Download ON SOME APPLICATIONS OF INTUITIONISTIC FUZZY π

Int Jr. of Mathematics Sciences & Applications
Vol. 2, No. 2, May 2012
Copyright  Mind Reader Publications
ISSN No: 2230-9888
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ON SOME APPLICATIONS OF INTUITIONISTIC FUZZY
π-β
β-LOCALLY CLOSED SETS
S.Padmapriya, M.K.Uma and E.Roja
Department of Mathematics,
Sri Sarada College for Women,
Salem, Tamil Nadu, India.
Abstract
The purpose of this paper is to study the new classes of an intuitionistic fuzzy π-locally closed
sets, intuitionistic fuzzy π-β-locally closed sets which are the weaker forms of the class of an
intuitionistic fuzzy locally closed sets. The concepts of an intuitionistic fuzzy π-β-local spaces,
intuitionistic fuzzy π-β -local border, intuitionistic fuzzy π-β-local frontier and intuitionistic
fuzzy π-β-local exterior are introduced and interesting properties are established. In this
connection, interrelations are discussed. Examples are provided wherever necessary.
Keywords: Intuitionistic fuzzy π-β-locally closed set, intuitionistic fuzzy π-β-local spaces,
intuitionistic fuzzy π-β -local border, intuitionistic fuzzy π-β-local frontier and intuitionistic
fuzzy π-β-local exterior.
2000 Mathematics subject classification 54A40-03E72
1. Introduction
The concept of fuzzy sets was introduced by Zadeh [9] and later Atanassov [1] introduced and studied
an intuitionistic fuzzy sets. On the other hand, Coker [5] introduced the notions of an intuitionistic fuzzy
topological spaces, intuitionistic fuzzy continuity and some other related concepts. The concept of an
intuitionistic fuzzy α-closed set was introduced by BiljanaKrsteshka and ErdalEkici [2]. The first step of locally
closedness was done by Bourbaki [3]. Ganster and Relly used locally closed sets in [6] to define LC-continuity
and LC-irresoluteness. The concept of π-open set in a topological space was introduced by V.Zaitsev [10].
M.Caldas, S.Jafari and T.Noiri [4] were introduced the topological properties of g-border, g-frontier, g-exterior
of a set using the concept of g-open sets. In this paper, the concepts of an intuitionistic fuzzy π-β-locally closed
set, intuitionistic fuzzy π-β-local spaces, intuitionistic fuzzy π-β-local border, intuitionistic fuzzy π-β-local
frontier and intuitionistic fuzzy π-β-local exterior are introduced and studied. Some interesting properties and
interrelations among sets and spaces are discussed with necessary examples.
2. Preliminaries
Definition 2.1 [ 1 ] Let X be a nonempty fixed set and I the closed interval [ 0, 1 ]. An intuitionistic fuzzy set
(IFS) A is an object of the following formA = { 〈 x, µA ( x ), γA ( x ) 〉 : x ∈X }, where the mapping µ A : X → I
and γ A : X → I denote the degree of membership (namelyµA( x )) and the degree of nonmembership (namelyγA(
x )) for each elementx ∈X to the set A, respectively, and 0 ≤ µ A (X) + γ A (X) ≤ 1 for eachx ∈X. Obviously,
every fuzzy set A on a nonempty set X is an IFS of the following form, A = { 〈 x, µA ( x ), 1 −µA ( x ) 〉 : x ∈X }.
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S.Padmapriya, M.K.Uma and E.Roja
For the sake of simplicity, we shall use the symbol A = 〈 x, µA , γA〉for the intuitionistic fuzzy set A = { 〈 x, µA (
x ), γA ( x ) 〉 : x ∈X }.
Definition 2.2 [ 1 ]Let A and B be IFSs of the form A = { 〈 x, µA ( x ), γA ( x ) 〉 : x ∈X }and B
= { 〈 x, µB ( x ), γB ( x ) 〉 : x ∈ X }. Then
(i) A ⊆ B if and only if µ A (X) ≤ µ B (X) and γ A (X) ≥ γ B (X)
(ii) A = { 〈 x, γA ( x ), µA ( x ) 〉 : x ∈ X }
(iii) A I B = { 〈 x, µA ( x ) ∧µB ( x ), γA ( x ) ∨γB ( x )〉 : x ∈ X }
(iv) A U B = { 〈 x, µA ( x ) ∨µB ( x ), γA ( x ) ∧γB ( x )〉 : x ∈ X }
Definition 2.3[1]The IFSs 0~ and 1~ are defined by 0~= {〈x, 0, 1〉:x∈ X} and1~ = {〈x, 1, 0〉:x ∈ X}.
Definition 2.4 [ 5 ]An intuitionistic fuzzy topology (IFT) in Coker’s sense on a nonempty set X is a family τ of
IFSs in X satisfying the following axioms:
(i)
0~, 1~∈τ ;
(ii)
(iii)
G1 I G2 ∈τfor any G1, G2 ∈τ ;
U Gi∈τ for arbitrary family { Gi| i∈I }⊆τ.
In this paper by ( X, τ ) or simply by X we will denote the Coker’s intuitionistic fuzzy topological
space (IFTS). Each IFS in τ is called intuitionistic fuzzy open set (IFOS) in X. The complement A of an IFOS A
in X is called an intuitionistic fuzzy closed set (IFCS) in X.
Definition 2.5 [ 5 ]Let A be an IFS in IFTS X. Then
int A =
U
{ G | G is an IFOS in X and G ⊆ A } is called an intuitionistic fuzzy interior of A;
cl A = I { G | G is an IFCS in X and G ⊇ A } is called an intuitionistic fuzzy closure of A.
Definition 2.6 [ 2 ]Let A be an IFS of an IFTS X. Then A is called
(i)
an intuitionisticfuzzy preopen set (IFPOS) if A ⊆int ( cl ( A ) );
(ii) anintuitionistic fuzzy semiopen set (IFSOS) if A ⊆ cl (int ( A ) ) ;
(iii) an intuitionistic fuzzy α-open set (IFαOS) if A ⊆int ( cl (int ( A ) ) );
(iv) an intuitionistic fuzzy β-open set (IFβOS) ifA ⊆ cl ( int ( cl ( A ) ) );
(v)
an intuitionisticfuzzy regular open set (IFROS) if A = int ( cl ( A ) ).
Definition 2.7 [ 2 ]Let A be an IFS of an IFTS X. Then A is called
(i)
anintuitionistic fuzzy preclosed set (IFPCS) if A is an IFPOS;
(ii)
an intuitionistic fuzzy semiclosed set (IFSCS) if A is an IFSOS;
(iii)
an intuitionistic fuzzy α-closed set (IFαCS) if A is an IFαOS;
(iv)
an intuitionistic fuzzy β-closed set (IFβCS) if A is an IFβOS;
(v) anintuitionistic fuzzy regular closed set (IFRCS) if A is an IFROS.
Remark 2.1Every intuitionisticfuzzy regular open set is an intuitionisticfuzzy open set.
Definition 2.8 [ 8 ]A fuzzy topological space ( X, T ) is said to be fuzzy β-T1/2 space if every gfβ-closed set in (
X, T ) is fuzzy closed in ( X, T ).
Definition 2.9 [ 7 ]A subset A of a space ( X, τ ) is called locally closed (briefly lc) if A = C
I D, where C is
open and D is closed in ( X, τ ).
Definition 2.10 [ 7 ]A subset A of a space ( X, τ ) is called ~
g -locally closed (briefly -lc) if
~
~
A = C I D, where C is g -open and D is g -closed in ( X, τ ).
Definition 2.11 [ 7 ]A subset A of a topological space ( X, τ ) is called ~
g -lc* if A = U
I F, where U is
~
g-
open and F is closed in ( X, τ ).
Definition 2.12 [ 10 ]The finite union of regular open sets is said to be π-open.The complement of π-open is
said to be π-closed set.
Proposition 2.1 [ 5 ]For any IFS A in ( X, τ ) we have (a) cl ( A ) = int(A) , (b) int ( A ) = cl (A) .
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ON SOME APPLICATIONS OF INTUITIONISTIC FUZZY…
3. Intuitionistic Fuzzy π-β
β-Locally Closed Sets and Intuitionistic Fuzzy π-β
β-Local Spaces
In this section, the concepts of an intuitionistic fuzzy locally closed set, intuitionistic fuzzy π-locally
closed set and intuitionistic fuzzy π-β-locally closed set are introduced and studied. The concepts of an
intuitionistic fuzzy π-locally closed intuitionistic fuzzy subspace, intuitionistic fuzzy π-β-local T1/2 space,
intuitionistic fuzzy π-β-local S space, intuitionistic fuzzy π-β-local pre space, intuitionistic fuzzy π-β-local α
space, intuitionistic fuzzy π-β-local β space, intuitionistic fuzzy π-β-local border, intuitionistic fuzzy π-β-local
frontier and intuitionistic fuzzy π-β-local exterior are introduced and studied. In this connection, interrelations
among sets and spaces are discussed with suitable examples.
Definition 3.1Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space ( X, T ). Then A is said to be an intuitionistic fuzzy locally
closed set ( in short, IFlcs ) if A = C
I
D, where C = 〈 x, µC, γC 〉 is an intuitionistic fuzzy open set and D = 〈 x,
µD, γD 〉 is an intuitionistic fuzzy closed set in ( X, T ).The complement of an intuitionistic fuzzy locally closed
set is said to be an intuitionistic fuzzy locally open set (in short, IFlos).
Definition 3.2Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space X. Then A is said to be an intuitionistic fuzzy π- open set if
n
A =
U Ai, where Ai =
i =1
X, µ A
i
, γA
i
is an intuitionistic fuzzy regular open set in an intuitionistic fuzzy
topological space ( X, T ).The complement of an intuitionistic fuzzy π- open set is said to be an intuitionistic
fuzzy π- closed set.
Definition 3.3Let ( X, T ) be an intuitionistic fuzzy topological space. LetA = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space ( X, T ). Then A is said to be an intuitionistic fuzzy πlocally closed set ( in short, IFπ-lcs ) if A = C
I
D, where C = 〈 x, µC, γC 〉 is an intuitionistic fuzzy π-open set
and D = 〈 x, µD, γD 〉 is an intuitionistic fuzzy closed set in ( X, T ).The complement of an intuitionistic fuzzy πlocally closed set is said to be an intuitionistic fuzzy π-locally open set (in short, IFπ-los).
Definition 3.4Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space X. The intuitionistic fuzzy π-locally closure of A is
denoted and defined by
IFπ-lcl( A ) =
I { B : B = 〈 x, µB, γB 〉 is an intuitionistic fuzzy π-locally closed set in X and A ⊆ B}
Definition 3.5Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space X. The intuitionistic fuzzy π-locally interior of A is
denoted and defined by
IFπ-lint ( A ) =
U { B : B = 〈 x, µB, γB 〉 is an intuitionistic fuzzy π-locally open set in X and B ⊆ A}
Definition 3.6Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space X. Then A is said to be an intuitionistic fuzzy π-β-locally
closed set ( in short,IFπ-β-lcs ) if A = B
I
C, where B is an intuitionistic fuzzy π-open set and C is an
intuitionistic fuzzy β-closed set.The complement of an intuitionistic fuzzy π-β-locally closedset is said to be an
intuitionistic fuzzy π-β-locally open set (in short, IFπ-β-los ).
Definition 3.7Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space X. The intuitionistic fuzzy π-β-locally closure of A is
denoted and defined by
IFπ-β-lcl(A) =
I { B: B =〈 x, µB, γB 〉 is an intuitionistic fuzzy π-β-locally closed set in X and A ⊆ B}
Definition 3.8Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set on an intuitionistic fuzzy topological space X. The intuitionistic fuzzy π-β-locally interior of A is
denoted and defined by
IFπ-β-lint(A) =
U { B : B = 〈 x, µB, γB 〉 is an intuitionistic fuzzy π-β-locally open set in X and B ⊆ A}
Proposition 3.3Every intuitionistic fuzzy π-locally closed set is an intuitionistic fuzzy locally closed set.
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S.Padmapriya, M.K.Uma and E.Roja
Remark 3.2 The converse of the Proposition 3.3 need not be true as shown in Example 3.1.Example 3.1 Let X
a b   a b
 a b  a b 
,
 ,  ,  〉 and B =〈 x,  ,  ,  ,  〉 be
 0.1 0.2   0.4 0.5 
 0.2 0.3   0.3 0.4 
= {a , b} be a nonempty set. Let A = 〈 x, 
intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic fuzzy topology on X.
Hence, E = A
a b   a b
I B = 〈 x,  ,  ,  ,  〉 is an intuitionistic fuzzy locally closed set. Since A is not an
 0.1 0.2   0.4 0.5 
intuitionistic fuzzy regular open set, E is not an intuitionistic fuzzy π-locally closed set. Hence, intuitionistic
fuzzy locally closed set need not be an intuitionistic fuzzy π-locally closed set.
Proposition 3.4Every intuitionistic fuzzy π-locally closed set is an intuitionistic fuzzy π-β-locally closed set.
Remark 3.3 The converse of the Proposition 3.4 need not be true as shown in Example 3.2.
 a b   a b
 a b 
,
 ,  ,  〉 and B = 〈 x,  ,  ,
 0.2 0.4   0.4 0.5 
 0.2 0.4 
Example 3.2 Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b 
 ,  〉 be an intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic fuzzy
 0.3 0.5 
 a b  a b 
,
 ,  ,  〉 be
 0.2 0.3   0.4 0.6 
topology on X. Therefore, B is an intuitionistic fuzzy π-open set.Let C =〈 x, 
intuitionisitc fuzzy β-closed set.Hence, E = B
I
 a b  a b 
,
 ,  ,  〉 is an intuitionistic fuzzy π-β 0.2 0.3   0.4 0.6 
C = 〈 x, 
locally closed set.But, E is not an intuitionistic fuzzy π-locally closed set. Hence, intuitionistic fuzzy π-β
βlocally closed set need not be an intuitionistic fuzzy π-locally closed set.
Remark 3.4 Intuitionistic fuzzy π-β-locally closed set and intuitionistic fuzzy locally closed set are independent
as see Example 3.3 and Example 3.4.
a b   a b 
 a b 
,
 ,  ,  〉 and B = 〈 x,  ,  ,
 0.1 0.2   0.3 0.5 
 0.2 0.2 
Example 3.3 Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b
 ,  〉 be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic fuzzy
 0.2 0.5 
 a b  a b 
,
 ,  ,  〉be an
 0.1 0.1   0.3 0.6 
topology on X. Therefore, B is an intuitionistic fuzzy π-open set.Let C = 〈 x, 
intuitionisitc fuzzy β-closedset.Hence, E = B
I
 a b  a b 
,
 ,  ,  〉 is an intuitionistic fuzzy π-β 0.1 0.1   0.3 0.6 
C = 〈 x, 
locally closed set. But, E is not an intuitionistic fuzzy locally closed set. Hence, intuitionistic fuzzy π-β
β -locally
closed set need not be an intuitionistic fuzzy locally closed set.
 a b  a b
 a b 
,
 ,  ,  〉 and B = 〈 x,  ,  ,
 0.2 0.5   0.4 0.5 
 0.3 0.5 
Example 3.4Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b
 ,  〉 be an intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B }is an
 0.4 0.5 
intuitionistic fuzzy topology on X. Hence, E = A
 a b  a b
I B = 〈 x,  ,  ,  ,  〉 is an intuitionistic fuzzy
 0.2 0.5   0.4 0.5 
locally closed set. Since A is not an intuitionistic fuzzy regular open set, Thus E is not an intuitionistic fuzzy πβ-locally closed set. Hence, intuitionistic fuzzy locally closed set need not be an intuitionistic fuzzy π-β
βlocally closed set.
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ON SOME APPLICATIONS OF INTUITIONISTIC FUZZY…
Definition 3.7 Let ( X, T ) be an intuitionistic fuzzy topological space and( Y, S ) be an intuitionistic fuzzy
subset of X. Then TY = ( A/Y | A ∈ T ) is an intuitionistic fuzzy topology on Y and is called the induced or
relative intuitionistic fuzzy topology. The pair ( Y, TY ) is called an intuitionistic fuzzy subspace of ( X, T ) : (
Y, TY ) is called an intuitionistic fuzzy open / intuitionistic fuzzy closed / intuitionistic fuzzy π-β-locally closed
intuitionistic fuzzy subspace if the intuitionistic fuzzy characteristic function of ( Y, TY ) vizχY is intuitionistic
fuzzy open / intuitionistic fuzzy closed / intuitionistic fuzzy π-β-locally closed set respectively.
Proposition 3.5 Let ( X, T ) be an intuitionistic fuzzy topological space. Suppose Z ⊆ Y ⊆ X and ( Y, TY ) is an
intuitionistic fuzzy π-β-locally closed intuitionistic fuzzy subspace of an intuitionistic fuzzy topological space
(X, T). If ( Z, TZ ) is an intuitionistic fuzzy π-β-locally closed intuitionistic fuzzy subspace in an intuitionistic
fuzzy topological space ( X, T ) ⇔ ( Z, TZ ) is an intuitionistic fuzzy π-β-locally closed intuitionistic fuzzy
subspace in an intuitionistic fuzzy topological space ( Y, TY ).
Definition 3.8 An intuitionistic fuzzy topological space ( X, T ) is said to be an intuitionistic fuzzy π-β-local
T1/2 space if for every intuitionistic fuzzy π-β-locally closed set is an intuitionisitc fuzzy closed set in an
intuitionistic fuzzy topological space ( X, T ).
Definition 3.9 An intuitionistic fuzzy topological space ( X, T ) is said to be an intuitionistic fuzzy π-β-local S (
resp. pre, α, β ) space, if for every intuitionistic fuzzy π-β-locally closed set is an intuitionistic fuzzy semi (resp.
pre, α, β) closed set in an intuitionistic fuzzy topological space ( X, T ).
Proposition 3.6 Everyintuitionistic fuzzy π-β-local T1/2 space is an intuitionistic fuzzy π-β-local S space.
Remark 3.6 The converse of the Proposition 3.6 need not be true as shown in Example 3.5.
 a b a b
 a b 
,
 ,  ,  〉, B = 〈 x,  ,  ,
 0.4 0.5   0.1 0.3 
 0.3 0.5 
Example 3.5 Let X = {a , b} be a nonempty set. Let A =〈 x, 
 a b 
a b   a b 
 ,  〉 and C = 〈 x,  ,  ,  ,  〉 be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼,
 0.2 0.4 
 0.1 0.2   0.4 0.6 
A, B, C } is an intuitionistic fuzzy topology on X. Therefore, C is an intuitionistic fuzzy π-open set.Let D =〈 x,
 a b   a b 
a b   a b 
 ,  ,  ,  〉 be an intuitionistic fuzzy β-closed set.Hence, E = C I D = 〈 x,  ,  ,  ,  〉
 0.2 0.2   0.3 0.6 
 0.1 0.2   0.4 0.6 
is an intuitionistic fuzzy π-β-locally closed set. Now, E is an intuitionistic fuzzy semi closed set. Hence, ( X, T )
is an intuitionistic fuzzy π-β
β -local S space. But, E is not an intuitionistic fuzzy closed set. Thus,( X, T ) is not
an intuitionistic fuzzy π-β
β -local T1/2 space. Hence, intuitionistic fuzzy π-β
β -local S space need not be an
intuitionistic fuzzy π-β
β -local T1/2 space.
Proposition 3.7Everyintuitionistic fuzzy π-β-local T1/2 space is an intuitionistic fuzzy π-β-local pre space.
Remark 3.7The converse of the Proposition 3.7 need not be true as shown in Example 3.6.
 a b   a b 
,
 ,  ,  〉 and B =〈 x,
 0.3 0.2   0.5 0.6 
Example 3.6Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b  a b 
 ,  ,  ,  〉be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic
 0.4 0.5   0.5 0.5 
 a b  a b 
,
 ,  ,  〉 be
 0.3 0.1   0.5 0.7 
fuzzy topology on X. Therefore, B is an intuitionistic fuzzy π-open set. Let C = 〈 x, 
any intuitionistic fuzzy β-closed set. Hence, E = B
I
 a b  a b 
,
 ,  ,  〉 is an intuitionistic
 0.3 0.1   0.5 0.7 
C = C = 〈 x, 
fuzzy π-β-locally closed set. Now, E is an intuitionistic fuzzy preclosed set. Hence, ( X, T ) is an intuitionistic
fuzzy π-β
β -local pre space. But, E is not an intuitionistic fuzzy closed set. Thus,( X, T ) is not an intuitionistic
fuzzy π-β
β -local T1/2 space. Hence, intuitionistic fuzzy π-β
β -local pre space need not be an intuitionistic
fuzzy π-β
β -local T1/2 space.
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S.Padmapriya, M.K.Uma and E.Roja
Proposition 3.8 Everyintuitionistic fuzzy π-β-local T1/2 space is an intuitionistic fuzzy π-β-local α space.
Remark 3.8 The converse of the Proposition 3.8 need not be true as shown in Example 3.7.
a b   a b 
 a b
,
 ,  ,  〉, B = 〈 x,  ,  ,
 0.1 0.2   0.4 0.4 
 0.3 0.3 
Example 3.7 Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b 
 a b a b 
 ,  〉 and C = 〈 x,  ,  ,  ,  〉 be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼,
 0.2 0.4 
 0.4 0.3   0.1 0.2 
A, B, C } is an intuitionistic fuzzy topology on X. Therefore, C is an intuitionistic fuzzy π-open set.Let D = 〈 x,
 a b   a b
 a b   a b
 ,  ,  ,  〉 be an intuitionistic fuzzy β-closed set.Hence, E = C I D = 〈 x,  ,  ,  ,  〉
 0.2 0.2   0.3 0.3 
 0.2 0.2   0.3 0.3 
is an intuitionistic fuzzy π-β-locally closed set.Now, E is an intuitionistic fuzzy α-closed set. Hence, ( X, T ) is
an intuitionistic fuzzy π-β
β -local α space. But, E is not an intuitionistic fuzzy closed set. Thus,( X, T ) is not an
intuitionistic fuzzy π-β
β -local T1/2 space. Hence, intuitionistic fuzzy π-β
β -local α space need not be an
intuitionistic fuzzy π-β
β -local T1/2 space.
Proposition 3.9 Everyintuitionistic fuzzy π-β-local T1/2 space is an intuitionistic fuzzy π-β-local β space.
Remark 3.9 The converse of the Proposition 3.9 need not be true as shown in Example 3.8.
a b   a b 
 a b
,
 ,  ,  〉 and B = 〈 x,  ,  ,
 0.1 0.2   0.3 0.4 
 0.2 0.3 
Example 3.8 Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b
 ,  〉 be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic fuzzy
 0.2 0.3 
 a b  a b 
,
 ,  ,  〉 be an
 0.1 0.1   0.3 0.5 
topology on X. Therefore, B is an intuitionistic fuzzy π-open set.Let C = 〈 x, 
intuitionistic fuzzy β-closed set.Hence, E = B
I
 a b  a b 
,
 ,  ,  〉 is an intuitionistic fuzzy π-β 0.1 0.1   0.3 0.5 
C = 〈 x, 
locally closed set.Now, E is an intuitionistic fuzzy β-closed set. Hence, ( X, T ) is an intuitionistic fuzzy π-β
βlocal β space. But, E is not an intuitionistic fuzzy closed set. Thus,( X, T ) is not an intuitionistic fuzzy π-β
βlocal T1/2 space. Hence, intuitionistic fuzzy π-β
β -local β space need not be an intuitionistic fuzzy π-β
β -local
T1/2 space.
Proposition 3.10 Every intuitionistic fuzzy π-β-local pre space is an intuitionistic fuzzy π-β-local β space.
Remark 3.10 The converse of the Proposition 3.10 need not be true as shown in Example 3.9.
 a b   a b 
 a b 
,
 ,  ,  〉 and B = 〈 x,  ,  ,
 0.4 0.4   0.6 0.6 
 0.4 0.6 
Example 3.9 Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b 
 ,  〉 be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic fuzzy
 0.6 0.4 
topology on X. Clearly, the intersection of any intuitionistic fuzzy π-open set and any intuitionistic fuzzy βclosed set in an intuitionistic fuzzy topological space ( X, T ) is an intuitionistic fuzzy π-β-locally closed set.
Thus, every intuitionistic fuzzy π-β-locally closed set is an intuitionistic fuzzy β-closed set. Hence,( X, T )
 a b 
,
,
 0.6 0.6 
intuitionistic fuzzy π-β
β-local β Space. Now, B is an intuitionistic fuzzy π-open set. Let C = 〈 x, 
 a b 
 a b   a b 
 ,  〉 be an intuitionistic fuzzy β-closed set. Hence, E = B I C = 〈 x,  ,  ,  ,  〉 is an
 0.4 0.4 
 0.4 0.6   0.6 0.4 
intuitionistic fuzzy π-β-locally closed set. Now, E is an intuitionistic fuzzy β-closed set. But, E is not
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ON SOME APPLICATIONS OF INTUITIONISTIC FUZZY…
anintuitionistic fuzzy preclosed set. Thus,( X, T ) is not an intuitionistic fuzzy π-β
β -local pre space. Hence,
intuitionistic fuzzy π-β
β -local β space need not be an intuitionistic fuzzy π-β
β -local pre space.
Proposition 3.11 Every intuitionistic fuzzy π-β-local α space is an intuitionistic fuzzy π-β-local S space.
Remark 3.11 The converse of the Proposition 3.11 need not be true as shown in Example 3.10.
 a b   a b 
 a b 
,
 ,  ,  〉 and B = 〈 x,  ,  ,
 0.3 0.4   0.7 0.6 
 0.4 0.4 
Example 3.10 Let X = {a , b} be a nonempty set. Let A = 〈 x, 
 a b 
 ,  〉 be intuitionistic fuzzy sets of X. Then the family T = { 0∼, 1∼, A, B } is an intuitionistic fuzzy
 0.3 0.5 
topology on X. Clearly, the intersection of any intuitionistic fuzzy π-open set and any intuitionistic fuzzy βclosed set in an intuitionistic fuzzy topological space ( X, T ) is an intuitionistic fuzzy π-β-locally closed set.
Thus, every intuitionistic fuzzy π-β-locally closed set is an intuitionistic fuzzy semiclosed set. Hence, ( X, T ) is
 a b 
,
,
 0.3 0.5 
an intuitionistic fuzzy π-β
β-local S space. Now, B is an intuitionistic fuzzy π-open set. Let C = 〈 x, 
 a b 
 , 〉
 0.4 0.4 
β-closed set. Hence, E = B
be
I
an
intuitionistic
fuzzy
 a b   a b
,
 ,  ,  〉 is an intuitionistic fuzzy π-β-locally closed
 0.3 0.4   0.4 0.5 
C = 〈 x, 
set.Now,E is an intuitionistic fuzzy semi closed set. But, E is not anintuitionistic fuzzy α-closed set. Thus,( X,
T ) is not an intuitionistic fuzzy π-β
β -local α space. Hence, intuitionistic fuzzy π-β
β -local S space need not be
an intuitionistic fuzzy π-β
β -local α space.
4. Intuitionistic fuzzy π-β
β -local border, Intuitionistic fuzzy π-β
β -local frontier and Intuitionistic fuzzy π-β
βlocal exterior
In this section, intuitionistic fuzzy π-β-local border, intuitionistic fuzzy π-β-local frontier and intuitionistic
fuzzy π-β-local exterior are introduced. Some of its properties are studied.
Definition 4.1 Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set in an intuitionistic fuzzy topological space ( X, T ). Then A
I IFπ-lcl ( A ) is said to be an
intuitionistic fuzzy π-local border of A and is denoted as IFπ-lb( A ).
Definition 4.2 Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set in an intuitionistic fuzzy topological space ( X, T ). Then A
I IFπ-β-lcl ( A ) is said to be
anintuitionistic fuzzy π-β-local border of A and is denoted as IFπ-β-lb( A ).
Definition 4.3 Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉be anintuitionistic
fuzzy set in an intuitionistic fuzzy topological space (X, T). Then IFπ-lcl(A) I IFπ-lc( A ) is said to be an
intuitionistic fuzzy π-local frontier of A and is denoted as IFπ-lFr ( A ).
Definition 4.4 Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set in an intuitionistic fuzzy topological space ( X, T ). Then IFπ-β-lcl( A )
I IFπ-β-lcl ( A ) is said to be
an intuitionistic fuzzy π-β-local frontier of A and is denoted as IFπ-β-lFr(A).Proposition 4.1 Let ( X, T ) be an
intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic fuzzy set in an intuitionistic fuzzy
topological space ( X, T ). Then the following statements hold
(i)IFπ-β-lb (A) ⊆ IFπ-lb (A) where IFπ-lb (A) denotes the intuitionistic fuzzy π-local border of A. (ii) IFπ-βlFr(A) ⊆IFπ-lFr(A) where IFπ-lFr(A) denotes the intuitionistic fuzzy π-local frontier of A.
(iii) IFπ-β-lb( A ) ⊆IFπ-β-lFr ( A )
(iv)IFπ-β-lFr( A ) = IFπ-β-lFr ( A )
(v)IFπ-β-lFr( IFπ-β-lint ( A ) ) ⊆ IFπ-β-lFr ( A )
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S.Padmapriya, M.K.Uma and E.Roja
Definition 4.5
Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉
be an intuitionistic fuzzy set in an intuitionistic fuzzy topological space ( X, T ). Then IFπ-lint ( A ) is said to be
an intuitionistic fuzzy π-local exterior of A and is denoted as IFπ-lExt( A ).
Definition 4.6 Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 be an intuitionistic
fuzzy set in an intuitionistic fuzzy topological space ( X, T ). Then IFπ-β-lint ( A ) is said to be an intuitionistic
fuzzy π-β-local exterior of A and is denoted as IFπ-β-lExt( A ).
Proposition 4.2 Let ( X, T ) be an intuitionistic fuzzy topological space. Let A = 〈 x, µA, γA 〉 and B = 〈 x, µB,
γB 〉 are intuitionistic fuzzy set in an intuitionistic fuzzy topological space ( X, T ). Then the following statements
hold
(i)IFπ-lExt(A)⊆IFπ-β-lExt(A),where IFπ-lExt(A)denotes the intuitionistic fuzzy π-local exterior of A.
(ii)IFπ-β-lExt( A ) = IFπ-β-lint ( A ) = IFπ − β − lcl(A)
(iii)IFπ-β-lExt( IFπ-β-lExt ( A ) ) = IFπ-β-lint ( IFπ-β-lcl ( A ) )
(iv)If A⊆ B then IFπ-β-lExt ( A ) ⊇ IFπ-β-lExt ( B )
(v)IFπ-β-lExt( 1~) = 0~
(vi)IFπ-β-lExt( 0~) = 1~
(vii)IFπ-β-lExt ( A
U B ) = ( IFπ-β-lExt ( A ) ) I ( IFπ-β-lExt ( B ) )
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