On Intuitionistic Fuzzy Semi-Generalized Closed. Sets and its Applications

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1 Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 34, On Intuitionistic Fuzzy Semi-Generalized Closed Sets and its Applications R. Santhi Department of Mathematics, Nallamuthu Gounder Mahalingam College, Pollachi , India santhifuzzy@yahoo.co.in K. Arun Prakash Department of Mathematics, Kongu Engineering College Perundurai , India Abstract. The purpose of this paper is to introduce and study the concepts of intuitionistic fuzzy semi-generalized closed set and intuitionistic fuzzy semigeneralized open sets in intuitionistic fuzzy topological space. Also we study the application of intuitionistic fuzzy semi-generalized closed sets namely intuitionistic fuzzy semi-t 1/2 space. Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy semi-generalized closed set, Intuitionistic fuzzy semi-generalized open set and Intuitionistic fuzzy semi-t 1/2 space. Mathematics Subject Classification: 54A40, 03F55 1. Introduction Fuzzy set (FS), proposed by Zadeh [13] in 1965, as a framework to encounter uncertainty, vagueness and partial truth, represents a degree of membership for each member of the universe of discourse to a subset of it. By adding the degree of non-membership to FS, Atanassov proposed intuitionistic fuzzy set (IFS) in 1983 [1] which looks more accurately to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. Later on fuzzy topology was introduced by Chang in 1967.After this, there have been several generalizations of notions of fuzzy sets and fuzzy topology. In last few years various concepts in

2 1678 R. Santhi and K. Arun Prakash fuzzy were extended to intuitionistic fuzzy sets. In 1997, Coker introduced the concept of intuitionistic fuzzy topological space. In this paper, we introduce one of the concept namely semi-generalized closed sets which were introduced initially by P.Bhattacharyya and B.K.Lahiri in General Topology in 1987 [2]. After this many researchers worked on this set and developed many interesting properties and applications. We have studied some of the basic properties regarding it. We also introduced application of intuitionistic fuzzy semigeneralized closed sets namely intuitionistic fuzzy semit 1/2 space and obtained some characterizations and several preservation theorems of such spaces. 2. Preliminaries Definition 2.1 [1] An intuitionistic fuzzy set (IFS) A in X is an object having the form A= { x, μ A (x), γ A (x) / x X } the functions μ A : X [0,1] and γ A : X [0,1] denote the degree of the membership (namely μ A (x)) and the degree of non- membership (namely γ A (x) ) of each element x X to the set A respectively, 0 μ A (x) + γ A (x) 1 for each x X. Definition 2.2 [1] Let A and B be IFS s of the forms A = { x, μ A (x), γ A (x) / x X} and B= { x, μ B (x), γ B (x) / x X}. Then (a) A B if and only if μ A (x) μ B (x) and γ A (x) γ B (x) for all x X. (b) A = B if and only if A B and B A. (c) Ā = { x, γ A (x), μ A (x) / x X}. (d) A B = { x, μ A (x) μ B (x), γ A (x) γ B (x) / x X}. (e) A B = { x, μ A (x) μ B (x), γ A (x) γ B (x) / x X}. (f) 0 = { x,0,1 : x X } and 1 = { x,1,0 : x X }. ~ ~ (g) 1 ~ = 0 and 0 1 ~ ~ =. ~ Definition 2.3 [5] An intuitionistic fuzzy point (IFP), written as p (α,β), is defined to be an IFS of X given by ( α, β ), x = p p( α, β ) = 0,1, otherwise ( ) Definition 2.4 [4] An intuitionistic fuzzy topology (IFT for short) on a nonempty X is a family τ of IFSs in X satisfying the following axioms: (i) 0 ~, 1 ~ τ, (ii) G 1 G 2 τ for any G 1, G 2 τ, (iii) G i τ, for any arbitrary family {G i i J} τ.

3 Intuitionistic fuzzy semi-generalized closed sets 1679 In this case the pair (X,τ) is called an intuitionistic fuzzy topological space (IFTS for short) and any IFS in τ is known as an intuitionistic fuzzy open set (IFOS for short) in X. The complement of an IFOS A in IFTS (X,τ) is called an IFCS ( IFCS for short ) in X. Definition 2.5 [4] Let (X,τ) be an IFTS and A = x,μ A,γ A be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A 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,τ), we have cl A int A int A = cl A ( ) = ( ) and ( ) ( ) Definition 2.6 An IFS A = { x, μ A (x), γ A (x) / x X} in an IFTS (X,τ) is called (i) an intuitionistic fuzzy semi open set (IFSOS) if A cl( int( A) ). [6] (ii) an intuitionistic fuzzy α-open set (IFαOS) if A int( cl( int( A) )). [6] (iii) an intuitionistic fuzzy pre-open set (IFαOS) if A int( cl( A) ). [6] (iv) an intuitionistic fuzzy regular open set (IFROS) if int ( cl ( A) ) = A. [6] (v) an intuitionistic fuzzy semi-pre open set (IFSPOS) if there exists B IFPO (X), such that B A cl( B). [11] (vi) an intuitionistic fuzzy generalized open set (IFGOS), if U int( A), whenever U A and U is an IFCS [9] (vii) an intuitionistic fuzzy regular generalized open set (IFRGOS), if U int( A), whenever U A and U is an IFRCS.[9] An IFS A is called intuitionistic fuzzy semiclosed set, intuitionistic fuzzy α-closed set, intuitionistic fuzzy pre-closed set, intuitionistic fuzzy regular closed set, intuitionistic fuzzy semi-pre-closed set, intuitionistic fuzzy generalized-closed set and intuitionistic fuzzy regular generalized-closed set (IFCS, IFαCS, IFPCS, IFRCS, IFSPCS, IFGCS and IFRGCS resp.), if the complement of is an IFSOS, IFαOS, IFPOS, IFROS, IFSPOS, IFGOS and IFRGOS respectively. The family of all intuitionistic fuzzy semi-open ( resp. intuitionistic fuzzy α-open, intuitionistic fuzzy pre-open, intuitionistic fuzzy regular-open, intuitionistic fuzzy semi-preopen, intuitionistic fuzzy generalized-open and intuitionistic fuzzy regular generalized-open ) sets of an IFTS (X,τ ) is denoted by IFSO(X) ( resp. IFαO(X), IFPO(X), IFRO(X), IFSPO(X), IFGO(X) and IFRGO(X) ). Definition.2.7 [5] An IFTS(X,τ) is said to be intuitionistic fuzzy T 1/2 space if every intuitionistic fuzzy g-closed set in X is intuitionistic fuzzy closed in X.

4 1680 R. Santhi and K. Arun Prakash Definition.2.8 [9] An IFTS(X,τ) is said to be intuitionistic fuzzy regular-t 1/2 space if every intuitionistic fuzzy rg-closed set in X is intuitionistic fuzzy regular closed in X. Definition.2.9 [10] Two IFSs A and B in an IFTS (X,τ) are called q-separated if 0 ~. 3. Intuitionistic fuzzy semi-generalized closed sets In this section, we introduce intuitionistic fuzzy semigeneralized closed sets and studied some of their basic properties. Definition 3.1 An IFS A of an IFTS (X, τ) is called an intuitionistic fuzzy semi-generalized closed (IFSGCS for short) set if sc1(a) U, whenever A U and U is an IFSOS. The complement of an IFSGCS is called an intuitionistic fuzzy semigeneralized open set (IFSGOS for short).the family of all IFSGCS (resp, IFSGOS) sets of an IFTS (X,τ) will be denoted by IFSGC(X) (resp. IFSGO(X)) Example 3.2 Let X= {a, b, c} Let.....,... Then τ = {0 ~, 1 ~, G 1, G 2 } is an IFT on X. 0 ~,1 ~,,,,, ; 0.3,1 0.1,1 0.4,1 0,0.3, 0,0.35 0, ,2,3,,,,,, 0 ~,1 ~,,,,, ; 0,0.3 0,0.35 0, ,1 0.1,1, 0.4,1 1 1,2,3 α 1,β H 1 α 2,β 2,α 3,β 3 a a, b, c =, b, c a, b, c α 1 α 2 α 3 β 1 β 2 β 3 Let A, be any IFS in X Then sc1 (A) = A.Hence A is an IFSGCS in X. Theorem 3.3 Every IFCS in (X,τ) is an IFSGCS, but not conversely. Proof: Obvious. Example 3.4 Let X = {a, b} and let Ax

5 Then τ = {0 ~, 1 ~, A} is an IFT on X. Intuitionistic fuzzy semi-generalized closed sets ~,1 ~,,, ; 0.3,1 0.4,1 0,0.2 0, ,2,,, 0 ~,1 ~,,, ; 0,0.2 0, ,1 0.4,1 1 1,2,, Let be any IFS in X. Then. Clearly, whenever, for all IFSOS U in X.Therefore B is an IFSGCS, but not an IFCS in X. Theorem 3.5 Every IFSCS in (X,τ) is an IFSGCS. Proof: Obvious. The converse of the Theorem.3.5 is not true as seen from the following example. Example 3.6 Let X = {a, b} and let Ax Then τ = {0 ~, 1 ~, A} is an IFT on X. 0 ~,1 ~,,, ;, 0.4,0.6, 0.4,0.6,, 1 1,2, 0 ~,1 ~,,, ;, 0.4,0.6, 0.4, ,2,, Let.., be any IFS in X.. Then. Clearly, whenever,... for all IFSOS U in X. Therefore B is an IFSGCS in X, but BIFSCS(X). Theorem 3.7 Every IFαCS in an (X,τ) is an IFSGCS, but not conversely. Proof: In [4], it has been proved that every IFαCS is an IFSCS. By Theorem 3.5, every IFSCS is an IFSGCS. Hence every IFαCS is an IFSGCS in X. Example 3.8 Let X = {a, b} and let A x.... Then τ = {0 ~, 1 ~, A} is an IFT on X.

6 1682 R. Santhi and K. Arun Prakash 0 ~,1 ~,,, ;, 0.4,0.6, 0.4,0.6,, 1 1,2, 0 ~,1 ~,,, ;, 0.4,0.6, 0.4, ,2,, Let.., be any IFS in X.. Clearly, whenever, for all IFSOS U in X. Therefore B is an IFSGCS in X., intclb inta A, cl intclb cla A. Clearly cl intclb A B.Hence B is not an IFαCS in X. Theorem 3.9 Every IFSGCS in an IFT (X,τ) is an IFSPCS, but not conversely. Proof: Follows from definitions. Example 3.10 Let X = {a, b} and let Ax Then τ = {0 ~, 1 ~, A} is an IFT on X. 0 ~,1 ~,,, ;, 0.1,0.4, 0.3, ,2,, Where, 0 ~,1 ~,,, ;, 0.3,0.5, 0.1, ,2 Where,, Let be any IFS in X. 0 ~,, 0 ~.. Hence B is an IFSPCS in X. sclb 1 ~, B. But sclb. Therefore B is not an IFSGCS in X. Theorem 3.11 Every IFRCS in an IFT (X,τ) is an IFSGCS, but not conversely. Proof: In [4], it has been proved that every IFRCS is an IFCS. By Theorem 3.3, every IFCS is an IFSGCS. Hence every IFRCS is an IFSGCS.

7 Intuitionistic fuzzy semi-generalized closed sets 1683 Example 3.12 In Example 3.6, the IFS B is an IFSGCS but not an IFRCS in X, since,. Remark 3.13 The concepts of IFGCS and IFSGCS are independent of each other as seen from the following two examples. Example 3.14 In Example 3.8, the IFS B is not an IFSGCS but an IFGCS in X, since 1 ~, B U, U is an IFOS, clb U. Hence B is an IFGCS in X. Example 3.15 In Example 3.10, the IFS B is an IFSGCS but not an IFRCS in X, since, B A, clb A. Hence B is an not IFGCS in X. Remark 3.16 The concepts of IFPCS and IFSGCS are independent of each other as seen from the following two examples. Example 3.17 Let X = {a, b} and let Ax Then τ = {0 ~, 1 ~, A} is an IFT on X. 0 ~,1 ~,,, ;, 0.2,0.4, 0.3, ,2,, Where, 0 ~,1 ~,,, ;, 0.3,0.6, 0.2, ,2 Where,, Let be any IFS in X. 1 ~ only, also 1 ~. Therefore B is an IFSGCS in X. 0 ~, 1 ~. Clearly. Therefore B is not an IFPCS in X. Example 3.18 In Example 3.17, consider an IFS.. 0 ~, 0 ~ 0 ~. Therefore C is an IFPCS in X. 1 ~, but. Hence C is not an IFSGCS in X.... Remark 3.19 The concepts of IFRGCS AND IFSGCS are independent of each other as seen from the following two examples.

8 1684 R. Santhi and K. Arun Prakash Example 3.20 In Example 3.17, consider an IFS B.. cl 1 ~ 1 ~ 1 ~. Therefore B is an IFRGCS in X ~, but. Hence B is an IFRGCS but B is not an IFSGCS in X. Example 3.21 Let X = {a, b} and let Ax Then τ = {0 ~, 1 ~, A} is an IFT on X. 0 ~,1 ~,,, ; 0.5,1 0.3,1 0,0.2 0, ,2,,, 0 ~,1 ~,,, ; 0,0.2 0, ,1 0.3,1 1 1,2,, Let be any IFS in X.... Then. Clearly, whenever, for all IFSOS U in X.Therefore B is an IFSGCS in X. cl, A is an IFROS. Therefore B is not an IFRGCS in X. Theorem 3.22 If A is an IFSGCS and A B scla, then B is an IFSGCS. Proof: Let U be an IFSOS such that B U. Since A B, A U. But A is an IFSGCS, so sc1 (A) U. By hypothesis, B sc1 (A), then sc1 (B) sc1 (A). This implies sc(b) U. Hence is an IFSGCS. Theorem 3.23 If sint(a) B A and A is an IFSGOS in an IFTS (X,τ).Then B is an IFSGOS. Proof: If, then. Since is an IFSGCS, then by Theorem 3.22, B is also an IFSGCS. Therefore B is an IFSGOS. The following diagram shows the relationships between IFSGCSs and some other sets. The reverse implications are not true in the following diagram.

9 Intuitionistic fuzzy semi-generalized closed sets 1685 IFGCS IFRGCS IFSGCS IFRCS IFCS IFαCS IFSCS IFSPCS IFPCS Theorem 3.24 An IFS A of an IFTS(X,τ) is an IFSGOS if and only if B sint(a), whenever B is an IFSCS and B A. Proof: Necessity: Assume that A is an IFSGOS in X. Also let B be an IFSCS in X, such that B A. Then is an IFSOS in X, such that A B. Since is an IFSGCS. But. Hence. This implies. Sufficiency: Assume that, whenever B is an IFSCS and B A. Then, whenever is an IFSOS and. Therefore is an IFSGCS.This implies A is an IFSGOS. Theorem 3.25 The concepts of IFSCS and intuitionistic fuzzy semi-open coincide if and only if every intuitionistic fuzzy subset of X is an IFSGCS. Proof: Sufficiency: Let A be an intuitionistic fuzzy subset of X, such that A U, U is an IFSOS. Then U is IFSCS such that sc1 (A) sc1 (U) = U. Hence sc1 (A) U. Therefore A is an IFSGCS. Necessity: Let U be an IFSOS. By hypothesis every intuitionistic fuzzy subset of X is an IFSGCS, therefore U is an IFSGCS. This shows that scl (U) U. Thus U is an IFCS. Next let us assume that B is an IFCS. Then is an IFSOS. By hypothesis is an IFSGCS, hence. This implies is an IFSCS. Therefore B is an IFSOS in X. Theorem 3.26 Let A be an IFSGCS subset of (X,τ). Then scl (A)-A does not contain any non-empty IFSCS. Proof: Assume that A is IFSGCS. Let F be an non-empty IFSCS, such that. (i.e). Therefore A. Since is an IFSOS,. But we have. So (i.e) φ.therefore F is empty. Theorem 3.27 Let A be an IFSGCS in an IFTS (X,τ). Then scl(a) - A contains no nonempty IFCS. Proof: Follows from the Theorem 3.26.

10 1686 R. Santhi and K. Arun Prakash Corollary 3.28 An IFSGCS A is an IFSCS if and if only if scl(a) - A is an IFSCS. Proof: Let A be an IFSGCS. If A is an IFSCS, then by Theorem 3.26, scl (A) - A=φ. Therefore scl (A) - A is an IFSCS. Assume that scl (A) - A is IFSCS. But A is an IFSGCS. By Theorem 3.21, scl (A) - A=φ. Hence scl (A) = A. A is an IFSCS. Theorem 3.29 If A is an IFSOS and IFSGCS in (X,τ), then A is an IFSCS in X. Proof: Since A is an IFSOS and IFSGCS in X, scl (A) A. But we have A scl(a). This implies scl(a) =A. Therefore A is an IFSCS in X. 4. Applications of Intuitionistic fuzzy semi-generalized closed sets In this section, we introduce intuitionistic fuzzy T 1/2 space, which utilizes intuitionistic fuzzy semi-generalized closed sets and its characterizations are proved. Definition 4.1 An IFTS (X, τ) is called an intuitionistic fuzzy semi-t 1/2 space, if every IFSGCS is an IFSCS. Theorem 4.2 An IFTS (X, τ) is an intuitionistic fuzzy semi-t 1/2 space if and only if IFSOS(X) = IFSGOS(X). Proof: Let A be an IFSGOS in X, then is an IFSGCS in X. By hypothesis is an IFSCS of X, and therefore A is an IFSOS of X. Hence IFSOS(X) = IFSGOS(X). Converse: Let A be an IFSGCS in X, then is an IFSGOS of X. By our assumption is an IFSOS in X, which in turn implies A is an IFSCS in X. Hence (X, τ) is an intuitionistic fuzzy semi-t 1/2 space. Theorem 4.3 For an IFTS(X,τ) the following conditions are equivalent: (i) (X,τ) is an intuitionistic fuzzy semi-t 1/2 space. (ii) Every singleton set of X is either an IFSCS or IFSOS. Proof: (i) => (ii). Assume that (X,τ) is an intuitionistic fuzzy semi-t 1/2 space. Suppose that {x} is not an IFSCS for some x X. Then X-{x} is not IFSOS and hence X is the only IFSOS containing X-{x}.Therefore, X-{x} is IFSGCS in (X,τ).Since (X,τ) is an intuitionistic fuzzy semi-t 1/2 space, then X-{x} is IFSCS or equivalently {x} is an IFSOS. (ii) => (i). Assume that every singleton set of X is either IFSCS or IFSOS. Let A be an IFSGCS of (X,τ). Let x X. We show that x A in two cases. Case(i): Assume that {x} is IFSCS. If x A, then x scl (A)-A. Now scl (A)- A contains a non - empty IFSCS. Since A is IFSGCS, by Theorem 3.21, we arrived to a contradiction. Hence x A.

11 Intuitionistic fuzzy semi-generalized closed sets 1687 Case (ii): Assume that {x}is IFSOS. Since x scl(a), then {x} A φ. So x A. Thus in any case x A. So scl(a) A. Therefore A = scl (A) or equivalently A is an IFSCS. Thus every IFSGCS is IFCS. Hence (X,τ) is intuitionistic fuzzy semi- T 1/2 space. Theorem 4.4 Every intuitionistic fuzzy T 1/2 space is an intuitionistic fuzzy semi- T 1/2 space. The following example supports that an intuitionistic fuzzy semi-t 1/2 space need not be an intuitionistic fuzzy T 1/2 space. Example 4.5 Let X ={a, b} Let Ax Then τ ={0 ~,1 ~,A} be an IFT on X. Clearly (X,τ) is an intuitionistic fuzzy semi- T 1/2 space, but not an intuitionistic fuzzy T 1/2 space. Theorem.4.6. Every intuitionistic fuzzy regular T 1/2 space is an intuitionistic fuzzy semi-t 1/2 space. Proof: In ([5].Remark.5.3), it has been proved that every intuitionistic fuzzy regular T 1/2 space is an intuitionistic fuzzy T 1/2 space. ByTheorem.4.3, every intuitionistic fuzzy T 1/2 space is an intuitionistic fuzzy semi-t 1/2 space. Hence every intuitionistic fuzzy regular T 1/2 space is an intuitionistic fuzzy semi-t 1/2 space Example 4.7 Let X ={a,b}and let Ax Then τ ={0 ~,1 ~,A} be an IFT on X. Clearly (X,τ) is an intuitionistic fuzzy semi- T 1/2 space, but not an intuitionistic fuzzy regular T 1/2 space. Theorem 4.9 Let (X, τ) be an IFTS and X is an intuitionistic fuzzy semi-t 1/2 space, then the following conditions are equivalent (i) AIFSGOX (ii) A clinta (iii) A IFRCX. Proof: (i) (ii): Let A be an IFSGOS in X. Since X is an intuitionistic fuzzy semi-t 1/2 space, A is an IFSOS in X. Hence by definition of IFSOS, A clinta. (ii) (iii): Assume that A clinta, then A clinta. Hence A is an IFRCS and A IFRCX. (iii) (i): Assume that, then A clinta. Since A clinta, A is an IFSOS and hence A is an IFSGOS, therefore. Theorem 4.10 Let A and B be q-separated IFSGOS s in (X, τ), then A B is also an IFSGOS in X is an intuitionistic fuzzy semi T 1/2 space. Proof: Let F be an IFSCS such that F A B. Then F cla A, since B cla 0 ~. Since A is an IFSOS in X, F cla sinta. Similarly F clb sintb. Now F FA B F cla FclB sinta

12 1688 R. Santhi and K. Arun Prakash sintb sintab. Therefore F sintab. Hence by theorem.3.19, A B is an IFSGOS. Theorem 4.11 Let A and B be two IFSGCS of an ITFS (X, τ) and suppose that and are q-separated. Then is an IFSGCS if X is an intuitionstic fuzzy semi-t 1/2 space. Proof: Assume that A and B are IFSGCS, then and are q-separated IFSGOS. By theorem 4.10, is an IFSGOS. Hence A B is an IFSGOS, which in turn implies that AB is an IFSGOS in X. References [1] K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), [2] P.Bhattacharyya and B.K.Lahiri, Semi-generalized Closed sets in Topology, Indian J.Pure Appl.Math 29(1987), [3] C.L.Chang, Fuzzy topological spaces, J.Math.Anal.Appl. 24 (1986), [4] D.Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy sets and systems, 88 (1997), [5] D.Coker and M.Demirci, On intuitionistic fuzzy points, Notes on Intuitionistic Fuzzy Sets 1(1995), [6] H.Gurcay and D.Coker, On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces, J.Fuzzy Math.5 (2) (1997), [7] M.E.EL.Shafei and A.Zakari, Semi-generalized Continuous Mappings in Fuzzy Topological Space, J.Egypt Math.Soc, Vol.15 (1) (2007), [8] J.K.Joen, Y.B.Jun and J.H.Park, Intuitionistic fuzzy alpha-continuity and intuitionistic fuzzy pre-continuity, IJMMS, 19 (2005), [9] S.S.Thakur and Rekka Chaturvedi, RG-Closed sets in intuitionistic fuzzy topological spaces, Universitatea Din Bacau Studii Si Cercertari Stiintifice, Nr.16 (2006), pg [10] N.Turanli and D.Coker, Fuzzy connectedness in Intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 116(2000), [11] Young Bae Jun and Seok-Zun Song, Intuitionistic fuzzy semi-pre open sets and Intuitionistic semi-pre continuous mappings, J. Appl. Math & Computing, 2005, [12] L.A.Zadeh, Fuzzy sets, Information Control, 8 (1965), [13] Zhen Guo Xu and Fu Gui Shi, Some weakly mappings on intuitionistic fuzzy topological space, Tamkang Journal of Mathematics, Vol39, No1, 25-32, Spring Received: January, 2010