IFquasi γ Supra Open and IFquasi γ Supra Closed Functions Inintuitionistic Fuzzy Supra Topological Spaces

IFquasi γ Supra Open and IFquasi γ Supra Closed Functions Inintuitionistic Fuzzy Supra Topological Spaces

R. Syed Ibrahim

and S. Chandrasekar

1

Assistant Professor, Department of Mathematics,

Sethu Institute of Technology, Kariapatti, Virudhunagar (DT),Tamilnadu, INDIA.

2

Assistant Professor,

PG and Research Department of Mathematics,

Arignar Anna Government Arts College, Namakkal (DT),Tamil Nadu, INDIA.

email: syedjanna@gmail.com, chandrumat@gmail.com.

(Received on: July 15, 2018)

ABSTRACT

NeclaTuranlintroduced the concept of Intuitionistic fuzzy supra topological space which is a special case of Intuitionisticfuzzy topological space. Aim of this paper is we introduced we studied about Intuitionistic fuzzy quasi supra γ open and Intuitionistic fuzzy quasi supra γ closed functions in Intuitionistic fuzzy supra topological spaces.

Keywords: IFγ supra open sets, IFγ supra closed set, Intuitionistic fuzzy quasi γsupra open functions, Intuitionistic fuzzy quasi γ supra closed functions.

1. INTRODUCTION

Topology is a classical subjects, as a generalization topological spaces many type of topological spaces introduced over the year. C.L. Chang

was introduced and developed fuzzy topological space by using L.A. Zadeh’s

fuzzy sets. Coker

introduced the notion of Intuitionistic fuzzy topological spaces by using Atanassov’s

Intuitionistic fuzzy set.

A.S. Mashhour et al. introducedand studied the supra topological spaces in the year 1983. M. E. AbdEl-Monsef et al.

introduced the fuzzy supra topological spaces and studied fuzzy supra-continuous functions and obtained some properties and characterizations. In 2003NeclaTuranl

introduced the concept of Intuitionistic fuzzy supra topological space.

In 1996 D. Andrijevic

introduced b open sets in topological space, Aim of this paper

is we introduced and studied about Intuitionistic fuzzy quasi supra γ open and Intuitionistic

fuzzy quasi supra γ closed functions in Intuitionistic fuzzy supra topological spaces.

Definition 2.1[3]

An Intuitionistic fuzzy set (IF for short) A is an object having the form A = {〈x, 𝜇

(x), 𝜈

(x)〉 : x ∈ X}

where the functions 𝜇

:X⟶[0,1] and 𝜈

:X⟶ [0,1] denote the degree of membership (namely 𝜇

(x)) and the degree of non-membership (namely 𝜈

(x)) of each element

x∈ X to the set A respectively, and 0 ≤ 𝜇

(x) + 𝜈

(x) ≤ 1 for each x ∈ X.

Obviously, every fuzzy set A on a nonempty set X is an IFS of the following form

A ={〈x, 𝜇

(x), 1- 𝜇

(x)〉: x ∈ X}.

Definition 2.2[3]

Let A and B be two Intuitionistic fuzzy sets of the form

A = {〈x, 𝜇

(x), 𝜈

(x)〉: x ∈ X} and B = {〈x, 𝜇

(x), 𝜈

(x)〉 : x∈X}. Then, (i) A ⊆ B if and only if 𝜇

(x)≤𝜇

(x) and 𝜈

(x)≥ 𝜈

(x) for all x ∈ X, (ii) A = B if and only if A ⊆ B and A ⊇ B,

(iii) A

= {〈x, 𝜈

(x), 𝜇

(x)〉: x ∈ X},

(iv) A∪B = {〈x, 𝜇

(x) ∨𝜇

(x), 𝜈

(x) ∧𝜈

(x)〉: x ∈ X}, (v) A∩B = {〈x, 𝜇

(x) ∧𝜇

(x), 𝜈

(x) ∨𝜈

(x)〉: x ∈ X}.

(vi) []A= {〈x, μ

(x), 1 − μ

(x)〉, x ∈X};

(vii) 〈〉A = {〈x, 1 − 𝜈

(x), 𝜈

(x)〉, x ∈X};

The Intuitionisticfuzzy sets 0~ = 〈x, 0, 1〉 and 1~ = 〈x, 1, 0〉 are respectively the empty set and the whole set of X

Definition 2.3.[3]

Let {Ai :i ∈J } be an arbitrary family of Intuitionistic fuzzy sets in X. Then (a) ∩A

= {〈x,∧μ

(x),∨𝜈

(x) 〉 : x ∈X};

(b) ∪A

= {〈x,∨μ

(x),∧𝜈

(x) 〉 : x ∈ X}.

Definition 2.4.[3]

Since our main purpose is to construct the tools for developing Intuitionistic fuzzy topological spaces, we must introduce the Intuitionistic fuzzy sets 0~and 1~in X as follows:

0~ = {〈x, 0, 1〉: x ∈X} and 1~ = {〈x, 1, 0〉 :x ∈X}.

Definition 2.5[3]

Let A,B,C be Intuitionistic fuzzy sets in X. Then

(i) A ⊆B and C ⊆D ⇒A ∪C ⊆B ∪D and A∩ C ⊆B ∩ D, (ii) A ⊆B and A ⊆C ⇒A ⊆B ∩ C,

(iii) A ⊆C and B ⊆C ⇒A ∪B ⊆C, (iv) A⊆B and B ⊆C ⇒A ⊆C, (v)𝐴 ∪ 𝐵 ̅̅̅̅̅̅̅̅̅ = 𝐴̅ ∩ 𝐵̅

(vi) 𝐴 ∩ 𝐵 ̅̅̅̅̅̅̅̅̅̅ = 𝐴̅ ∪ 𝐵̅,

(vii) A ⊆B ⇒𝐵̅⊆𝐴̅,

(viii) (𝐴̅) ̅̅̅̅̅= A,

(ix) 1̅∼ = 0∼, (x) 0̅∼ = 1∼.

Definition 2.6[5]

Let f be a mapping from an ordinary set X into an ordinary set Y ,

If B ={〈y, μ

(y), ν

(y)〉: y ∈Y } is an IFST in Y , then the inverse image of B under fis an IFST defined by f

(B) = {〈x, f

(μ

)(x), f

(ν

)(x)〉: x ∈X}

The image of IFST A = {〈y, μ

(y), ν

(y)〉:y ∈Y } under f is an IFST defined by f(A) = {〈y, f(μ

)(y), f(ν

)(y)〉: y ∈Y }.

Definition 2.7[3, 5]

Let A, Ai (i ∈J) be Intuitionistic fuzzy sets in X, B, Bi (i ∈K) beIntuitionistic fuzzy sets in Y and f:X → Y is a function. Then

(i) A

⊆A

⇒f (A

) ⊆f (A

), (ii) B

⊆B

⇒f

(B

) ⊆f

(B

),

(iii) A ⊆f

(f (A)) { If f is injective, then A = f

(f (A))}, (iv) f (f

(B)) ⊆B { If f is surjective, then f (f

(B)) = B }, (v) f

(∪ 𝐵

) =∪ f

(𝐵

)

(vi)f

(∩ 𝐵

) =∩ f

(𝐵

) (vii) f(∪ 𝐵

) =∪ f(𝐵

)

(viii) f(∩ 𝐵

) ⊆∩ f(𝐵

){ Iff is injective, then f(∩ 𝐵

) =∩ f(𝐵

)}

(ix) f

(1∼) = 1∼ , (x) f

(0∼) = 0∼,

(xi) f (1∼) = 1∼ , if f is surjective (xii) f (0∼) = 0∼,

(xiii) f (A) ̅̅̅̅̅̅̅⊆f (𝐴̅), if f is surjective, (xiv) f

(𝐵̅) = f ̅̅̅̅̅̅̅̅̅̅.

(B)

Definition 2.8[21]

A family τ

Intuitionistic fuzzy sets on X is called an Intuitionistic fuzzy supra topology (in short, IFST) on X if 0~∈τ

,1~∈τ

and τ

is closed under arbitrary suprema.

Then we call the pair (X,τ

) an Intuitionistic fuzzy supra topological space (in short, IFSTS).

Each member of τ

is called an Intuitionistic fuzzy supra open set and the complement of an Intuitionistic fuzzy supra open set is called an Intuitionistic fuzzy supra closed set.

Definition 2.9[21]

The Intuitionistic fuzzy supra closure of a set A is denoted by S-cl(A) and is defined as S-cl(A)=∩{B : B is Intuitionistic fuzzy supra closed and A ⊆ B} .

The Intuitionistic fuzzy supra interior of a set A is denoted by S-int(A) and is defined as S-int(A) = ∪{B : B is Intuitionistic fuzzy supra open and A ⊇ B}

Definition 2.10[21]

(i). ¬ (AqB) A ⊆B

.

(ii). A is an Intuitionistic fuzzy supra closed set in X S-cl (A)=A.

(iii). A is an Intuitionisticfuzzy supra open set in X S-int (A)=A.

(iv). S-cl(A

) = (S-int(A))

. (v). S-int(A

) = (S-cl(A))

. (vi). A ⊆ B ⇒ S-int(A) ⊆S-int(B).

(vii). A ⊆ B ⇒ S-cl(A) ⊆S-cl(B).

(viii). S-cl(A ∪ B) =S-cl(A)∪S-cl(B).

(ix). S-int(A ∩ B) = S-int(A)∩S-int(B).

Definition 2.11

Let (X, τμ) is a Intuitionistic fuzzy supra topological spaceand A ⊆X . Then A is said to be Intuitionistic fuzzy γ supra open (briefly IFγs -open) set if A⊆ s-cl(s-int(A))∪ s-int(s-cl(A)).

The complement of Intuitionistic fuzzy γ supra open set is called Intuitionistic fuzzy γ supra closed set (briefly IFγs closed).

Definition 2.12

The Intuitionistic fuzzy γ supra closure of a set A is denoted by IFγS-cl(A) and is defined as

IFγS -cl (A) = ∩{B : B is Intuitionistic fuzzy γ supra closed and A ⊆ B} .

The Intuitionistic fuzzy γ supra interior of a set A is denoted by IFγS -int(A) and is defined as

IFγS -int(A) = ∪{B : B is Intuitionistic fuzzy γ supra open and A ⊇ B}

Definition 2.13

Let f : (X, τ

) → (Y, σ

))Intuitionistic fuzzy supra topological spaces and. Then f is said to be

(i) Intuitionistic fuzzy supra continuous if the pre image of each Intuitionistic fuzzy supra open set of Y is an Intuitionistic fuzzy open set in X.

(ii) Intuitionistic fuzzy supra closed if the image of each Intuitionistic fuzzy supra closed set in X is an Intuitionistic fuzzy supra closed set in Y.

(iii) Intuitionistic fuzzy supra open if the image of each Intuitionistic fuzzy supra open set in X is an Intuitionistic fuzzy supra open set in Y.

3. INTUITIONISTIC FUZZY QUASI IF SUPRAγ SUPRA OPEN FUNCTIONS In this section we introduce a new definition is called Intuitionistic Fuzzy Quasi γ supra open function and its properties are discussed

Definition 3.1

A function f : (X, τ

) → (Y, σ

) is said to be Intuitionistic Fuzzy Quasi γ supra open if the

image of every IFγ supra open set in space X is Intuitionistic Fuzzy supra open in space Y.

It is evident that, the concepts of Intuitionistic Fuzzy Quasi γ supra openness and IFγS- continuity coincide if the function is a bijection.

Theorem 3.2

A function f : (X, τ

) → (Y, σ

) is Intuitionistic Fuzzy Quasi γ supra open if and only if for every subset U of space X, f(IFγS-int(U)) S-int(f(U)).

Proof

Let f be a Intuitionistic Fuzzy Quasi γ supra open function and U be a subset of space X. Now, we have IFγS-int(U) U and IFγS-int(U) is a IFγ supra open set. Hence, we obtain that f(IFγS- int(U))  f(U). As f(IFγS-int(U)) is IF supra open, f(IFγS-int(U))S-int(f(U)). Conversely, assume that U is a IFγ supra open set in space X.Then, f(U)=f(IFγS-int(U))S-int(f(U)) but S-int(f(U))f(U). Consequently, f(U)=S-int(f(U)) and f(U) is IF supra open in space Y. Hence f is Intuitionistic Fuzzy Quasi γ supra open .

Lemma 3.3

If a function f : (X, τ

) → (Y, σ

) is Intuitionistic Fuzzy Quasi γ supra open , then IFγS- int(f

(G)) f

(S-int(G)) for every subset G of space Y.

Proof

Let G be any arbitrary subset of space Y. Then, IFγS-int(f

(G)) is a IFγ supra open set in space X and since f is Intuitionistic Fuzzy Quasi γ supra open , then f (IFγS-int(f

(G))) S- int(f(f

(G))) IFS int(G). Thus,IFγS-int(f

(G))f

(S-int(G)).

Recall that a subset S is called a IFγS-neighbourhood of a point x of space X if there exists a IFγ supra open set U such that x  U  S.

Theorem 3.4

For a function f: (X, τ

) →(Y, σ

) the following are equivalent:

(i) f is Intuitionistic Fuzzy Quasi γ supra open ;

(ii) For each subset U of space X, f (IFγS-int(U))IFS-int(f (U));

(iii) For each xX and each IFγS-neighbourhood U of x in space X, there exists a neighbourhoodf(U) of f(x) in space Y such that f(V)  f(U).

Proof

(i)  (ii): It follows from Theorem 3.2.

(ii)  (iii): Let x X and U be an arbitrary IFγS-neighbourhood of x in space X. Then there exists aIFγ supra open set V in space X such that x V U. Then by (ii), we have f(V)=f(IFγS- int(V))S-int(f(V)) and hence f(V)=S-int(f(V)). Therefore, it follows that f(V) is IF supra open in space Y such that f(x)  f(V)  f(U).

(iii)  (i): Let U be an arbitrary IFγ supra open set in space X such that x  U. Then for each f(x) = y  f (U), by (iii) there exists a neighbourhoodVy of y in space Y such that Vyf (U).

As Vy is a neighbourhood of y, there exists an IF supra open set Wy in space Y such that y

WyVy . Thus f (U) = {Wy : y f (U)} which is an IF supra open set in space Y. This

implies that f is Intuitionistic Fuzzy Quasi γsupra open function.

Theorem 3.6

A function f : (X, τ

) → (Y, σ

) is Intuitionistic Fuzzy Quasi γ supra open if and only if f

(S-cl(B)) IFγS-cl(f

(B)) for every subset B of space Y.

Proof

Suppose that f is Intuitionistic Fuzzy Quasi γ supra open. For any subset B of space Y, f

(B)

IFγS-cl(f^-1

(B)). Therefore by Theorem 3.5, there exists a IF supra closed set H in Y such that B  H and f

(H) IFγS-cl(f

(B)). Therefore, we obtain f

(S-cl(B))  f

(H) IFγS-cl(f

1

(B)).

Conversely, let B  Y and H be a IFγ supra closed of space X containing

f

(B). Put W = S-clY(B), then we have B  W and W is IF supra closed in space Y and f

1

(W) IFγS-cl(f

(B))  H. Then, by Theorem 3.5, f is Intuitionistic Fuzzy Quasi γ supra open.

Lemma 3.7

Let f : (X, τ

) → (Y, σ

) and g : (Y, σ

) → (Z, ρ

) be two functions such that g o f : (X, τ

) → (Z, ρ

) is Intuitionistic Fuzzy Quasi γ supra open. If g is IF supra continuous and injective, then f is Intuitionistic Fuzzy Quasi γ supra open.

Proof

Let U be a IFγ supra open set in space X. Then (g o f)(U) is open in Z since

g o f is Intuitionistic Fuzzy Quasiγ supra open .Again g is an injectiveIF supra continuous function,f(U) = g

(g o f (U))) is IF supra open in space Y. This shows that f is Intuitionistic Fuzzy Quasi γ supra open .

4. INTUITIONISTIC FUZZY QUASI γ SUPRA CLOSED FUNCTIONS

In this section we introduce a new definition is called Intuitionistic Fuzzy Quasi γ supra closed function and its properties are discussed

Definition 4.1

A function f : (X, τ

) → (Y, σ

) is said to be Intuitionistic Fuzzy Quasi γ supra closed if the image of each IFγ supra closed set in space X is IF supra closed in space Y.

Clearly, every Intuitionistic Fuzzy Quasi γ supra closed function is IF supra closed as well as IFγS-closed.

Remark 4.2

Every IFγ supra closed (resp. IF supra closed ) function need not be Intuitionistic Fuzzy Quasi γ supra closed as shown by the following example.

Example 4.3

Let X = {a, b}, Y = {u, v} and

A

= {<0.6, 0.3> ,<0.4, 0.5>},

A

= {<0.4, 0.5> ,<0.7, 0.6>},

A

= {<0.6, 0.5> ,<0.4, 0.5>}, A

= {<0.4, 0.5> ,<0.7, 0.6>}.

Then τ ={0~, 1~, A

, A

, A

∪ A

} be an Intuitionistic fuzzy supra topology on X. Then the Intuitionistic fuzzy supra topology σ on Y is defined as follows:

σ = {0~,1~, A

, A

, A

∪A

}.

Defined a mapping f (X, τ

) → (Y, σ

) by

f(a) = u and f(b) = v. Then f is IFγ supra closed as well as IF supra closed but not Intuitionistic Fuzzy Quasi IFγS-closed.

Lemma 4.4

If a function f : (X, τ

) → (Y, σ

) is Intuitionistic Fuzzy Quasi IFγS-closed, then f

(S-cl(B))

IFγS-cl(f^-1

(B)) for every subset B of space Y.

Proof

This proof is similar to the proof of Lemma 3.3.

Theorem 4.5

A function f : (X, τ

) → (Y, σ

) is Intuitionistic Fuzzy Quasi γ supra closed if and only if for any subset B of space Y and for any IFγ supra open set G of space X containing f

(B), there exists an IF supra open set U of space Y containing B such that f

(U)  G.

Proof

This proof is similar to that of Theorem 3.5.

Definition 4.6

A function f : (X, τ

) → (Y, σ

) is called IFγ*S-closed if the image of every IFγ supra closed subset of space X is IFγ supra closed in space Y.

Theorem 4.7

If f : (X, τ

) → (Y, σ

) and g : (Y, σ

) → (Z, ρ

) are two Intuitionistic Fuzzy Quasi γ supra closed functions, then g o f : (X, τ

)→(Z, ρ

) is a Intuitionistic Fuzzy Quasi γ supra closed function.

Proof Obvious.

Furthermore, we have the following theorem.

Theorem 4.8

Let f : (X, τ

) → (Y, σ

) and g : (Y, σ

) → (Z, ρ

) be any two functions. Then

(i) if f is IFγ supra closed and g is Intuitionistic Fuzzy Quasi IFγS-closed, then g o f is IF supra closed ;

(ii) if f is Intuitionistic Fuzzy Quasi γ supra closed and g is IFγS-closed, then g o f is IFγ*S-closed;

(iii) if f is IFγ*S -closed and g is Intuitionistic Fuzzy Quasi IFγS-closed, then g o f is

Intuitionistic Fuzzy Quasi IFγS-closed.

Proof Obvious.

Theorem 4.9

Let f : (X, τ

) → (Y, σ

) and g : (Y, σ

) → (Z, ρ

) be two functions such that g o f : (X, τ

) → (Z, ρ

) is Intuitionistic Fuzzy Quasi IFγS-closed. Then

(i) if f is IFγS-irresolute surjective, then g is IF supra closed . (ii) if g is IFγS-continuous injective, then f is IFγ*S-closed.

Proof

(i) Suppose that U is an arbitrary IF supra closed set in space Y. As f is IFγS-irresolute, f

(U) is IFγ supra closed in space X. Since g o f is IntuitionisticFuzzy Quasi γ supra closed and f is surjective, (g o f (f

(U))) = g(U), which is IF supra closed in space Z. This implies that g is a IF supra closed function.

(ii) Suppose that U is any IFγ supra closed set in space X. Since g o f is Intuitionistic Fuzzy Quasi IFγS-closed, (g o f)(U) is IF supra closed in space (Z, ρ

) . Again g is a IFγS- continuous injective function, g

(g o f (U)) = f (U), which is IFγ supra closed in spaceY. This shows that f is IFγ*S-closed.

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