Regular Generalized Semipreopen Sets in Intuitionistic Fuzzy Topological Spaces

Regular Generalized Semipreopen Sets in Intuitionistic Fuzzy Topological Spaces

Vaishnavy V

MSc Mathematics, Avinashilingam University, Coimbatore, ]

Tamil Nadu, India

Jayanthi D

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

Abstract In this paper, we introduce the notion of intuitionistic fuzzy regular generalized semipreopen sets. Furthermore, we investigate some of the properties and theoretical applications of the intuitionistic fuzzy regular generalized semipreopen sets.

Keywords Intuitionistic fuzzy topology, Intuitionistic fuzzy semipreopen sets, Intuitionistic fuzzy regular generalized semipreclosed sets, Intuitionistic fuzzy regular generalized semipreopen sets.

1. INTRODUCTION

   The establishment of fuzzy sets was made by Zadeh [12] in 1965. Later the introduction of fuzzy topology was given by Chang [2] in 1967. This was followed by the introduction of intuitionistic fuzzy sets by Atanassov [1]. Using this notion, Coker [3] constructed the basic concepts of intuitionistic fuzzy topological spaces. Subsequently this was followed by the introduction of intuitionistic fuzzy regular generalized semipreclosed sets by Vaishnavy V and Jayanthi D [10] in 2015. We now extend our idea towards intuitionistic fuzzy regular generalized semipreopen sets and study some of their properties and applications.

2. PRELIMINARIES

Definition 2.1 [1]: An intuitionistic fuzzy set (IFS in short) A is an object having the form

A={x, A(x), A(x) : x X}

where the function A : X [0,1] and A : X [0,1] denote the degree of membership (namely A(x)) and the degree of non membership (namely A(x)) of each element x X to the set A, respectively, and 0 A(x) + A(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, A, A instead of denoting A={x, A(x), A(x) : x X}.

Definition 2.2 [1] : Let A and B be two IFSs of the form A={x, A(x), A(x) : x X}

and

B={x, B(x), B(x) : x X}.

Then,

1. A B if and only if A(x) B(x) and A(x) B(x) for all x X;

2. A = B if and only if A B and A B;

(c) Ac = {x, A(x), A(x) : x X};

(d) A B = {x, A(x) B(x), A(x) B(x) : x X};

(e) AB = {x, A(x) B(x), A(x) B(x) : x X}.

The intuitionistic fuzzy 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] : An intuitionistic fuzzy topology (IFT in short) on X is a family of IFSs in X satisfying the following axioms:

(i) 0 , 1

1. G1 G2 for any G1 ,G2

2. Gi for any family {Gi : i J} .

In this case the pair (X, ) is called the intuitionistic fuzzy topological space (IFTS in short) and any IFS in is known as an intuitionistic fuzzy open set (IFOS in short) in

X. The compliment Ac of an IFOS A in IFTS (X,) is called an intuitionistic fuzzy closed set (IFCS in short) in X.

Definition 2.4 [3] : Let (X, ) be an IFTS and A= x, A, A be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by

int(A) = {G | G is an IFOS in X and G A} cl(A) = {K | K is an IFCS in X and A K}.

Note that for any IFS A in (X, ), we have cl(Ac) = (int(A))c and int(Ac) = (cl(A))c.

Definition 2.5 [4] : An IFS A = x, A, A in an IFTS (X,) is said to be an

1. intuitionistic fuzzy semi open set (IFSOS in short) if A cl(int(A))

2. intuitionistic fuzzy pre open set (IFPOS in short) if A int(cl(A))

3. intuitionistic fuzzy open set (IFOS in short) if A int(cl(int(A)))

4. intuitionistic fuzzy open set (IFOS in short) if A cl(int(cl(A))).

Definition 2.6 [11] : An IFS A = x, A, A in an IFTS (X,) is said to be an intuitionistic fuzzy semi-pre open set (IFSPOS in short) if there exists an IFPOS B such that B A cl(B).

Definition 2.7 [9] : An IFS A is an

1. intuitionistic fuzzy regular closed set (IFRCS in short) if A = cl(int(A))

2. intuitionistic fuzzy regular open set (IFROS in short) if A= int(cl(A))

3. intuitionistic fuzzy generalized closed set (IFGCS in short) if cl(A) U whenever A U and U is an IFOS

4. intuitionistic fuzzy regular generalized closed set (IFRGCS in short) if cl(A) U whenever A U and U is IFROS.

Definition 2.8 [5] Let A be an IFS in an IFTS (X, ). Then the semi-pre interior and the semi-pre closure of A are defined as

spint (A) = {G | G is an IFSPOS in X and G A}, spcl (A) = {K | K is an IFSPCS in X and A K}.

Definition 2.9 [7] An IFTS (X,) is said to be an IFT1/2 space if every IFGCS in (X,) is an IFCS in (X,).

Definition 2.10 [10] An IFS A in an IFTS (X,) is said to be an intuitionistic fuzzy regular generalized semipreclosed set (IFRGSPCS in short) if spcl(A) U whenever A U and U is an IFROS in (X,).

III. INTUITIONISTIC FUZZY REGULAR GENERALIZED SEMIPREOPEN SETS

In this section we introduce the notion of intuitionistic fuzzy regular generalized semipreopen sets and study some of their properties.

Definition 3.1 The complement Ac of an IFRGSPCS A in an IFTS (X,) is called an Intuitionistic fuzzy regular generalized semipreopen set (IFRGSPOS in short) in X.

The family of all IFRGSPOSs of an IFTS (X,) is denoted by IFRGSPO(X).

Theorem 3.2 Every IFOS, IFGOS, IFSOS, IFPOS, IFSPOS,

IFOS, IFOS, IFROS is an IFRGSPOS but the converses are not true in general.

Proof: Straightforward.

Example 3.3 Let X = {a, b} and G = x, (0.5, 0.4), (0.5,

0.6) where G(a) = 0.5, G(b) = 0.4, G(a) = 0.5, G(b) = 0.6.

Then = {0~, G, 1~} is an IFT on X. LET A = x, (0.6, 0.7), (0.4, 0.2) be an IFS in X. Then, IFPC(X) = {0~, 1~,

a [0,1], b [0,1], a [0,1], b [0,1] / either b 0.6 or b < 0.4 whenever a 0.5, a + a 1 and b + b 1}. Therefore, IFSPC(X) = {0~, 1~, a [0,1], b [0,1], a [0,1], b [0,1] / a + a 1 and b + b 1}. As spcl(Ac) = Ac, we have Ac Gc implies spcl(Ac) Gc, where G is an IFROS in X. This implies that Ac is an IFRGSPCS in X and hence A is an IFRGSPOS. Now since int(A) = G A, A is not an IFOS in X. Also Gc A but Gc int(A). Therefore A is not an IFGOS in X. Now int(cl(A)) = int(1~) = 1~ A. Therefore A is not an IFROS in

X. Hence A is an IFRGSPOS but not IFOS, IFGOS, IFROS.

Example 3.4 Let X = {a, b} and G = x, (0.5, 0.6), (0.5,

0.4) where G(a) = 0.5, G(b) = 0.6, G(a) = 0.5, G(b) = 0.4.

Then = {0~, G, 1~} is an IFT on X. Let A = x, (0.5, 0.3), (0.5, 0.7) be an IFS in X. Then, IFPC(X) = {0~, 1~,

a [0,1], b [0,1], a [0,1], b [0,1] / b< 0.6 whenever a 0.5, a < 0.5 whenever b 0.6, a + a 1 and b + b 1}. Therefore, IFSPC(X) = {0~, 1~, a [0,1], b [0,1], a [0,1], b [0,1] / b < 0.6 whenever a 0.5, a < 0.5 whenever b 0.6, a + a 1 and b + b 1}. As spcl(Ac) = 1~, we have Ac 1~ implies spcl(Ac) 1~, where 1~ is an IFRCS. This implies that Ac is an IFRGSPCS in X. Hence A is an IFRGSPOS in X. Now since A int(cl(A)) = int(Gc) = 0~, we get A is not an IFPOS in X. Further A cl(int(cl(A))) = cl(int(Gc)) = cl(0~) = 0~. Hence A is not an IFOS in X. Also A cl(int(A)) = cl(0~) = 0~. Thus A is not an IFSOS in X. Now since A int(cl(int(A))) = int(cl(0~)) = int(0~) = 0~,A is not an IFOS in X. Further there exists no IFPOS B such that A B cl(A). Therefore A is not an IFSPOS in X. Hence A is an IFRGSPOS but not IFPOS, IFOS, IFSOS, IFOS, IFSPOS.

Theorem 3.5 Let (X, ) be an IFTS. Then for every A IFRGSPO(X) and for every B IFS(X), spint(A) B A B IFRGSPO(X).

Proof: Let A be any IFRGSPOS of X and B be any IFS of

X. Let spint(A) B A. Then Ac is an IFRGSPCS and Ac Bc spcl(Ac). Therefore Bc is an IFRGSPCS[10] which implies B is an IFRGSPOS in X. Hence B IFRGSPO(X).

Theorem 3.6 Let (X, ) be an IFTS. Then for every A IFS(X) and for every B IFPO(X), B A cl(int(B)) A IFRGSPO(X).

Proof: Let B be an IFPOS. Then B int(cl(B)). By hypothesis, A cl(int(B)) cl(int(int(cl(B)))) = cl(int(cl(B))) cl(int(cl(A))) as B A. Therefore A is an IFOS and by Theorem 3.2, A is an IFRGSPOS. Hence A

IFRGSPO(X).

Theorem 3.7 An IFS A of an IFTS (X, ) is an IFRGSPOS if and only if F spint(A) whenever F is an IFRCS and F A. Proof: Necessity: Suppose A is an IFRGSPOS. Let F be an IFRCS such that F A. Then Fc is an IFROS and Ac Fc. By hypothesis Ac is an IFRGSPCS, we have spcl(Ac) Fc. Therefore F spint(A).

Sufficiency: Let F be an IFRCS such that F A, then F spint(A). That is (spint(A))c Fc. This implies spcl(Ac) Fc where Fc is an IFROS. Therefore Ac is an IFRGSPCS. Hence A is an IFRGSPOS.

Theorem 3.8 Let (X, ) be an IFTS then for every A IFSPO(X) and for every IFS B in X, A B cl(A) B IFRGSPO(X).

Proof: Let A be an IFSPOS in X. Then by Definition 2.6, there exists an IFPOS, say C such that C A cl(A). By hypothesis, A B. Therefore C B. Since A cl(C), cl(A) cl(C) and B cl(C). Thus C B cl(C). This implies B is an IFSPOS in X. Then by Theorem 3.2, B is an IFRGSPOS. That is B IFRGSPO(X).

IV. APPLICATIONS

The concept of intuitionistic fuzzy semipre T1/2 space was introduced by Santhi, R. and Jayanthi, D [7] in 2009. In this section we have discussed some applications of intuitionistic fuzzy regular generalized semipreclosed sets.

Definition 4.1 If every IFRGSPCS in (X, ) is an IFSPCS in (X, ), then the space can be called as an intuitionistic fuzzy regular semipre T1/2 space (IFRSPT1/2 in short).

1. (iii) Let int(cl(int(A))) A. Then int(A) int(int(cl(int(A)))) = int(cl(int(A))) int(int(A)) = int(A). Therefore int(cl(int(A))) = int(A). Hence int(A) IFRO(X).

2. (i) Since int(A) is an IFROS, int(A) = (int(cl(int(A))) and since int(A) (A), int(cl(int(A))) A. Therefore A is an IFCS which implies Ac is an IFOS. Hence by Theorem 3.2, Ac is an IFRGSPOS. Therefore A IFRGSPC(X).

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Definition 4.7 An IFTS (X, ) is said to be an intuitionistic

Theorem 4.2 An IFTS (X, ) is an IFRSPT1/2 space if and only if IFSPO(X) = IFRGSPO(X).

Proof: Necessity: Let A be an IFRGSPOS in (X, ), then Ac

fuzzy regular semipre T* space (IFRSPT*

every IFRGSPCS is an IFRCS in (X, ).

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in short) if

is an IFRGSPCS in (X, ). By hypothesis, Ac is an IFSPCS in

Remark 4.8 Every IFRSPT* space is an IFRSPT

1/2 space

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(X, ) and therefore A is an IFSPOS in (X, ). Hence IFSPO(X) = IFRGSPO(X).

Sufficiency: Let A be an IFRGSPCS in (X, ). Then Ac is an IFRGSPOS in (X, ). By hypothesis Ac is an IFSPOS in (X, ) and therefore A is an IFSPCS in (X, ). Hence (X, ) is an IFRSPT1/2 space.

Remark 4.3 Not every IFRSPT1/2 space is an IFT1/2 space. This can be seen easily by the following example.

Example 4.4 Let X = {a, b} and G = x, (0.5, 0.4), (0.5,

0.6) where G(a) = 0.5, G(b) = 0.4, G(a) = 0.5, G(b) = 0.6.

Then = {0~, G, 1~} is an IFT on X. Then, IFPC(X) = {0~, 1~, a [0,1], b [0,1], a [0,1], b [0,1] / either b

0.6 or b < 0.4 whenever a 0.5, a + a 1 and b + b 1}. Therefore, IFSPC(X) = {0~, 1~, a [0,1], b [0,1],

[0,1], [0,1] / + 1 and + 1}. Since all

but not conversely.

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Proof: Let (X, ) be an IFRSPT* space. Let A be an IFRGSPCS in (X, ). By hypothesis, A is an IFRCS. Since every IFRCS is an IFSPCS, A is an IFSPCS in (X, ). Hence (X, ) is an IFRSPT1/2 space.

Example 4.9 Let X = {a, b} and G = x, (0.5, 0.4), (0.5,

0.6) where G(a) = 0.5, G(b) = 0.4, G(a) = 0.5, G(b) = 0.6.

Then = {0~, G, 1~} is an IFT on X. Then, IFPC(X) = {0~, 1~, a [0,1], b [0,1], a [0,1], b [0,1] / either b

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0.6 or b < 0.4 whenever a 0.5, a + a 1 and b + b 1}. Therefore, IFSPC(X) = {0~, 1~, a [0,1], b [0,1], a [0,1], b [0,1] / a + a 1 and b + b 1}. Since all IFRGSPCS in X are IFSPCS, (X, ) is an IFRSPT* space since if A = x, (0.5, 0.7), (0.5, 0.3), then spcl(A) = A 1~

whenever A 1~. Hence A is an IFRGSPCS in X but since cl(int(A)) = cl(G) = Gc A, A is not an IFRCS in X.

a b a a b b

Therefore (X, ) is not an IFRSPT* space.

IFRGSPCS in X are IFSPCS in X, (X, ) is an IFRSPT1/2 space. But it is not a IFT1/2 space since if A = x, (0.5, 0.7), (0.5, 0.3), then cl(A) = 1~ 1~ whenever A 1~. Hence A is an IFGCS in X but cl(A) = 1~ A, A is not an IFCS in X. Therefore (X, ) is not an IFT1/2 space.

Theorem 4.5 Let (X, ) be an IFTS and X is an IFRSPT1/2 space, then the following conditions are equivalent:

1. A IFRGSPO(X)

2. A cl(int(cl(A)))

3. cl(A) IFRC(X).

Proof: (i) (ii) Let A be an IFRGSPOS. Then since X is an IFRSPT1/2 space, A is an IFSPOS. Since every IFSPOS is an IFOS [5] we get, A cl(int(cl(A))).

1. (iii) Let A cl(int(cl(A))). Then cl(A) cl(cl(int(cl(A)))) = cl(int(cl(A))) cl(cl(A)) = cl(A). Therefore cl(A) = cl(int(cl(A))). Hence cl(A) IFRC(X).

2. (i) Since cl(A) is an IFRCS, cl(A) = cl(int(cl(A))) and since A cl(A), A cl(int(cl(A))). Therefore A is an IFOS. Hence by Theorem 3.2, A IFRGSPO(X).

Theorem 4.6 Let (X, ) be an IFTS and X is an IFRSPT1/2

space, then the following conditions are equivalent:

1. A IFRGSPC(X)

2. int(cl(int(A))) A

3. int(A) IFRO(X).

Proof: (i) (ii) Let A be an IFRGSPCS. Then since X is an IFRSPT1/2 space, A is an IFSPCS. Since every IFSPCS is an IFCS [5] we get, int(cl(int(A))) A.

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