Download sb* - CLOSED SETS AND CONTRA sb

Poongothai et al., International Journal of Advanced Engineering Technology
E-ISSN 0976-3945
Research Paper
sb* - CLOSED SETS AND CONTRA sb* - CONTINUOUS
MAPS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES
A. Poongothai*, R. Parimelazhagan, S. Jafari
Address for Correspondence
Department of Science and Humanities, Karpagam College of Engineering,
Coimbatore -32, India
Department of Science and Humanities, Karpagam College of Engineering, Coimbatore -32, India
College of Vestsjaelland, South Herrestraede 11, 4200 Slagelse, Denmark
ABSTRACT:
In this paper, we introduce and study the concept of intuitionistic fuzzy sb * - closed sets and also we study the intuitionistic
fuzzy contra sb* - continuous maps and its properties in intuitionistic fuzzy topological spaces.
AMS Classification (2000) MSC: 54A40.
KEYWORDS: Intuitionistic Fuzzy sb*-closed set, Intuitionistic Fuzzy b - closed set, Intuitionistic Fuzzy sb*-open sets,
Intuitionistic Fuzzy contra sb* - continuous maps.
1. INTRODUCTION
Zadeh [21] introduced the concept of fuzzy sets.
Using fuzzy sets Chang [4] introduced fuzzy
topological spaces. Since then various authors have
contributed to the development of fuzzy topology.
The concept of intuitionistic fuzzy sets was
introduced by Atanassov[1,2] as a generalization of
fuzzy sets. In the last 30 years various concept of
Fuzzy mathematics has been extended for
intuitionistic fuzzy sets. Coker [5] introduced the
concept of intuitionistic fuzzy topological space.
Intuitionistic fuzzy compactness [6], Intuitionistic
fuzzy connectedness [20], Intuitionistic fuzzy
separation axioms [3], Intuitionistic fuzzy continuity
[9], Intuitionistic fuzzy g- closed sets [18] and
Intuitionistic fuzzy g-continuity [17] have been
generalized for intuitionistic fuzzy topological spaces.
Dontchev[7] introduced the concept of contra
continuous mapping. Ekici and Kerre[8] introduced
the concept of fuzzy contra continuous mapping.
Krsteska and Ekici[12] introduced the concept of
intuitionistic fuzzy contra continuous mapping and
fuzzy contra strongly pre continuous mapping. The
authors [13, 14, 15] introduced sb* - closed sets, sb* continuous maps, sb* irresolute and homeomorphisms
and studied some of their basic properties in
topological spaces.
In the present paper, we introduce the concept of
intuitionistic fuzzy sb*-closed set and intuitionistic
fuzzy contra sb* - continuous maps and also we
obtain some of their characterization and properties.
2. PRELIMINARIES
In this section, we recall some basic notions,
definitions and properties.
Throughout this paper, X and Y denotes the
intuitionistic fuzzy topological spaces (X,τ ) and (Y,
σ) respectively. For any subset A of a space (X, τ ) ,
the closure of A, the interior of A and the complement
of A are denoted by cl(A) , int(A) and Ac , A (or) XA respectively.
Definition 2.1[1]: Let X be a non empty fixed set. An
intuitionistic fuzzy set A in X is an object having the
form A = {<x, µA(x), γA(x)> : x  X} where the
functions µA : X →[0,1] and γA : X →[0,1] denote
the degree of membership µA(X) and the degree of
non membership γA(x) of each element x  X to the
set A respectively, and 0≤ µA(x) + γA(x) ≤ 1 for each
x  X.
~
Definition 2.2[1]: The intuitionistic fuzzy sets 0 =
{<x, 0, 1>, x  X} and
~
1 ={<x, 1, 0>, x  X} are
Int J Adv Engg Tech/Vol. VII/Issue I/Jan.-March.,2016/663-667
respectively called empty and whole intuitionistic
fuzzy set on X. Two intuitionistic fuzzy sets A = {<x,
µA(x), γA(x)> : x ∈ X} and B = {<x, µB (x) γB (x)> :
x ∈ X} are said to be q-coincident (AqB in short) if
and only if there exists an element x ∈X such that
µA(x) > γ B (x) (or) γA(x) < µB (x).
Definition 2.3[2]: Let X be a non empty set, and let
the intuitionistic fuzzy sets be A and B. Let {Aj : j 
J} be an arbitrary family of intuitionistic fuzzy sets in
X. Then
(a) A ⊆ B iff µA(x) ≤ µB(x) and γA(x) ≥ γB(x) for
all x ∈ X;
(b) A = {<x, γA(x), µA(x)> : x ∈ X} ;
(c) ∩Aj = {<x, ∧µAj (x), ∨γAj (x) > : x ∈ X };
(d) ∪Aj = {<x, ∨µAj (x), ∧γAj (x) > : x ∈ X} ;
(e) [ ]A = {<x, µA(x), 1 − µA(x)> : x ∈ X} ;
(f) < >A= {<x, 1 − γA(x), γA(x)> : x ∈ X} ;
(g)
~ ~
~ ~
A =A, 1 = 0 and 0  1 ;
(h) A  B  A  B, A  B  A  B
Definition 2.4[5]: An intuitionistic fuzzy topology τ
on a non empty set X is a family τ of intuitionistic
fuzzy sets in X satisfying the following axioms:
~ ~
(T1) 0 , 1  τ .
(T2) G1∩G2  τ for any G1, G2  τ .
(T3) ∪ Gi∈ τ for any arbitrary family {Gi : i ∈ I} ⊆ τ .
In this case the pair (X, τ ) is called an intuitionistic
fuzzy topological space and each intuitionistic fuzzy
set in τ is known as an intuitionistic fuzzy open set in
X.
Definition 2.5 [5]: Let A be an intuitionistic fuzzy set
in an intuitionistic fuzzy topological space X. Then
int A = ∪ {G /G is an intuitionistic fuzzy open set in
X and G ⊆ A} is called intuitionistic fuzzy interior of
A;
cl A = ∩ {G / G is an intuitionistic fuzzy closed set in
X and G ⊇ A} is called an intuitionistic fuzzy closure
of A.
Lemma 2.6[5]: Let A and B be any two intuitionistic
fuzzy sets of an intuitionistic fuzzy topological space
(X,τ ), then
(a) ¬(AqB) ⇔ A ⊆ Bc.
(b) A is an intuitionistic fuzzy closed set in X ⇔
cl(A) = A.
(c) A is intuitionistic open set in X ⇔ int(A) = A.
(d) cl(Ac) = (int(A))c.
(e) int(Ac) = (cl(A))c.
(f) A⊆ B ⇒ int(A) ⊆ int (B).
(g) A⊆ B ⇒ cl(A) ⊆ cl (B).
Poongothai et al., International Journal of Advanced Engineering Technology
(h) cl(A∪B)=cl(A) ∪ cl(B).
(i) int(A∩B) = int(A) ∩ int(B).
Definition 2.7[6]:Let X be a non empty set and c∈ X
a fixed element in X. If α ∈ (0,1] and β ∈[0,1) are two
real numbers such that α + β ≤ 1 then
(a) C(α, β) = <x, Cα, C1−β > is called an intuitionistic
fuzzy point in X, where α denotes the degree of
membership if c(α, β) and β denote the degree of non
membership of c(α, β).
(b) C(β) = <x, 0, 1 − C1−β > is called a vanishing
intuitionistic fuzzy point in X, where β denote the
degree of non membership of C(β).
Definition 2.8[9]: Let A be an intuitionistic fuzzy set
of an intuitionistic fuzzy topological space X. Then A
is called
(i) an intuitionistic fuzzy α-open set (briefly IFαOS)
if A⊆int(cl(int(A))),
(ii) an intuitionistic fuzzy semi open set (briefly
IFSOS) if A ⊆ cl(int(A)).
Definition 2.9[9]: Let A be an intuitionistic fuzzy set
of an intuitionistic fuzzy topological space X. Then A
is called
(i) an intuitionistic fuzzy α- closed set (briefly
IFαCS) if Ac is an intuitionistic fuzzy α-open set,
(ii) an intuitionistic fuzzy semi- closed set (briefly
IFSCS) if Ac is an intuitionistic fuzzy semi open set.
Definition 2.10[19]: An intuitionistic fuzzy set A of
an intuitionistic fuzzzy topological space X is called
intuitionistic fuzzy w-closed if cl(int(A))⊆O
whenever A ⊆ O and O is intuitionistic fuzzy semi
-open.
Definition 2.11[11]: An intuitionistic fuzzy set A in
an intuitionistic fuzzy topological space X is said to
be intuitionistic fuzzy b- closed (briefly IFbCS) if
cl(int(A))∩ int(cl(A)) ⊆ A.
Definition 2.12[16]:An intuitionistic fuzzy set A is an
intuitionistic fuzzy weakly generalized closed set
(briefly IFwgCS) if cl(int(A))⊆U whenever A ⊆U , U
is intuitionistic fuzzy open in X.
Definition 2.13[13]: A subset A of a topological
space (X, τ ) is called a strongly b∗-closed set (briefly
sb*- closed) if cl(int(A)) ⊆ U whenever A ⊆ U and U
is b open in X.
Definition 2.14[14]: Let X and Y be topological
spaces. A map f: X → Y is called strongly b* continuous (briefly sb*- continuous) if the inverse
image of every open set in Y is sb* - open in X.
Definition 2.15[15]: Let X and Y are topological
spaces. A map f: (X,τ ) → (Y, σ) is said to be sb*
irresolute if the inverse image of every sb* - closed
set in Y is
sb* - closed set in X.
Definition 2.16[15]: A bijection f : (X, τ ) → (Y, σ) is
called a sb* - homeomorphism if f is both sb* continuous and sb* - open.
Definition 2.17[5]: Let X and Y be two non empty
sets and f: X → Y be a function.
(i). If B = {<y, µB (y), γB (y)> : y ∈ Y } ; is an
intuitionistic fuzzy set in Y, then the pre image of B
under f is denoted and defined by f −1(B) = {<x, f
−1
(µB )(x), f −1(γB )(x) > : x ∈ X} ;.
(ii). If A = {<x, λA(x), υA(x)> : x ∈ X} ; is an
intuitionistic fuzzy set in X, then the image of A
under f is denoted and defined by f(A) ={<y, f
(λA)(y), f − (υA)(y) > : y ∈ Y } , where f-1(υA) = 1 − f
(1 − υA).
In (i), (ii), since µB , γB , λA, υA are fuzzy sets, we
explain that f −1(µB )(x)a = µB (f (x)), and f (λA)(y) =
Int J Adv Engg Tech/Vol. VII/Issue I/Jan.-March.,2016/663-667
E-ISSN 0976-3945
{supλA(x) if f −1(y)  φ, = 0 otherwise}.
Definition 2.18[5]: Let A, Ai(i∈ J)be an intuitionistic
fuzzy sets in X and B, Bj (j∈ K) be an intuitionistic
fuzzy sets in Y and f: X → Y be a function. Then
(i)
f−1 (∪Bj ) = ∪ f −1 (Bj );
(ii)
f −1 (∩Bj ) = ∩f −1 (Bj );
(iii)
~
f −1 ( 1 ) =
~
~ −1 ~
1 ;f (0 )= 0 ;
1
 
B ;
(iv)
f−1 ( B ) = f
(v)
f(∪Ai) = ∪f(Ai).
Definition 2.19[5,10]: Let X and Y be two
intuitionistic fuzzy topological spaces and f : X →
Y be a function. Then
(i) f is intuitionistic fuzzy continuous iff the pre image
of each intuitionistic fuzzy open set in Y is an
intuitionistic fuzzy open set in X.
Definition 2.20[10]: Let f: X → Y be a function. The
graph g: X → X × Y of f is defined by g(x) = (x, f(x)),
∀ x ∈ X.
Lemma 2.21[10]: Let g: X → X × Y be the graph of
a function f: X → Y. If A is an intuitionistic fuzzy set
of X and B is an intuitionistic fuzzy set of Y, then
g−1 (A× B)(x) = (A ∩ f −1(B)(x)).
Definition 2.22[9]: Let f: X → Y be mapping. Then f
is said to be intuitionistic fuzzy continuous if f −1(B)
is an intuitionistic fuzzy open set in X for every
intuitionistic fuzzy open set B in Y.
Definition 2.23[8]:A mapping f: X → Y is called
intuitionistic fuzzy contra continuous if f −1(B) is an
intuitionistic fuzzy open set in X, for each
intuitionistic fuzzy closed set B in Y.
3. INTUITIONISTIC FUZZY sb*- CLOSED
SETS
In this section, we introduce a new class of closed sets
called intuitionistic fuzzy sb*- closed set in an
intuitionistic fuzzy topological space. Also we discuss
some of its properties.
Definition 3.1: An intuitionistic fuzzy set A in an
intuitionistic fuzzy topological space (X, τ) is called
an intuitionistic fuzzy sb∗ - closed set(briefly IFsb*closed) if cl(int(A))⊆ O whenever A⊆O and O is
intuitionistic fuzzy b-open.
Remark 3.2: Every intuitionistic fuzzy closed set is
intuitionistic fuzzy sb* - closed set but the converse is
not true.
~ ~
Example 3.3: Let X = {a, b} and τ = 0 , 1 ,U be an
intuitionistic fuzzy topology on X, where U = {<a,
0.5, 0.5> , <b, 0.4, 0.6>}. Then the intuitionistic fuzzy
set A = {<a, 0.5, 0.5> , <b, 0.5, 0.5>} is intuitionistic
fuzzy sb∗ -closed but it is not intuitionistic fuzzy
closed
Remark 3.4: Every intuitionistic fuzzy sb∗ -closed set
is intuitionistic fuzzy b-closed but the converse is not
true.
Example 3.5: Let X = {a, b} and τ = ~0 , ~1 , U be an
intuitionistic fuzzy topology on X, where U = {<a,
0.4, 0.6>, <b, 0.3, 0.7>}. Then the intuitionistic fuzzy
set A = {<a, 0.4, 0.6>, <b, 0.3, 0.7>} is intuitionistic
fuzzy b -closed but it is not intuitionistic fuzzy sb*closed.
Remark 3.6: Every intuitionistic fuzzy w-closed set
is intuitionistic fuzzy sb∗-closed but the converse is
not true.
~ ~
Example 3.7: Let X = {a, b} and τ = 0 , 1 ,U be an
intuitionistic fuzzy topology on X, where U = {<a,
0.7, 0.3>, <b, 0.6, 0.4>}. Then the intuitionistic fuzzy






Poongothai et al., International Journal of Advanced Engineering Technology
set A = {<a, 0.6, 0.4>, <b, 0.7, 0.3>} is intuitionistic
fuzzy sb ∗ -closed but it is not intuitionistic fuzzy wclosed.
Remark 3.8: Every intuitionistic fuzzy α-closed set is
intuitionistic fuzzy sb∗-closed but the converse is not
true.
~ ~
Example 3.9: Let X = {a, b} and τ = 0 , 1 , U be an
intuitionistic fuzzy topology on X, where U = {<a,
0.3, 0.7>, <b, 0.2, 0.7>}. Then the intuitionistic fuzzy
set A = {<a, 0.6, 0.3>, <b, 0.6, 0.4>} is intuitionistic
fuzzy sb∗-closed but it is not intuitionistic fuzzy α closed.
Remark 3.10: Every intuitionistic fuzzy sb∗-closed
set is intuitionistic fuzzy wg-closed but the converse
is not true.
Example 3.11: Let X= {a, b} and τ = ~0 , ~1 , U be an
intuitionistic fuzzy topology on X where U = {<a, 0.9,
0.1>, <b, 0.7, 0.2>}. Then the intuitionistic fuzzy set
A = {<a, 0.4, 0.6>, <b, 0.3, 0.7>} is intuitionistic
fuzzy wg-closed but it is not intuitionistic fuzzy sb∗closed.
Theorem 3.12: Let (X, τ ) be an intuitionistic fuzzy
topological space and A is an intuitionistic fuzzy set
of X. Then A is intuitionistic fuzzy sb∗ closed if and
only if ¬(AqF ) ⇒ ¬(cl(int(A))qF)for every
intuitionistic fuzzy b closed set F of X.
Proof:
Necessity: Suppose A be an intuitionistic fuzzy sb*closed set.Let F be an intuitionistic fuzzy b closed set
of X and ¬(AqF ). Then by lemma 2.6(a), A ⊆ F c and
Fc is intuitionistic fuzzy b open in X. Since A is
intuitionistic fuzzy sb* - closed, cl(int(A)) ⊆ F c.
Hence by lemma 2.6(a) ¬ (cl(int(A))qF).
Sufficiency: Let O be an intuitionistic fuzzy b open
set of X such that A ⊆O(ie.,) A (Oc)c. Therefore by
lemma 2.6(a), ¬(AqOc) and Oc is an intuitionistic
fuzzy b closed set in X. Hence by hypothesis
¬(cl(int(A))qOc).Therefore
by
lemma
2.6(a),
cl(int(A)) ⊆ (Oc)c .(i.e.,) cl(int(A)) ⊆O. Hence A is
intuitionistic fuzzy sb∗ -closed in X.
Theorem 3.13:Let A be an intuitionistic fuzzy sb∗closed set in an intuitionistic fuzzy topological space
(X,τ ) and C(α, β) be an intuitionistic fuzzy point of X
such that
C(α, β)q cl(int(A)) then cl(int(C(α,
β)))qA.
Proof: If ¬cl(int(C(α, β)))qA, then by lemma (2.5)(a),
cl(int(C(α, β))) ⊆ Ac which implies that A⊆
(cl(int(C(α, β))))c and so cl(int(A)) ⊆ (cl(int(C(α,
β))))c ⊆ (C(α, β))c, since A is intuitionistic fuzzy sb∗closed in X. Therefore by lemma 2.6(a),
¬(cl(int(A))q(C(α, β))) which is a contradiction.
Theorem 3.14: Let A and B be two intuitionistic
fuzzy sb∗-closed sets in an intuitionistic fuzzy
topological space, then A ∩B is intuitionistic fuzzy
sb∗ -closed.
Proof: Let O be an intuitionistic fuzzy b- open set in
X such that A∩B ⊆O. Then A⊆ O and B⊆O. So
cl(int(A))⊆ O and cl(int(B)) ⊆ O. Therefore
cl(int(A∩B))⊆ cl(int(A)) ∩ cl(int(B)) ⊆O. Hence A
∩B is intuitionistic fuzzy sb *-closed.
Remark 3.15: The union of any two intuitionistic
fuzzy sb* closed sets need not be an intuitionistic sb*
closed set in general as seen from the following
example.
Example 3.16: Let X = {a, b} and τ = ~0 , ~1 , U  is an
intuitionistic fuzzy topology on X, Where U = {<a,




Int J Adv Engg Tech/Vol. VII/Issue I/Jan.-March.,2016/663-667
E-ISSN 0976-3945
0.6, 0.4>, <b, 0.8, 0.2>}. The intuitionistic fuzzy set
A = {<a, 0.1, 0.9> , <b, 0.8, 0.2> and B = {<a, 0.6,
0.4> , <b, 0.7, 0.3>} are intuitionistic fuzzy sb*
closed sets but the union A∪B is not an intuitionistic
fuzzy sb∗-closed set.
Theorem 3.17: Let A be an intuitionistic fuzzy sb*closed in an intuitionistic fuzzy topological space (X,τ
) and A⊆ B ⊆ cl(int(A)). Then B is intuitionistic
fuzzy sb*-closed in X.
Proof: Let A be an intuitionistic fuzzy sb*-closed set
such that A⊆ B ⊆ cl(int(A)). Let U be an
intuitionistic fuzzy b-open set of X such that B ⊆U.
Since A is intuitionistic fuzzy sb∗-closed, we have
cl(int(A))⊆U and A ⊆ B ⊆ U. Since B ⊆ cl(int(A)),
cl(int(B)) ⊆cl(int(A)) ⊆U. Therfore cl(int(B)) ⊆ U.
Thus B is intuitionistic fuzzy
sb*-closed set in X.
4. INTUITIONISTIC FUZZY CONTRA sb∗
CONTINUOUS MAPS
In this section, we introduce a new definition called
intuitionistic fuzzy contra sb∗ -continuity in
intuitionistic fuzzy topological space. Also we discuss
some of its basic properties.
Definition 4.1: (i) A mapping f: X → Y is called
intuitionistic fuzzy contra strongly b* continuous if f
−1
(B) is an intuitionistic fuzzy sb* open set in X, for
each intuitionistic fuzzy closed set B in Y.
(ii) A mapping f: X → Y is called intuitionistic fuzzy
contra sb*- irresolute if the inverse image of every
intuitionistic fuzzy sb* - closed set in Y is
intuitionistic fuzzy sb* - open set in X.
Remark 4.2: From the above definition, the
following implication is true.
Intuitionistic fuzzy contra continuity ⇒ intuitionistic
fuzzy contra sb* - continuity.
The converse of the above remark is not true as seen
from the following example.
Example 4.3: Let X = {a, b}, Y = {x, y} and τ =
~~
~ ~
0 , 1 , U and σ = 0 , 1 , V are the intuitionistic fuzzy
topologies on X and Y, where U = {<a, 0.5, 0.5>, <b,
0.4, 0.6>}and V = {<x, 0.5, 0.5> , <y, 0.5, 0.5>}
respectively. Then the mapping f: X → Y is defined
by f(a) = x and f(b) = y is intuitionistic fuzzy contra
sb* continuous but not intuitionistic fuzzy contra
continuous.
Theorem 4.4: Let f: X→Y be a mapping from an
intuitionistic fuzzy topological space X into an
intuitionistic fuzzy topological space Y. Then the
following statements are equivalent:
i.
f is an intuitionistic fuzzy contra sb* continuous mapping,
ii.
f −1(B) is an intuitionistic fuzzy sb* - closed
set in X, for each intuitionistic fuzzy open set
B in Y.
Proof :
(i) ⇒ (ii): Let f be any intuitionistic fuzzy contra sb* continuous mapping and let B be an intuitionistic
fuzzy open set in Y. Then, Bc is an intuitionistic fuzzy
closed set in Y. By assumption f −1(Bc)is intuitionistic
fuzzy sb* open set in X. (f −1(B))c is an intuitionistic
fuzzy sb* open set in X, since f −1(Bc)=(f −1(B))c .
Hence f −1(B) is intuitionistic fuzzy sb* - closed set in
X.
(ii) ⇒ (i): Let B be an intuitionistic fuzzy closed set in
Y. Bc is an intuitionistic fuzzy open set in Y. By (ii), f
−1
(Bc) = (f −1(B))c is an intuitionistic fuzzy sb* closed set in X. Hence f −1 (B) is an intuitionistic
fuzzy sb* -open set in X. Therefore f is intuitionistic




Poongothai et al., International Journal of Advanced Engineering Technology
fuzzy contra sb* - continuous mapping.
Theorem 4.5: Let f: X →Y and g: Y→Z, where X, Y
and Z are intuitionistic fuzzy topological spaces. Then
the following statements hold.
i.
If f: X→Y is an intuitionistic fuzzy contra
sb* - continuous mapping and g: Y→Z is an
intuitionistic fuzzy contra continuous
mapping, then g◦f: X→Z is an intuitionistic
fuzzy sb* - continuous mapping.
ii.
If f: X→ Y is an intuitionistic fuzzy sb*
irresolute mapping and g: Y→Z is an
intuitionistic fuzzy contra sb* - continuous
mapping, then g◦f: X→ Z is an intuitionistic
fuzzy contra sb* -continuous mapping.
iii.
If f : X →Y is an intuitionistic fuzzy sb* closed, surjective mapping and g◦f: X →Z is
an intuitionistic fuzzy contra sb* continuous mapping, then g: Y → Z is an
intuitionistic fuzzy contra sb*-continuous
mapping.
iv.
If g◦f: X→Z is an intuitionistic fuzzy contra
sb*- continuous mapping and g : Y→ Z is an
intuitionistic fuzzy open, injective mapping,
then f: X →Y is a intuitionistic fuzzy contra
sb* - continuous mapping.
Proof:
i.
Let A be an intuitionistic fuzzy open set in Z.
Since g is intuitionistic fuzzy contra
continuous, g−1(A) is intuitionistic fuzzy
closed set in Y . Since f is intuitionistic fuzzy
contra sb* - continuous, f −1(g−1(A)) = (g ◦ f
)−1(A) is intuitionistic fuzzy sb* - open set in
X. Hence g ◦f is an intuitionistic fuzzy sb* continuous mapping.
ii.
Let A be an intuitionistic fuzzy open set in Z.
Since g is intuitionistic fuzzy contra sb* continuous, g−1 (A) is intuitionistic fuzzy sb*
- closed in X. Since f is intuitionistic fuzzy
sb* - irresolute, f −1(g−1(A)) = (g ◦ f )−1(A) is
an intuitionistic fuzzy sb* - closed set in X.
Hence g◦f is an intuitionistic fuzzy contra
sb* continuous mapping.
iii.
From the fact that for any surjective mapping
g: Y→Z, g−1(A) = f(g◦f)−1(A) holds for each
intuitionistic fuzzy set A in Z.
iv.
From the fact that for any open, injective
mapping g: Y→Z, f −1(B) = (g◦f)−1 (g(B))
holds for each intuitionistic fuzzy set B in Y.
Theorem 4.6: Let f: X → Y be a function and g: X →
X × Y be the graph of the function f, then f is
intuitionistic fuzzy contra sb* continuous.
Proof: Let B be any intuitionistic fuzzy open set in Y.
~
~
~
Then f −1 (B) = f−1 ( 1 × B) = 1 ∧f −1 (B) = g −1 ( 1
×B). Since B is an intuitionistic fuzzy open set in Y,
~
1 ×B is an intuitionistic fuzzy open set in X × Y.
Since g is intuitionistic fuzzy contra sb* - continuous,
−1
~
( 1 ×B) is an intuitionistic fuzzy sb* - closed set
g
in X. Hence
f −1 (B) is an intuitionistic fuzzy sb*
closed set in X and so f is an intuitionistic fuzzy
contra sb* - continuous mapping.
Theorem 4.7: If f: X → Y is an intuitionistic fuzzy
contra sb* - continuous function and g: Y → Z is an
intuitionistic fuzzy continuous function, then g◦f: X
→ Z is intuitionistic fuzzy contra sb* - continuous.
Proof: Let V be an intuitionistic fuzzy open set in Z.
Then g−1(V ) is intuitionistic fuzzy open in Y. Since f
Int J Adv Engg Tech/Vol. VII/Issue I/Jan.-March.,2016/663-667
E-ISSN 0976-3945
is intuitionistic fuzzy contra sb* - continuous, f −1 (g−1
(V)) = (g◦f)−1(V ) is intuitionistic fuzzy sb* - closed in
X. Therefore g◦f : X →Z is intuitionistic fuzzy contra
sb* - continuous.
Theorem 4.8: Let X and Y be intuitionistic fuzzy
topological spaces. Let f: X → Y is any mapping. If
the graph g: X → X × Y of f is intuitionistic fuzzy
contra sb* irresolute then f is also intuitionistic fuzzy
contra sb* irresolute.
Proof: Let A be an intuitionistic fuzzy sb* - open set
~
~
in Y. By definition, f −1 (A) = 1 ∧f −1 (A) = g −1 ( 1 ×
A). Since g is intuitionistic fuzzy contra sb* -
~
irresolute, g −1 ( 1 × A) is intuitionistic fuzzy sb* closed set in X. Hence f −1 (A) is intuitionistic fuzzy
sb* - closed in X. Thus f is intuitionistic fuzzy contra
sb* - irresolute.
Theorem 4.9: An intuitionistic fuzzy continuous
mapping f: X → Y is an intuitionistic fuzzy contra
sb* - continuous mapping if IFsb*OS(X) =
IFsb*CS(X).
Proof: Let A be an intuitionistic fuzzy open set in Y.
By hypothesis, f −1 (A) is an intuitionistic fuzzy open
set in X. Since every intuitionistic fuzzy open set is an
intuitionistic fuzzy sb* open set, f −1(A) is an
intuitionistic fuzzy sb* open set in X. Thus f −1 (A) is
an intuitionistic fuzzy sb* closed set in X, by
hypothesis. Hence f is an intuitionistic fuzzy contra
sb* - continuous mapping.
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