International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 192
ISSN 2229-5518
Topological Structures on Intuitionistic
Fuzzy Multisets
Shinoj T K, Sunil Jacob John
Abstract – In this paper, we introduced the concept of Intuitionistic Fuzzy Multiset Topology. Mapping functions are defined to connect Intuitionistic Fuzzy Multisets defined on different sets. Subspaces and Continuous functions are discussed to study the topological structures of Intuitionistic Fuzzy Multisets and their various properties are discussed.
Keywords-- Intuitionistic Fuzzy Multiset, Intuitionistic Fuzzy Multiset Topology, Continuous functions, Subspaces.
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introduced algebraic structures on Intuitionistic Fuzzy
HE theory of sets, one of the most powerful tools in modern mathematics is usually considered to have begun with Georg Cantor (1845-1918). Considering the uncertainty factor, Lotfi A. Zadeh [1] introduced Fuzzy sets in 1965, in which a membership function assigns to each element of the universe of discourse, a number from the unit interval [0,1] to indicate the degree of belongingness to
the set under consideration.
If repeated occurrences of any object are allowed in a set, then the mathematical structure is called as multiset [11,12]. As a generalization of multiset, Yager [2] introduced fuzzy multisets and suggested possible applications to relational databases. An element of a Fuzzy Multiset can occur more than once with possibly the same or different membership values.
In 1983, Atanassov [3,10] introduced the concept of
Intuitionistic Fuzzy sets. An Intuitionistic Fuzzy set is characterized by two functions expressing the degree of membership and the degree of nonmembership of elements of the universe to the Intuitionistic Fuzzy set. Among the various notions of higher-order Fuzzy sets, Intuitionistic Fuzzy sets proposed by Atanassov provide a flexible framework to explain uncertainty and vagueness.
The concept of Intuitionistic Fuzzy Multiset is introduced in [4] by combining the all the above concepts. Intuitionistic Fuzzy Multiset has applications in medical diagnosis and robotics [13,14].
In [5] Shinoj T K, Anagha Baby and Sunil Jacob John
————————————————
• Shinoj T K. E-mail: shinojthrissur@gmail.com. Department of
Mathematics, National Institute of Technology,Calicut-673 601, Kerala,
India.
• Sunil Jacob John: sunil@nitc.ac.in. Department of Mathematics, National
Institute of Technology,Calicut-673 601, Kerala, India.
Multiset.
General topology was the first field of pure mathematics
where the concepts and ideas of fuzzy sets took strong roots. In 1968, Chang [9] introduced Fuzzy topological spaces. And as a continuation of this, in 1997, Dogan Coker [6] introduced the concept of Intuitionistic fuzzy topological spaces. In our work, we generalized this concept into Intuitionistic Fuzzy Multiset. First we discuss the basic operations and in the subsequent sections we introduce the concept of functions on Intuitionistic Fuzzy Multiset and followed by this the topological structures and its various properties are discussed.
2.1 Definition [1] Let X be a nonempty set. A Fuzzy set A
drawn from X is defined as A = {< x : µA (x) > : x ϵ X}.Where :
X →[0,1] is the membership function of the Fuzzy Set A.
(FMS) A drawn from X is characterized by a function,
‘count membership’ of A denoted by CMA such that CMA : X → Q where Q is the set of all crisp multisets drawn from the unit interval [0,1]. Then for any x ∈ X, the value CMA (x) is a crisp multiset drawn from [0,1]. For each x ∈X, the
membership sequence is defined as the decreasingly
ordered sequence of elements in CMA (x). It is denoted by
(µ1 A (x), µ2 A (x),...,µP A (x)) where µ1 A (x)> µ2 A (x) >,..., > µP A (x).
A complete account of the applications of Fuzzy Multisets in various fields can be seen in [9].
Fuzzy Set (IFS) A is an object having the form A = {< x:
µA (x), vA (x) >: x ϵ X}, where the functions µA : X→ [0,1] and vA : X→[0,1] define respectively the degree of membership
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and the degree of non membership of the element x∈X to the set A with 0 < µA (x) + vA (x) < 1 for each x ϵ X.
2.4 Remark Every Fuzzy set A on a nonempty set X is obviously an IFS having the form
A = {< x : µA (x), 1 - µA (x) > : x ϵ X}
∇A = {< x : (v1 A (x),...,vP A (x)), (µ1 A (x),……. ...,µP A (x))
> : x ϵ X}
3. Union (A ∪ B)
In A ∪ B the membership and non membership
values are obtained as follows.
µ A∪B (x) = µ A (x) ∨ µ B (x)
j j j
Using the definition of FMS and IFS, a new generalized concept can be defined as follows:
2.5 Definition [4] Let X be a nonempty set. An Intuitionistic Fuzzy Multiset A denoted by IFMS drawn from X is characterized by two functions : ‘count membership’ of A (CMA ) and ‘count non membership’ of A (CNA ) given
respectively by CMA : X→Q and CNA : X→ Q where Q is
the set of all crisp multisets drawn from the unit interval [0,
1] such that for each x ϵ X, the membership sequence is defined as a decreasingly ordered sequence of elements in CMA(x) which is denoted by (µ1 A (x), µ2 A (x),...,µP A (x)) where (µ1 A (x) > µ2 A (x) >,... >,µP A (x) and the corresponding non membership sequence will be denoted by (v1 A (x), v2 A (x),...,vP A (x)) such that 0 < µiA (x) + vi A (x) < 1 for every x ϵ X and i = 1,2,...,p.
An IFMS A is denoted by
A = {< x : (µ1 A (x), µ2 A (x),...,µP A (x)), (v1 A (x), v2 A (x), ... ,vP A (x))
> : x ϵ X}
2.7 Definition [4] For any two IFMSs A and B drawn from a set X, the following operations and relations will hold. Let A = {< x : (µ1 A (x), µ2 A (x),...,µP A (x)), (v1 A (x), v2 A (x),...,vP A (x)) >
: x ϵ X} and
B = {< x : (µ1 B (x), µ2 B (x),...,µP B (x)), (v1 B (x), v2 B (x),...,vP B (x)) > :
x ϵ X} then
1. Inclusion
A ⊂ B ⇔ µj A (x) < µj A (x) and vj A (x) > vj B (x);
vj A∪B (x) = vj A (x)∧vj B (x)
j = 1, 2,...,L(x), x ∈ X.
4. Intersection (A ∩ B)
In A∩B the membership and non membership
values are obtained as follows.
µj A∩B (x) = µj A (x)∧ µj B (x)
vj A∩B (x) = vj A (x) ∨ vj B (x)
j = 1, 2,...,L(x), x ∈ X.
5. Addition (A ⊕ B)
In A⊕B the membership and non membership
values are obtained as follows.
µj A⊕B (x) = µj A (x) + µj B (x) - µj A (x). µj B (x)
vj A⊕B (x) = vj A (x). vj B (x)
j = 1, 2,...,L(x), x ϵ X.
6. Multiplication (A ⊗ B)
In A⊗B the membership and non membership
values are obtained as follows.
µj A⊗B (x) = µj A (x). µj B (x)
vj A⊗B (x) = vj A (x) + vj B (x) - vj A (x). vj B (x)
j = 1, 2,...,L(x), x ϵ X.
here ∨, ∧ , . , +, - denotes maximum, minimum, multiplication, addition, subtraction of real numbers respectively.
2.8 Definition [5] Let X and Y be two nonempty sets and f : X →Y be a mapping. Then
a) The image of the FMS A ∈ FM(X) under the
mapping f is denoted by f(A) or
f[A], where
j = 1,2,..., L(x), x ϵ X
CMf [A] (y) = �˅f(x)=y CMA(x) ; f
−1 (y) ≠ ∅
A = B ⇔ A ⊂ B and B ⊂ A
2. Complement
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0 otherwise
b) The inverse image of the FMS B ∈ FM(Y) under
the mapping f is denoted by
f -1(B) or f -1[B], where 𝐶𝑀𝑓−1 [𝐵] (𝑥) = 𝐶𝑀𝐵 𝑓[𝑥].
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2.9 Definition [6] Let X and Y be two nonempty sets and f :
X →Y be a mapping. Then
a) If B = {< y, µB (y), vB (y) >: y ϵ Y} is an IFS in Y, then
the preimage of B under f, denoted by f −1 (B), is
the IFS in X defined by
d) f −1 [∪i∈I Bi ] = ∪i∈I f −1 [Bi]
e) f −1 [∩i∈I Bi ] = ⋂i∈I f −1 [Bi]
f) g[f(Ai)] = [gf](Ai ) and f −1 �g−1 �Bj �� = [gf]−1(Bj )
Proof:
a) Let A1 ⊆ A2 .
CMA1 (x) ≤ CMA2 (x) ∀ x ∈ X
f −1 (B) = {< x, f −1 �µ )(𝑥), f −1 (𝑣𝐵 )(𝑥)� >: 𝑥ϵ X}
Then
⇒ ⋁f(x)=y CMA1 (x)
≤ ⋁f(x)=y CMA2 (x)
b) If A = {< x, λA (x), γA (x) >: x ϵ X} is an IFS in X, then the image of A under f, denoted by f(A), is the IFS in Y defined by,
f(A) = {< 𝑦, 𝑓(𝜆𝐴 )(𝑦), f−1 (𝛾𝐴 )(𝑦)) >: yϵY} where
supxϵf−1 (y) λA(x) ; f −1 (y) ≠ ∅
f(λA )(y) = � 0 otherwise
−1
⇒ CMf(A1 ) (y) ≤ CMf(A2) (y) ∀ y = f(x)
………………….(1)
Also CNA1 (x) ≥ CNA2 (x) ∀ x ∈ X
⇒ ⋀𝑓 (𝑥)=𝑦 C𝑁𝐴1 (x) ≥ ⋀𝑓 (𝑥)=𝑦 C𝑁𝐴2 (x)
⇒ CNf(A1) (y) ≥ CNf(A2 ) (y) ∀ y = f(x)
………………….(2)
(1) and (2) ⇒ f (A1 ) ⊆ f (A2 )
infxϵf−1 (y) γA (x) ; f
(y) ≠ ∅
f−1 (γA )(y) = � 0 otherwise
b) Let B1
⊆ B2
⇒ CMB1 �f(x)� ≤ CMB2 (f(x) and CNB1 �f(x)� ≥ CNB2 (f(x))
In the next section we amalgamate the above two
definitions of functions on FMS and IFS to define the
⇒ CMf−1
(B1 )
(x) ≤ CMf−1
(B2 )
(x) and CNf−1
(B1 )
(x) ≥
functions on IFMS.
CNf−1 (B2 ) (x) by 3.1(b)
⇒ f −1 (B1 ) ⊆ f −1 (B2).
c) Let A = Ai
So CMf(⋃ A ) (y) = CMf(A) (y)
3.1 Definition Let X and Y be two nonempty sets and f : X →Y be a mapping. Then
a) The image of an IFMS A in X under the mapping f is denoted by f(A) is defined as
−1
i∈I i
= ⋁f(x)=y CMA (x)
= ⋁f(x)=y{⋁i∈I CMAi (x)}
= ⋁i∈I{⋁f(x)=y CMAi (x)}
= ⋁i∈I CMf(Ai) (y)
CM⋃ [f(A )] (y)
CMf [A] (y) = �˅f(x)=y CMA(x) ; f
(y) ≠ ∅
= i∈I i
0 otherwise
−1
……………….(1)
CNf [A] (y) = �˄f(x)=y CNA (x) ; f
(y) ≠ ∅
And CN
(y) = CN
(y)
1 otherwise
b) The inverse image of the IFMS B in Y under the
mapping f is denoted by 𝑓−1 (𝐵) where
𝐶𝑀𝑓−1 [𝐵] (𝑥) = 𝐶𝑀𝐵 𝑓[𝑥], 𝐶𝑁𝑓−1 [𝐵] (𝑥) = 𝐶𝑁𝐵 𝑓[𝑥]
f(⋃i∈I Ai)
f(A)
= ⋀𝑓 (𝑥)=𝑦 C𝑁𝐴 (x)
= ⋀f(x)=y{⋀i∈ICNAi (x)}
= ⋀i∈I {⋀f(x)=yCNAi (x)}
In the next proposition we discuss some of the properties of functions which we have proved for FMS in [5].
and f : X → Y and g : Y → Z be two mappings. If
A, Ai ∈ IFMS(X), B, Bi ∈IFMS(Y), C ∈ IFMS(Z) ; i∈
I then
a) A1 ⊆ A2 ⇒ f (A1 ) ⊆ f (A2 )
b) B1 ⊆ B2 ⇒ f −1 (B1 ) ⊆ f −1 (B2 )
c) f[∪i∈I Ai ] = ∪i∈I f[Ai ]
= ⋀i∈ICNf(Ai) (x)}
= CN⋃i∈I[f(Ai )] (y)
……………….(2)
From (1) and (2) f[∪i∈I Ai ] = ∪i∈I f[Ai ]
d) Let ⋃i∈I Bi = B
CMf−1(⋃i∈I Bi ) (x) = CMf−1 (B) (x)
= CMB f(x)
= ⋁i∈I CMBi f(x)
= ⋁i∈I CMf−1 [Bi] (x)
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= CM⋃i∈I f−1 (Bi) (x)
…………………(1)
And CNf−1(⋃i∈I Bi ) (x) = CNf−1 (B) (x)
= CNB f(x)
= ⋀i∈I CNBi f(x)
= ⋀i∈I CNf−1(Bi ) x
= CN⋃i∈I f−1 (Bi ) (x)
…………………(2)
So f −1 [∪i∈I Bi ] = ∪i∈I f −1 [Bi ].
e) Let ⋂i∈I Bi = B
CMf−1 (⋂i∈I Bi ) (x) = CMf−1 (B) (x)
= CMB f(x)
= ⋀i∈I CMBi f(x)
= ⋀i∈I CMf−1 (Bi) (x)
= CM⋂i∈I f−1 (Bi ) (x)
………………….(1)
And CNf−1 (⋂i∈I Bi) (x) = CNf−1 (B) (x)
= CNB f(x)
= ⋁i∈I CNBi f(x)
= ⋁i∈ICNf−1 Bi x
= CN⋂i∈I f−1 (Bi) (x)
………………….(2)
Using the first part and 3.1(b) we can prove the second part.
In this section we introduced the concept of Intuitionistic Fuzzy multiset Topology (IFMT). Here we extend the concept of Intuitionistic fuzzy topological spaces introduced by Dogan Coker in [6] to the case of
Intuitionistic fuzzy multisets.
For this first we introduced ⇁0 and ⇁1 in a nonempty set X as follows.
Let ⇁0 = {< x: (0,0,……..,0), (1,1,………,1): x ϵ X }
⇁1 = {< x: (1,1,…….1), (0,0,……….,0) : x ϵ X }
(IFMT) on X is a family ґ of intuitionistic fuzzy multisets
(IFMSs) such that
1. ⇁0, ⇁1 ϵ ґ
1 ∩G2 ϵ ґ for any G1 , G 2 ϵ ґ
So f −1 [∩i∈I Bi ] = ⋂i∈I f −1[Bi ]
f) Let A ∈ IFMS(X) and z ∈ Z
Then CMg[f(A)] (z) = ⋁g(y)=z CMf(A) (y) ; y ∈ Y
= ⋁g(y)=z {⋁f(x)=y CMA (x)} ; y ∈ Y and x ∈
X
= ⋁{⋁{ CMA (x) ; x ∈ X and f(x) = y } ; y ∈
Y and g(y) = z}
= ⋁{ CMA(x) ; x ∈ X and [gf](x) = z }
= ⋁[gf](x)=z CMA (x) ; x ∈ X
= CM[gf](A) (z).
………………….(1)
And CNg[f(A)] (z) = ⋀g(y)=z CNf(A) (y) ; y ∈ Y
= ⋀g(y)=z {⋀f(x)=yCNA (x)} ; y ∈ Y and x ∈ X
= ⋀{⋀{ CNA(x) ; x ∈ X and f(x) = y } ; y ∈
Y and g(y) = z}
= ⋀{CNA (x) ; x ∈ X and [gf](x) = z}
= ⋀[gf](x)=z CN(x) ; x ∈ X
= CN[gf](A) (z).
……………….(2)
(1) and (2) ⇒ g[f(A)] = [gf](A).
2. G
3. UG i ϵ ґ for any arbitrary family {G i : i ϵ I} in ґ
Then the pair (X, ґ) is called Intuitionistic Fuzzy multiset topological space (IFMT for short) and any IFMS in ґ is known as an open intuitionistic fuzzy multiset (OIFMS in short) in X.
For n ϵ N+ , p ϵ N
Gn = {<1: (n/n+1, n+1/n+2,…, n+p/n+p+1), (1/n+2,
1/n+3,…,1/n+p+2)>,
<2: (n+1/n+2, n+2/n+3,…,n+p+1/n+p+2), (1/n+3,
1/n+4,…,1/n+p+3)> }
Let ґ = {⇁0, ⇁1}U{Gn }
Then (X, ґ) forms an IFMT. Note that here membership
values forms a monotonically increasing sequence and
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ISSN 2229-5518
nonmembership values forms a monotonically decreasing sequence.
Here we construct Intuitionistic fuzzy multiset topology from a given IFMT. Dogan Coker in [6] has constructed these topologies for IFS.
Consider a nonempty set X. Let A = {< x : (µ1 A (x),
µ2 A (x),...,µP A (x)), (v1 A (x), v2 A (x),...,vP A (x)) > : x ϵ X} be an
IFMS. Define
[]A = {< x : (µ1 A (x), µ2 A (x),...,µP A (x)),(1-µ1 A (x),1-µ2 A (x),...,1-
µPA (x)) > : x ϵ X}
4.5 Proposition Let (X,ᴦ) be an IFMT on X. Then ᴦ0,1 = {[]A: A ϵ ᴦ} is an IFMS.
Proof
It is obvious that ⇁0, ⇁1ϵ ᴦ0,1.
Let A1 , A1 ϵ ᴦ0,1.
Then A1 = {[]A1 ′ ; A1 ′ ϵ ᴦ} and A2 = {[]A2 ′ ; A2 ′ ϵ ᴦ}
⇒ A1 = {< x : (µ1 A1′ (x), µ2 A1′ (x),..., µp A1′ (x)),(1- µ1 A1′ (x),1-
µ2 A1′ (x),...,1- µp A1′ (x)) > : x ϵ X} and
A2 = {< x : (µ1 A2′ (x), µ2 A2′ (x),..., µp A2′ (x)),(1- µ1 A2′ (x),1-
µ2 A2′ (x),...,1- µp A2′ (x)) > : x ϵ X}
Then A1 ∩ A2 = {< x : (µ1 A1′ (x) ∧ µ1 A2′ (x),..., µpA1′ (x) ∧
µpA2′ (x)),((1- µ1 A1′ (x)) ∨ (1- µ1 A2′ (x),...,1- µp A1′ (x) ∨ (1- µp A2′ (x))
> : x ϵ X}
[](A1 ′ ∩ A2 ′) = []{< x : (µ1 A1′ (x) ∧ µ1 A2′ (x),..., µpA1′ (x) ∧ µp A2′ (x)),(
v1 A1′ (x) ∨ v1 A2′ (x),..., vp A1′ (x) ∨ vp A2′ (x)) > : x ϵ X}
Since 1- {µi A1′ (x) ∧ µiA2′ (x)} = {1- µi A1′ (x)} ∨ {1- µiA2′ (x)} A1 ∩ A2 = [](A1 ′ ∩ A2 ′) ϵ ᴦ0,1
Now If Ai ϵ ᴦ0,1 ; i ϵ I Then Ai = {[]Ai ′ ; Ai ′ ϵ ᴦ}
UA i = U([]Ai′) = [](UAi′) ϵ ᴦ0,1.
Hence the proof.
Then closure of A denoted by cl(A) is defined as cl(A) =
∩{M: M is closed in X and A ⊆ M}.
Then interior of B is denoted by
int(B) is defined as int(B) = U{N: N is open in X and N ⊆ B}.
4.8 Definition Let (X, ґ1 ) and (X, ґ2 ) be two IFMTs on X.
Then ґ1 is coarser (weaker) than ґ2 if Aϵґ2 for each Aϵґ1. It is
denoted as ґ1⊆ ґ2.
4.9 Proposition Let {ґi: I ϵ I} be a family of IFMTs on X.
Then ∩ґi is an IFMT on X and it is the coarsest IFMT on X
containing all the ґi’s.
Proof follows from the definitions 4.2 and 4.8.
X. Then cl(A) is a CIFMS. Proof:
cl(A) = ∩{M: M is closed in X and A ⊆ M}
⇒ ∇cl(A) is the union of all open sets and hence it is open.
⇒ cl(A) is a CIFMS.
X. Then int(A) is an OIFMS. Proof:
Since int(A) is the union of all OIFMS which is contained in
A, the proof follows from definition 4.2.3.
From the above two propositions and by the definitions it is clear that cl(A) is the smallest CIFMS which contains A and int(A) is the largest open set which is contained in A.
Then cl(∇A) = ∇(int(A))
Proof:
Let A = {< x : (µ1 A (x),µ2 A (x),...,
µpA (x)),((v1 A (x),v2 A (x),...,vP A (x)) >) > : x ϵ X}
Let Ai = {< x : (µ1 Ai (x), µ2 Ai (x),...,
µpAi (x)),((v1 Ai(x),v2 Ai(x),...,vP Ai(x)) >) > : x ϵ X, i ϵ I} be the family of OIFMSs which is contained in A.
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Then µj Ai ≤ µj A and vj Ai ≥ vj A ; i ϵ I, j = 1,2,…,p
…………(1)
And
int(A) = {<
Since A is a CIFMS, cl(A) ⊆ A
………..(2)
From (1) and (2) cl(A) = A.
𝑥: (⋁µ1
(x), … . , ⋁µp
(x)), (⋀v1A (x), … . , ⋀vp A (x)) >: 𝑥ϵ X}
Now assume the converse. Hence by proposition 4.10, A is
Ai Ai i i
⇒
a CIFMS.
∇(int(A)) = {<
𝑥: �⋀v1 A (x), … . , ⋀vp A (x)� , (⋁µ1
(x), … . , ⋁µp
(x)) >
Hence the proof.
i i Ai Ai
: 𝑥ϵ X} ……(2)
Now ∇A = {< x : ((v1 A (x),v2 A (x),...,vP A (x)), (µ1 A (x),µ2 A (x),...,
µpA (x)) > : x ϵ X}
Also from (1) it is clear that {∇Ai; i ϵ I} is the family of
CIFMTs containing ∇A.
⇒
cl(∇A) = {<
In the same way we can prove the next proposition.
X. Then A is an OIFMS if and only if int(A) = A.
function f : X →Y is said to be Continuous if and only if
inverse image of each OIFMS in ф is an OIFMS in ᴦ.
𝑥: �⋀v1 A (x), … . , ⋀vp A (x)� , (⋁µ1
(x), … . , ⋁µp
(x)) >
i i Ai Ai
: 𝑥ϵ X} …….(3)
From (2) and (3) cl(∇A) = ∇(int(A)).
Then int(∇A) = ∇(cl(A))
Proof is similar to proposition 4.12.
The following properties can be easily derived from the definitions.
OIFMSs. Then the following properties hold.
I. int(A) ⊆ A II. A ⊆ cl(A)
III. A ⊆ B ⇒ int(A) ⊆ int(B)
IV. A ⊆ B ⇒ cl(A) ⊆ cl(B) V. int(⇁1) = ⇁1
VI. cl(⇁0) = ⇁0
X. Then A is a CIFMS if and only if cl(A) = A.
Next we discuss some of the equivalence relations of
Continuous functions.
Proof:
Assume that f is continuous and C be a CIFMS in ф.
To prove f-1(C) is closed, it is enough to show that ⇁ f-1(C) is an OIFMS in ᴦ.
Now the inverse image of the IFMS C in Y under the
mapping f is denoted by 𝑓−1 (𝐶) where
𝐶𝑀𝑓−1 [𝐶] (𝑥) = 𝐶𝑀𝐶 𝑓[𝑥], 𝐶𝑁𝑓−1 [𝐶] (𝑥) = 𝐶𝑁𝐶 𝑓[𝑥]
⇒ �f −1 (C) = {𝑥: < 𝐶𝑀𝐶 𝑓[𝑥], 𝐶𝑁𝐶 𝑓[𝑥] >}
⇒ � ⇁ f −1 (C) = {𝑥: < 𝐶𝑁𝐶 𝑓[𝑥], 𝐶𝑀𝐶 𝑓[𝑥] >}
…………..(1)
Proof:
Assume that A is a CIFMS. From 4.14.2, A ⊆ cl(A)
………..(1)
Now f −1 (⇁ C) = f −1 {𝑦: < 𝐶𝑁𝐶 (𝑦), 𝐶𝑀𝐶 (𝑦) >}
⇒ �f −1 (⇁ C) = {𝑥: < 𝐶𝑁𝐶 𝑓[𝑥], 𝐶𝑀𝐶 𝑓[𝑥] >}
…………..(2)
Where y = f(x) From (1) and (2)
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⇁ f −1 (C) = f −1 (⇁ C)
…………..(3)
But by proposition 4.14.2, f-1(C) ⊆ cl(f-1(C))
⇒ f-1(C) = cl(f-1(C))
Since f is continuous and ⇁ C is open, definition 4.15
f −1 (⇁ C) is open and hence f-1(C) is closed.
Now assume that inverse image of each CIFMS in ф is a
CIFMS in ᴦ.
To prove f is continuous, it is enough to prove f-1(O) is open
Hence the proof.
⇒ f-1(C) is closed, By proposition 4.10.
⇒ f is continuous.
for every OIFMS O in ф.
O is an OIFMS ⇒ ⇁O is a CIFMS.
⇒ f-1(⇁O) is a CIFMS. (By assumption)
Since (3) is true for any OIFMS, ⇁f-1(O) is a CIFMS and
hence f-1(O) is an OIFMS.
Thus f is continuous. Hence the proof.
Proof:
Assume f is continuous.
For any IFMT A in X, f(A) ⊆ cl(f(A)) by proposition 4.14.2
⇒ A = f-1(f(A)) ⊆ f-1(cl[f(A)]) by proposition 3.2.b
Since f is continuous and cl[f(A)] is closed, f-1(cl[f(A)]) is closed.
⇒ cl(A) ⊆ f-1(cl[f(A)]), since cl(A)
is the smallest CIFMS contains A.
⇒ f(cl(A)) ⊆ cl[f(A)]. Conversely assume the given condition.
To prove f is continuous, let C be a CIFMS in ф.
Then by assumption, f(cl(f-1(C))) ⊆ cl[f(f-1(C))] = cl(C) = C, By proposition 4.15.
Thus cl(f-1(C)) ⊆ f-1(C)
function f: X →Y is Continuous if and only if for each IFMT B in Y, cl[f-1(B)] ⊆ f-1[cl(B)]
Proof:
Replace A by f-1(B) in theorem 4.19.
Proof:
Assume that f is continuous.
For any IFMT A in X, int[f(A)] ⊆ f(A) by proposition 4.14.1
⇒ f-1[int(f(A))] ⊆ A by proposition 2.2.b
Since f is continuous and int[f(A)] is open, f-1[int(f(A))] is open. But int(A) is the largest OIFMS
contained in A.
⇒ f-1[int(f(A))] ⊆ int(A)
⇒ int[f(A)] ⊆ f[int(A)].
Conversely assume the given condition.
To prove f is continuous, let O be an OIFMS in ф.
Then by assumption, int[f(f-1(O))] ⊆ f[int(f-1(O))]
⇒ int(O) ⊆ f[int(f-1(O))]
⇒ O ⊆ f[int(f-1(O))] by proposition 4.16
⇒ f-1(O) ⊆ int(f-1(O))
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But by 4.14.1, int(f-1(O)) ⊆ f-1(O)
⇒ f-1(O) = int(f-1(O))
⇒ f-1(O) is an OIFMS, By proposition
4.11.
⇒ f is continuous. Hence the proof.
4.22 Theorem Let (X, ґ) and (Y, ф) be two IFMTs. Then the function f: X →Y is Continuous if and only if for each IFMT B in Y, f-1[int(B)] ⊆ int[f-1(B)]
Proof:
Replace A by f-1(B) in theorem 4.21.
4.23 Definition Let (X, ґ) and (Y, ф) be two IFMTs. The topological space Y is called a subspace of the topological space X if Y⊆ X and if the open subsets of Y are precisely the subsets O′ of the form
O′ = O ∩ Y
for some open subsets O of X. Here we may say that each open subset O′ of Y is the restriction to Y of an open subset O of X. O′ is also called relative open in Y.
For n ϵ N+, p ϵ N
Gn = {<1: (n/n+1, n+1/n+2,…, n+p-1/n+p), (1/n+2,
1/n+3,…,1/n+p+1)>,
<2: (n+1/n+2, n+2/n+3,…,n+p+1/n+p+1), (1/n+3,
1/n+4,…,1/n+p+2)>}
Let ґ′ = {⇁0, ⇁1}U{Gn }
Then (X, ґ′) is a subspace of (X, ґ) example 4.4.
where O is an OIFMS in (X, ґ). Then (Y U ⇁1, ф) is an IFMT
and a subspace of (X, ґ)
Proof:
We have ⇁0 = ⇁0 ∩ Y.
Suppose O′ 1 , O′ 2 ,….., O′ n ϵ ф
Then O′ i = O i ∩ Y for some O i ϵ ґ
Then O′ 1 ∩ O′ 2 ∩…..∩ O′ n = (O1 ∩ O2 ∩ ……. ∩ On ) ∩ Y ϵ ф
Since (O1 ∩ O2 ∩ ……. ∩ On ) ϵ ґ
Finally, suppose that for each α ϵ I, O′α ϵ ф.
Thus for each α ϵ I, O′α = Oα ∩ Y for some Oα ϵ ґ.
Then UO′α = U(Oα ∩ Y) = UOα ∩ Y ϵ ф since UOα ϵ ґ
Hence (Y U ⇁1, ф) is an IFMT and therefore a subspace of
(X, ґ).
subset C′ of Y is relatively closed in Y if and only if
C′ = C ∩ Y
for some CIFMS C of X. Proof:
First assume that C′ is relatively closed. Then ⇁ Y (C′) is relatively open, where ⇁ Y denotes the complement w.r.t
Y.
Thus ⇁ Y (C′) = O ∩ Y for some OIFMS O in X.
⇒ C′ = ⇁ Y (O ∩ Y) = ⇁ X (O) ∩ Y
where ⇁ X (O) is a CIFMS in X.
Conversely suppose that C′ = C ∩ Y for some CIFMS C of X.
Then ⇁ Y (C′) = ⇁ X (C) ∩ Y.
4.25 Theorem Let (X, ґ) be an IFMT and let Y be a subset of X. Define the collection ф of subsets of Y as the collection of subsets O′ of Y of the form
O′ = O ∩ Y
Hence ⇁ Y (C′) is relatively open in Y and therefore C′ is
relatively closed in Y.
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ISSN 2229-5518
correspondence i(x) = x for each x ϵ A is called an inclusion
mapping or function.
Then the inclusion mapping i : Y →X is continuous. Proof:
For each IFMS A of X, i-1(A) = A ∩ Y.
Thus if O is an OIFMS in X, then i-1(O) = O ∩ Y is a relatively
open subset of Y. Hence i is continuous.
In this work we studied the topological structures of Intuitionistic Fuzzy Multisets. We introduced the concept of Intuitionistic Fuzzy Multiset Topology. We defined the concept of functions between Intuitionistic Fuzzy Multisets. Open and closed sets are defined. The Closure, Interior, Subspace topology and Continuous functions are defined and their various properties are discussed. The foundations which we made through this paper can be extended and studied to get an insight into the topological structures Intuitionistic Fuzzy Multisets.
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