On intuitionistic fuzzy

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ISSN (e): 2250 – 3005 || Volume, 06 || Issue, 03||March – 2016 || International Journal of Computational Engineering Research (IJCER)

On intuitionistic fuzzy đ?œˇ generalized closed sets Saranya M1, Jayanthi D2 1

MSc Mathematics, Avinashilingam University, Coimbatore, Tamil Nadu, India Assistant Professor of Mathematics, Avinashilingam University, Coimbatore, Tamil Nadu, India

2

ABSTRACT In this paper, we have introduced the notion of intuitionistic fuzzy đ?›˝ generalized closed sets, and investigated some of their properties and characterizations

KEYWORDS:Intuitionistic fuzzy topology, intuitionistic fuzzyđ?›˝ closed sets, intuitionistic fuzzy đ?›˝ generalized closed sets.

I. Introduction The concept of fuzzy sets was introduced byZadeh [12] and later Atanassov [1] generalized this idea to intuitionistic fuzzy sets using the notion of fuzzy sets. On the other hand Coker [3] introduced intuitionistic fuzzy topological spaces using the notion of intuitionistic fuzzy sets. In this paper, we have introduced the notion of intuitionistic fuzzy đ?›˝ generalized closed sets, and investigated some of their properties and characterizations.

II. Preliminaries Definition 2.1: [1] An intuitionistic fuzzy set (IFS for short) A is an object having the form A = {⌊x, đ?œ‡ A(x), đ?œˆ A(x)âŒŞ : x ∈ X} where the functions đ?œ‡ A : Xâ&#x;ś [0,1] and đ?œˆ A : Xâ&#x;ś [0,1] denote the degree of membership (namely đ?œ‡ A(x)) and the degree of non-membership (namely đ?œˆ A(x)) of each element x đ?œ– X to the set A respectively, and0 ≤ đ?œ‡ A(x) + đ?œˆA(x)≤1 for each x ∈ X. Denote by IFS (X), the set of all intuitionistic fuzzy sets in X. An intuitionistic fuzzy set A in X is simply denoted by A=⌊x, đ?œ‡ A ,đ?œˆ AâŒŞinstead of denotingA = {⌊x, đ?œ‡ A(x), đ?œˆ A(x)âŒŞ: x ∈ X}.

Definition 2.2: [1] Let A and B be two IFSs of the form A = {⌊x, đ?œ‡A(x), đ?œˆA(x)âŒŞ: x ∈ X} and B = {⌊x, đ?œ‡B(x), đ?œˆ B(x)âŒŞ : x đ?œ– X}. Then, (a) (b) (c) (d) (e)

A ⊆ B if and only if đ?œ‡ A(x) ≤ đ?œ‡ B(x) and đ?œˆ A(x) ≼ đ?œˆB(x) for all x ∈ X, A = B if and only if A ⊆ B and A ⊇ B, Ac = {⌊x, đ?œˆ A(x),đ?œ‡ A(x)âŒŞ: x ∈ X}, AâˆŞB = {⌊x, đ?œ‡ A(x) ∨ đ?œ‡ B(x), đ?œˆ A(x) ∧ đ?œˆ B(x)âŒŞ: x∈X}, A∊B = {⌊x, đ?œ‡ A(x) ∧ đ?œ‡ B(x), đ?œˆ A(x) ∨ đ?œˆ B(x)âŒŞ: x∈X}.

The intuitionistic fuzzy sets 0~ = ⌊x, 0, 1âŒŞ and 1~ =⌊x, 1, 0âŒŞ are respectively the empty set and the whole set of X.

Definition 2.3: [3]An intuitionistic fuzzy topology (IFT in short) on X is a family đ?œ? of IFSsin X satisfying the following axioms: (i) (ii) (iii)

0~, 1~ ∈ đ?œ? G1 ∊ G2∈ đ?œ? for any G1,G2∈ đ?œ? âˆŞGi∈ đ?œ? for any family {Gi:i∈ J} ⊆ đ?œ?.

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On intuitionistic fuzzyđ?›˝ generalized closed‌ In this case the pair (X,đ?œ?) is called intuitionistic fuzzy topological space (IFTS in short) and any IFS in đ?œ? is known as an intuitionistic fuzzy open set (IFOS in short) in X. The complement A c of an IFOS A in an IFTS (X,đ?œ?) is called an intuitionistic fuzzy closed set (IFCS in short) in X.

Definition 2.4:[5] An IFS A = ⌊x, đ?œ‡A, đ?œˆA âŒŞ in an IFTS (X,đ?œ?) is said to be an (i) (ii)

intuitionistic fuzzy � closed set (IF�CS for short) if int(cl(int(A))) ⊆ A, intuitionistic fuzzy � open set (IF�OS for short) if A ⊆ cl(int(cl(A))).

Definition 2.5: [6]Let A be an IFS in an IFTS (X,đ?œ?). Then the đ?›˝-interior and đ?›˝-closure of A are defined as đ?›˝int(A) = âˆŞ {G / G is an IFđ?›˝OS in X and G ⊆ A}. đ?›˝cl(A) = ∊{K / K is an IFđ?›˝CS in X and A ⊆ K}. Note that for any IFS A in (X,đ?œ?), we have đ?›˝cl(Ac) = (đ?›˝int(A))c andđ?›˝int(Ac) = (đ?›˝cl(A))c.

Result 2.6: Let A be an IFS in (X,đ?œ?), then (i) (ii)

đ?›˝cl(A)⊇ A âˆŞ int(cl(int(A))) đ?›˝int(A) ⊆ A ∊ cl(int(cl(A)))

Proof: (i)Now int(cl(int(A))) ⊆ int(cl(int(đ?›˝cl(A))) ⊆ đ?›˝cl(A), since A ⊆ đ?›˝cl(A) and Therefore A âˆŞ int(cl(int(A))) ⊆ đ?›˝cl(A). (ii) can be proved easily by taking complement in (i).

đ?›˝cl(A) is an IFđ?›˝CS.

III. Intuitionistic fuzzyđ?œˇgeneralized closed sets In this section we have introduced intuitionistic fuzzy đ?›˝ generalized closed sets and studied some of their properties.

Definition 3.1:An IFS A in an IFTS (X,đ?œ?) is said to be an intuitionistic fuzzy đ?›˝ generalized closed set (IFđ?›˝GCS for short) if đ?›˝cl(A) ⊆ U whenever A⊆ U and U is an IFđ?›˝OS in (X,đ?œ?). The complement Ac of an IFđ?›˝GCS A in an IFTS (X,đ?œ?) is called an intuitionistic fuzzy đ?›˝ generalized open set (IFđ?›˝GOS in short) in X. The family of all IFđ?›˝GCSs of an IFTS (X,đ?œ?) is denoted by IFđ?›˝GC(X).

Example 3.2:Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ. Then đ?œ? = {0~, G, 1~} is an IFT on X. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ a đ?œ–[0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ ađ?œ–[0,1], đ?œˆ bđ?œ– [0,1] / 0≤ đ?œ‡ a+đ?œˆ a≤1 and0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. We have A ⊆ G. As đ?›˝cl(A) = A,đ?›˝cl(A) ⊆ G, where G is an IFđ?›˝OS in X. This implies that A is an IFđ?›˝GCS in X.

Theorem 3.3:EveryIFCSin (X, đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof:Let A be an IFCS. Thereforecl(A) = A. Let A ⊆ U and U be an IFđ?›˝OS. Since đ?›˝cl(A) ⊆ cl(A) =A ⊆ U, we have đ?›˝cl(A) ⊆ U. Hence A is an IFđ?›˝GCS in (X, đ?œ?).

Example 3.4:Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ. Then đ?œ? = {0~, G, 1~} is an IFT onX. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ ađ?œ–[0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ ađ?œ–[0,1], đ?œˆ bđ?œ– [0,1] / 0≤ đ?œ‡ a+đ?œˆ a≤ 1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. We have A ⊆ G. As đ?›˝cl(A) = A,đ?›˝cl(A) ⊆ G, where G is an IFđ?›˝OS in X. This implies that A is an IFđ?›˝GCS in X, but not an IFCS, sincecl(A) = Gc≠A.

Theorem 3.5:EveryIFRCSin (X,đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof: Let A be an IFRCS[10]. Since every IFRCS is an IFCS [9], by Theorem 3.3, A is an IFđ?›˝GCS.

Example 3.6:Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ.Then đ?œ? = {0~, G, 1~} is an IFT onX. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ ađ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ ađ?œ– [0,1], đ?œˆ bđ?œ– [0,1] / 0 ≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}.

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On intuitionistic fuzzy� generalized closed‌ We have A ⊆ G. As �cl(A) = A,�cl(A) ⊆ G, where G is an IF�OS inX. This implies that A is an IF�GCS in X, but not an IFRCS, sincecl(int(A)) = cl(0~) = 0~ ≠A.

Theorem 3.7:EveryIFSCSin (X,đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof:Assume A is an IFSCS [5]. Let A ⊆ U and U be an IFđ?›˝OS. Sinceđ?›˝cl(A) ⊆scl(A) = A and A ⊆ U, by hypothesis, we haveđ?›˝cl(A) ⊆ U. Hence A is an IFđ?›˝GCS.

Example 3.8:Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ.Then đ?œ? = {0~, G, 1~} is an IFT on X. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ a đ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ ađ?œ– [0,1], đ?œˆ bđ?œ–[0,1] / 0≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. We have A ⊆ G.As đ?›˝cl(A) = A, đ?›˝cl(A) ⊆ G, where G is an IFđ?›˝OS in X. This implies that A is an IFđ?›˝GCS in X, but not an IFSCS, since int(cl(A)) = int(Gc) = G ⊈ A.

Theorem 3.9:EveryIFđ?›źCSin (X,đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof:Assume A is an IFđ?›źCS [5]. Let A ⊆ U and U be an IFđ?›˝OS. Sinceđ?›˝cl(A) ⊆ đ?›źcl(A) =A and A ⊆ U, by hypothesis, we have đ?›˝cl(A) ⊆ U. Hence A is an IFđ?›˝GCS.

Example 3.10: Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ. Thenđ?œ? = {0~, G, 1~} is an IFT on X. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ a đ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] / 0≤ đ?œ‡ a+đ?œˆ a≤ 1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. We have A ⊆ G. As đ?›˝cl(A) = A, đ?›˝cl(A) ⊆ G, where G is an IFđ?›˝OS in X. This implies that A is an IFđ?›˝GCS in X, but not an IFđ?›źCS, since cl(int(cl(A))) = cl(int(Gc)) = cl(G) = Gc⊈ A.

Theorem 3.11:EveryIFPCSin (X,đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof: Assume A is an IFPCS [5]. LetA ⊆ U and U be an IFđ?›˝OS.Sinceđ?›˝cl(A) ⊆ pcl(A) = Aand A⊆ U, by hypothesis, we have đ?›˝cl(A) ⊆ U. Hence A is an IFđ?›˝GCS.

Example 3.12: Let X = {a, b} and G = ⌊x, (0.5a, 0.6b), (0.5a, 0.4b)âŒŞ, Then đ?œ? = {0~, G, 1~} is an IFT on X. Let A = ⌊x, (0.5a, 0.7b), (0.5a, 0.3b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ a đ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] /đ?œ‡ b <0.6 whenever đ?œ‡ a≼0.5, đ?œ‡ a <0.5 whenever đ?œ‡ b ≼ 0.6,0≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. NowA⊆ 1~. As đ?›˝cl(A) = 1~ ⊆ 1~, we have A is an IFđ?›˝GCS in X, but not an IFPCS since cl(int(A)) = cl(G) =1~ ⊈ A.

Remark 3.13: Every IFGCS and every IFđ?›˝GCS are independent to each other. Example 3.14: Let X = {a, b} and G1 = ⌊x, (0.5a, 0.5b), (0.5a, 0.5b)âŒŞ and G2 = ⌊x,(0.3a, 0.1b),(0.7a, 0.8b)âŒŞ. Then đ?œ? = {0~, G1, G2,1~} is an IFT on X. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then A ⊆ G1 andcl(A) = G1c⊆ G1.Therefore A is an IFGCS in X. Now IFđ?›˝C(X) = {0~, 1~, đ?œ‡ a đ?œ–[0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] / either đ?œ‡ a ≼0.5 and đ?œ‡ b ≼0.5 orđ?œ‡ a< 0.3 and đ?œ‡ b<0.1, 0≤ đ?œ‡ a+đ?œˆ a ≤ 1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. Since A ⊆ G1 where G1 is an IFđ?›˝OS in X, but đ?›˝cl(A) =⌊x, (0.5a, 0.5b), (0.5a, 0.5b)âŒŞ ⊈ A, A is not an IFđ?›˝GCS.

Example 3.15: Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ. Then đ?œ? = {0~, G, 1~} is an IFT on X. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ ađ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] /0 ≤ đ?œ‡ a+đ?œˆ a ≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b ≤1}. We have A ⊆ G. As đ?›˝cl(A) = A,đ?›˝cl(A) ⊆ G, where G is an IFđ?›˝OS in X. This implies that A is an IFđ?›˝GCS in X, but not an IFGCS in X, sincecl(A) = Gc⊈ G.

Theorem 3.16:EveryIFđ?›˝CSin (X,đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof: Assume A is an IFđ?›˝CS [5] then đ?›˝cl(A) = A. Let A ⊆ U and U be an IFđ?›˝OS. Thenđ?›˝cl(A) ⊆ U, by hypothesis. Therefore A is an IFđ?›˝GCS.

Example 3.17:Let X = {a, b} and G = ⌊x, (0.5a, 0.7b), (0.5a, 0.3b)âŒŞ.Thenđ?œ? = {0~, G, 1~} is an IFT on X. LetA = ⌊x, (0.5a, 0.8b), (0.5a, 0.2b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ a đ?œ–[0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] / provided đ?œ‡ b <0.7 wheneverđ?œ‡ a≼0.5,đ?œ‡ a <0.5 whenever đ?œ‡ b ≼0.7, 0≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}.

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On intuitionistic fuzzyđ?›˝ generalized closed‌ Now A ⊆ 1~ and đ?›˝cl(A) = 1~ ⊆ 1 ~. This implies that A is an IFđ?›˝GCS in X,but not an IFđ?›˝CS,sinceint(cl(int(A))) = int(cl(G)) = int(1 ~) = 1 ~ ⊈ A.

Theorem 3.18:Every IFSPCS in (X,đ?œ?) is an IFđ?›˝GCS in (X,đ?œ?) but not conversely. Proof: Assume A is an IFSPCS[11]. Since every IFSPCS is an IFđ?›˝CS [7], by Theorem 3.16, A is an IFđ?›˝GCS.

Example 3.19:Let X = {a, b} and G = ⌊x, (0.5a, 0.4b), (0.5a, 0.6b)âŒŞ. Then đ?œ? = {0~, G, 1~} is an IFT on X. Let A = ⌊x, (0.4a, 0.3b), (0.6a, 0.7b)âŒŞ be an IFS in X. Then, IFđ?›˝C(X) = {0~, 1~, đ?œ‡ ađ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] /0≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. Here A is an IFđ?›˝CS in X. As int(cl(int(A))) = 0~ ⊆ A. Therefore A is an IFđ?›˝GCS in X. Since IFPC(X) = {0~, 1~, đ?œ‡ a đ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ– [0,1], đ?œˆ bđ?œ– [0,1] /either đ?œ‡ b≼ 0.6 or đ?œ‡ b < 0.4 whenever đ?œ‡ a≼ 0.5, 0≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b ≤ 1}. But A is not an IFSPCS in X, as we cannot find any IFPCS B such that int(B) ⊆ A ⊆ B in X. In the following diagram, we have provided relations between various types of intuitionistic fuzzy closedness.

IFRCS

IFCS

IFGCS

IFÎąCS

IFPCS

IFSCS

IFSPCS

IFβGCS

IFβCS

The reverse implications are not true in general in the above diagram.

Remark 3.20:The union of any two IFđ?›˝GCS is not an IFđ?›˝GCS in general as seen from the following example. Example 3.21:Let X = {a, b} and đ?œ? = {0~, G1, G2, 1~} whereG1 = ⌊x, (0.7a, 0.8b), (0.3a, 0.2b)âŒŞand G2 = ⌊x, (0.6a, 0.7b), (0.4a, 0.3b)âŒŞ.Then the IFSsA = ⌊x, (0.6a, 0.5b), (0.4a, 0.3b)âŒŞ and B = ⌊x, (0.4a, 0.8b), (0.4a, 0.2b)âŒŞare IFđ?›˝GCSs in (X,đ?œ?) but A âˆŞ B is not an IFđ?›˝GCS in (X,đ?œ?). Then IFđ?›˝C(X) = {0~, 1~, đ?œ‡ ađ?œ– [0,1], đ?œ‡ b đ?œ– [0,1], đ?œˆ a đ?œ–[0,1], đ?œˆ bđ?œ– [0,1] / provided đ?œ‡ b <0.7 wheneverđ?œ‡ a≼0.6, đ?œ‡ a <0.6 whenever đ?œ‡ b≼0.7, 0≤ đ?œ‡ a+đ?œˆ a≤1 and 0 ≤ đ?œ‡ b+đ?œˆ b≤ 1}. As đ?›˝cl(A) = A, we have A is an IFđ?›˝GCS in X and đ?›˝cl(B) = B, we haveB is an IFđ?›˝GCS in X. NowA âˆŞ B = ⌊x, (0.6a, 0.8b), (0.4a, 0.2b)âŒŞ ⊆ G1, where G1 is an IFđ?›˝OS, but đ?›˝cl(A âˆŞ B) = 1~ ⊈ G1.

Theorem 3.22:Let (X,đ?œ?) be an IFTS. Then for every A đ?œ– IFđ?›˝GC(X) and for every B đ?œ– IFS(X), A ⊆ B ⊆ đ?›˝cl(A) ⇒ B đ?œ– IFđ?›˝GC(X). Proof:Let B ⊆ U and U be an IFđ?›˝OS. Then since, A ⊆ B, A ⊆ U. By hypothesis, B ⊆ đ?›˝cl(A). Therefore đ?›˝cl(B) ⊆ đ?›˝cl(đ?›˝cl(A)) = đ?›˝cl(A) ⊆ U, since A is an IFđ?›˝GCS. Hence B đ?œ– IFđ?›˝GC(X).

Theorem 3.23:An IFS A of an IFTS (X,đ?œ?) is an IFđ?›˝GCS if and only ifAq c F ⇒ đ?›˝cl(A) q c F for every IFđ?›˝CS F of X. Proof: (Necessity): Let F be an IFđ?›˝CS and A q c F, then A ⊆ Fc [9], where Fc is an IFđ?›˝OS. Then đ?›˝cl(A) ⊆ Fc, by hypothesis. Hence again [9] đ?›˝cl(A) qc F. Sufficiency: Let U be an IFđ?›˝OS such that A ⊆ U. Then Uc is an IFđ?›˝CS and A ⊆ (Uc)c. By hypothesis, Aqc Uc⇒ đ?›˝cl(A) qc Uc. Hence by [9], đ?›˝cl(A) ⊆ (Uc)c = U. Therefore đ?›˝cl(A) ⊆ U. Hence A is an IFđ?›˝GCS.

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On intuitionistic fuzzyđ?›˝ generalized closed‌ Theorem 3.24:Let (X,đ?œ?) be an IFTS. Then every IFS in (X,đ?œ?) is an IFđ?›˝GCS if and only if IFđ?›˝O(X) = IFđ?›˝C(X). Proof: (Necessity):Suppose that every IFS in (X,đ?œ?) is an IFđ?›˝GCS. Let U đ?œ– IFđ?›˝O(X), and by hypothesis, đ?›˝cl(U) ⊆ U ⊆ đ?›˝cl(U). This implies đ?›˝cl(U) = U. Therefore U đ?œ– IFđ?›˝C(X). Hence IFđ?›˝O(X) ⊆ IFđ?›˝C(X). Let Ađ?œ– IFđ?›˝C(X), then Acđ?œ– IFđ?›˝O(X) ⊆ IFđ?›˝C(X). That is, Acđ?œ– IFđ?›˝C(X). Therefore Ađ?œ– IFđ?›˝O(X). Hence IFđ?›˝C(X) ⊆ IFđ?›˝O(X). Thus IFđ?›˝O(X) = IFđ?›˝C(X). Sufficiency: Suppose that IFđ?›˝O(X) = IFđ?›˝C(X). Let A ⊆ U and U be an IFđ?›˝OS. By hypothesis đ?›˝cl(A) ⊆ đ?›˝cl(U) =U, since U đ?œ– IFđ?›˝C(X). Therefore A is an IFđ?›˝GCS in X.

Theorem 3.25:If A is an IFđ?›˝OS and an IFđ?›˝GCS in (X,đ?œ?) then A is an IFđ?›˝CS in (X,đ?œ?). Proof: Since A ⊆ A and A is an IFđ?›˝OS, by hypothesis, đ?›˝cl(A) ⊆ A. But A ⊆ đ?›˝cl(A). Therefore đ?›˝cl(A) = A. Hence A is an IFđ?›˝CS.

Theorem 3.26: Let A be an IFđ?›˝GCS in (X,đ?œ?) and p(Îą,β)be an IFP in X such that int(p(Îą,β))qđ?›˝cl(A), then int(cl(int(p(Îą,β)))) q A. Proof: Let A be an IFđ?›˝GCSand let (int(p(Îą,β)))qđ?›˝cl(A). Suppose int(cl(int(p(Îą,β)))) qc A, since by [9] A ⊆ [int(cl(int(p(Îą,β))))]c.This implies [int(cl(int(p(Îą,β))))]c is an IFđ?›˝OS. Then by hypothesis, đ?›˝cl(A) ⊆[int(cl(int(p(Îą,β))))]c = cl(int(cl[(p(Îą,β))]c. ⊆ cl(cl[(p(Îą,β))]c. = cl[(p(Îą,β))]c. =(int(p(Îą,β)))c. This impliesint(p(Îą,β))qcđ?›˝cl(A),which is a contradiction to the hypothesis. Hence int(cl(int(p(Îą,β)))) q A.

Theorem 3.27:Let F ⊆ A ⊆ X where A is an IFđ?›˝OS and an IFđ?›˝GCS in X. Then F is an IFđ?›˝GCS in A if and only if F is an IFđ?›˝GCS in X. Proof: Necessity: Let U be an IFđ?›˝OS in X and F ⊆ U. Also let F be an IFđ?›˝GCS in A. Then clearly F ⊆ A ∊ U and A ∊ U is an IFđ?›˝OS in A. Hence the đ?›˝ closure of F in A, đ?›˝clA(F) ⊆A ∊ U. By Theorem 3.25, A is an IFđ?›˝CS. Therefore đ?›˝cl(A) = A and the đ?›˝ closure of F in X, đ?›˝cl(F) ⊆ đ?›˝cl(F) ∊ đ?›˝cl(A) = đ?›˝cl(F) ∊ A = đ?›˝clA(F) ⊆ A ∊ U ⊆ U. That is, đ?›˝cl(F) ⊆ U whenever F ⊆ U. Hence F is an IFđ?›˝GCS in X. Sufficiency: Let V be an IFđ?›˝OS in A such that F ⊆ V. Since A is an IFđ?›˝OS in X, V is an IFđ?›˝OS in X. Therefore đ?›˝cl(F) ⊆ V, since F is an IFđ?›˝GCS in X. Thus đ?›˝clA(F) =đ?›˝cl(F) ∊ A ⊆ V ∊ A ⊆ V. Hence F is an IFđ?›˝GCS in A.

Theorem 3.28:For an IFS A, the following conditions are equivalent: (i)

A is an IFOS and an IFđ?›˝GCS

(ii)

A is an IFROS

Proof: (i) ⇒ (ii) Let A be an IFOS and an IF�GCS. Then �cl(A) ⊆ A and A ⊆ �cl(A) this implies that �cl(A) = A. Therefore A is an IF�CS, since int(cl(int(A))) ⊆ A. Since A is an IFOS, int(A) = A. Therefore int(cl(A)) ⊆ A. Since A is an IFOS, it is an IFPOS. Hence A ⊆int(cl(A)). Therfore A = int(cl(A)). Hence A is an IFROS. (ii) ⇒ (i) Let A be an IFROS. Therefore A = int(cl(A)). Since every IFROS in an IFOS and A⊆ A. This implies int(cl(A)) ⊆ A. That is int(cl(int(A))) ⊆ A. Therfore A is an IF�CS. Hence A is an IF�GCS.

Theorem 3.29: For an IFOS A in (X,đ?œ?), the following conditions are equivalent. (i) (ii)

A is an IFCS A is an IFđ?›˝GCS and an IFQ-set

Proof: (i) ⇒ (ii) Since A is an IFCS, it is an IF�GCS. Now int(cl(A)) = int(A) = A = cl(A) = cl(int(A)), by hypothesis. Hence A is an IFQ-set[8].

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On intuitionistic fuzzy� generalized closed‌ (ii) ⇒ (i) Since A is an IFOS and an IF�GCS, by Theorem 3.28, A is an IFROS. ThereforeA = int(cl(A)) = cl(int(A)) = cl(A), by hypothesis. A is an IFCS.

Theorem3.30:Let (X,đ?œ?) be an IFTS, then for every A ∈ IFSPC(X) andfor every B in X, int(A) ⊆ B ⊆ A ⇒B ∈ IFđ?›˝GC(X). Proof: Let A be an IFSPCS in X. Then there exists an IFPCS, (say) C such that int(C) ⊆ A ⊆ C. By hypothesis, B ⊆A. Therefore B ⊆ C. Since int(C) ⊆ A, int(C) ⊆int(A) and int(C) ⊆ B, by hypothesis. Thus int(C) ⊆ B ⊆ C and by [5], B ∈ IFSPC(X). Hence by Theorem 3.18,B ∈ IFđ?›˝GC(X).

References [1] [2] [3] [4] [5] [6]

[7] [8] [9]

[10] [11] [12]

K. Atanassov. Intuitionistic Fuzzy Sets,Fuzzy Sets and Systems,1986, 87-96. C. Chang. Fuzzy Topological SpacesJ. Math. Anal. Appl., 1968, 182-190. D. Coker. An Introduction to Intuitionistic Fuzzy Topological Space,Fuzzy Sets and Systems,1997, 8189. D. Coker and M.Demirci. On Intuitionistic Fuzzy Points,Notes on Intuitionistic Fuzzy Sets1(1995), 7984. H. Gurcay, D. Coker andEs. A.Haydar.On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces, The J. Fuzzy Mathematics, 1997,365-378. D. Jayanthi.Generalized � Closed Sets in Intuitionistic Fuzzy Topological Spaces,International Journal of Advance Foundation and Research in Science & Engineering (IJAFRSE) Volume1, Issue 7, December 2014. D. Jayanthi.Relation BetweenSemipre Closed Sets andBeta Closed sets in Intuitionistic Fuzzy Topological Spaces, Notes on Intuitionistic Fuzzy Sets,Vol. 19, 2013, 53-56. R. Santhi, and D. Jayanthi. Generalized Semi- Pre Connectedness in Intuitionistic Fuzzy Topological Spaces,Annals of Fuzzy Mathematics and Informatics, 3, 2012, 243-253. S.S. Thakurand Rekha Chaturvedi. Regular Generalized Closed Sets in Intuitionistic Fuzzy Topological Spaces, Universitatea Din Bacau, Studii SiCercetariStiintifice,Seria: Mathematica, 2006, 257-272. S.S. Thakur and S. Singh.On Fuzzy Semi-pre Open Sets and Fuzzy Semi-pre Continuity, Fuzzy Sets and Systems, 1998, 383-391. Young Bae Jun &seok – Zun Song. Intuitionistic Fuzzy Semi-pre Open Sets and Intuitionistic Semi-pre Continuous Mappings, jour. of Appl. Math & computing, 2005, 465-474. L.A. Zadeh.Fuzzy Sets, Information and control,1965, 338-353.

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