Proof: We have, S
×
∗,◊ (T
U) ={(X
×
Y,A
1(x
1,
x
2,...,
x
n, t)∗
mini{A
2 (y
1,
y
2,...,
y
n, t),A
3(y
1,
y
2,...,
y
n,t)}◊
)
,
,...,
,
(
1 21
