Intuitionistic Fuzzy Neutrosophic Soft Topological Spaces

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1 ISSNOnlin : ISSN Print : n ISO 3297: 2007 ertified Organization Intuitionistic Fuzzy Neutrosophic Soft Topological Spaces R.Saroja 1, Dr..Kalaichelvi 2 Research Scholar, Department of Mathematics, vinashilingam Institute for Home Science and Higher ducation for Women,oimbatore, Tamilnadu, India 1. ssociate Professor, Department of Mathematics, vinashilingam Institute for Home Science and Higher ducation for Women, oimbatore, Tamilnadu, India 2. STRT: The aim of this paper is to construct a topology on an intuitionistic fuzzy neutrosophic soft set. The concepts of intuitionistic fuzzy neutrosophic soft closure, intuitionistic fuzzy neutrosophic soft interior, intuitionistic fuzzy neutrosophic soft exterior, intuitionistic fuzzy neutrosophic soft boundary are introduced and some of its properties are studied. KYWORDS: Intuitionistic fuzzy neutrosophic soft set,intuitionistic fuzzy neutrosophic soft topology,intuitionistic fuzzy neutrosophic soft interior,intuitionistic fuzzy neutrosophic soft closure,intuitionistic fuzzy neutrosophic soft exterior, Intuitionistic fuzzy neutrosophic soft boundary. I. INTRODUTION In 1999, Molodtsov initiated the theory of soft sets as a new mathematical tool for dealing uncertainty. The topological structures of set theory dealing with uncertainties were first studied by hang in 1968.hang[4] introduced the notion of fuzzy topology and also studied some of its basic properties. Shabir and Naz[10]introduced the notion of the soft topology and studied some basic concepts such as soft interior, soft closure and soft sub base. Maji, Roy and iswas [5,6] initiated the concept of fuzzy soft set which is a combination of fuzzy set and soft set. tanassov[3] defined the concept of intuitionistic fuzzy set which is more general than the fuzzy set and Maji et.al[6] introduced the concept of intuitionistic fuzzy soft set. Smarandache[11] initiated the concept of neutrosophic set and Maji[7] defined the notion of neutrosophic soft set. rockiarani et.al [1]defined the notion of fuzzy neutrosophic set and fuzzy neutrosophic soft set. II. RLTD WORK Fuzzy soft topological spaces are introduced by TridivJyotiNeog et.al[12].intuitionistic fuzzy soft topological spaces are introduced by Sadiayramov and Gigdemgunduz[8]. Neutrosophic topological spaces are introduced by Salama and lblowi[9]. Fuzzy neutrosophic topological spaces and Fuzzy neutrosophic soft topological spaces are introduced by rockiarani, and MartinaJency[1,2]. In this paper intuitionistic fuzzy neutrosophic soft topological spaces are introduced.lso the concepts of intuitionistic fuzzy neutrosophic soft closure, intuitionistic fuzzy neutrosophic soft interior, intuitionistic fuzzy neutrosophic soft exterior, intuitionistic fuzzy neutrosophic soft boundary are introduced and some important theorems are established. DFINITION:3.1 Let ={ III.PRLIMINRIS x : T x, I x, F x, xu }where the functionst I, F, :U, 1 0 define respectively the degree of membership,the degree of indeterminacy and the degree of non-membership of the elements opyright to IJIRST DOI: /IJIRST

2 ISSNOnlin : ISSN Print : n ISO 3297: 2007 ertified Organization to the set with the condition, 0 T x I x F x 3. The neutrosophic set takes the value from real standard or non-standard subsets of 0,1 so instead of 0,1 we need to take the interval [0,1] for technical applications DFINITION:3.2 fuzzy neutrosophic set on the universe of discourse U is defined as { x : T x, I x, F x, xu} wheret, I, F : U [0,1 ] 0 T x I x F x 3 DFINITION:3.3 and satisfies the condition n intuitionistic fuzzy neutrosophic set on the universe of discourse U is defined as { x : T x, I x, F x, x U} where min[ F x, T x ] 0. 5,min[ I x, T x ] 0. 5 min[ F x, I x ] 0. 5 DFINITION:3.4,for all x U, with the condition 0 T x I x F x 2 Let U be an initial universe set and be a set of parameters.let PU denotes the power set of U. onsider a non-empty set. The pair F, is called soft set over U, where F is a mapping given by F: PU.The soft set F,is denoted as F DFINITION:3.5 Let U be an initial universe set. Let be a set of parameters and a non-empty set.let IFNU denotes the set of all intuitionistic fuzzy neutrosophic sets of U. The pair F, is called intuitionistic fuzzy neutrosophic soft setin short IFNSSover U, where F is a mapping given by F:IFNU and IFNSSF, is denoted as DFINITION:3.6 Let F and G be two intuitionistic fuzzy neutrosophic soft sets over the common universe U. Then F is G if and only if said to be an intuitionistic fuzzy neutrosophic soft sub set of e We denote this relationship by F is an intuitionistic fuzzy neutrosophic subset of G Or T x T x, I x I x, F x F x, e, x U F G F G F G F G and F is said to be an intuitionistic fuzzy neutrosophic soft super set ofg if G is an intuitionistic fuzzy neutrosophic soft subset of F. We denote this relationship by F G. F opyright to IJIRST DOI: /IJIRST

3 ISSNOnlin : ISSN Print : DFINITION:3.7 Let ={, e2,..., e } 1 n as ={ e 1, e 2,.., e n } where n ISO 3297: 2007 ertified Organization e be a set of parameters. The NOT set of is denoted by is defined e i not e. DFINITION:3.8 The complement of an intuitionistic fuzzy neutrosophic soft set F is denoted by F defined as F F, where F : IFN U is a mapping given by F e neutrosophic soft complement with T x F x, I F F x I x and F F F x T x F F x U, e. DFINITION:3.9 Let soft union of i, i and is intuitionistic fuzzy F and G be two IFNSSs over the common universe U. Intuitionistic fuzzy neutrosophic F and G denoted by F G and is defined as F G K, K are as follows. membership, indeterminacy membership and falsity membership of T x max T x, T x, I min, K F G x I x I x K F G F x min F x, F x, e, x U K F G DFINITION:3.10 Let soft intersection of where = and the truth F and G be two IFNSSs over the common universe U. Intuitionistic fuzzy neutrosophic F and G denoted by F G and is defined as F G K, where K and the truth membership, indeterminacy membership and falsity membership of T x min T x, T x, I min, K F G x I x I x K F G F x max F x, F x, e, x U. K F G DFINITION:3.11 n intuitionistic fuzzy neutrosophic soft set neutrosophic soft set with respect to the parameter if T x 0, I x 0, F x 1 e, F F F T x 0, I x 0, F x e 1 DFINITION:3.12 n intuitionistic fuzzy neutrosophic soft set fuzzy neutrosophic soft set with respect to the parameter if T x 1, I x 0, F x 0e, F F F T x 1, I x 0, F x e NOT: 0 1 and 1 = 0. are as follows. F over U is said to be empty intuitionistic fuzzy xu.it is denoted by 0. F over U is said to be the Universe intuitionistic xu.it is denoted by 1 opyright to IJIRST DOI: /IJIRST

4 ISSNOnlin : ISSN Print : n ISO 3297: 2007 ertified Organization IV.INTUITIONISTI FUZZY NUTROSOPHI SOFT TOPOLOGIL SPS DFINITION:4.1 Let U be an initial universe set. be a set of parameters. n intuitionistic fuzzy neutrosophic soft topology in short IFNST on U is a family of intuitionistic fuzzy neutrosophic soft sets over U satisfying the following properties: i 0, 1 ii If iii If F, G F i Then F G,i I, I be any arbitrary index set, then F. i ii The triplet U,, is said to be an intuitionistic fuzzy neutrosophic soft topological spacein short IFNSTS. DFINITION:4.2 If is an intuitionistic fuzzy neutrosophic soft topology on U, the triplet U,, is said to be an intuitionistic fuzzy neutrosophic soft topological space.ach member of is called an intuitionistic fuzzy neutrosophic soft open set in U,,. n intuitionistic fuzzy neutrosophic soft set is called intuitionistic fuzzy neutrosophic soft closed if its complement is intuitionistic fuzzy neutrosophic soft open. DFINITION:4.3 Let U,, be an intuitionistic fuzzy neutrosophic soft topological space.let over U. Then an intuitionistic fuzzy neutrosophic soft closure of intersection of all intuitionistic fuzzy neutrosophic soft closed sets containing l F G = { : G is intuitionistic fuzzy neutrosophic soft closed and IFNS closed and l F. NOT: F i l 0 = 0 and l 1 1 ii F denoted by l F F. F is intuitionistic fuzzy neutrosophic soft closed if and only if F l iii l l F l F THORM:4.1 Let U,, be IFNSTS. Let l F G l F l G F F and G are two IFNSS over U.Then, F be an IFNSS and is defined as the F G }.It is also clear that l F is opyright to IJIRST DOI: /IJIRST

5 ISSNOnlin : ISSN Print : F Since F F n ISO 3297: 2007 ertified Organization G, G F G l l F G, l G l F G l l F G 1 So, F l G F l F F and G l F l F l G F G G l G l G is IFNS closed set containing F G l l F l G 2 From 1 & 2 F G DFINITION :4.4 l = l F l G over U. Thus Let U,, be an intuitionistic fuzzy neutrosophic soft topological space. Let the intuitionistic fuzzy neutrosophic soft interior of intuitionistic fuzzy neutrosophic soft open sets contained in int F = { : G It is also clear that int NOT: int = 0 i 0 ii F denoted by int F. G is intuitionistic fuzzy neutrosophic soft open and G } F be an IFNSS over U. Then F and is defined as the union of all F F is intuitionistic fuzzy neutrosophic soft open and int F F and int 1 1 F is intuitionistic fuzzy neutrosophic soft open if and only if F int iii intint F int F F THORM:4.2 Let U,, be IFNSTS. Let F =int F int G int G F and G are two IFNSS over U. Then opyright to IJIRST DOI: /IJIRST

6 ISSNOnlin : ISSN Print : Since int G F G F, F G G n ISO 3297: 2007 ertified Organization F int F and int F G int G F int F int G 1 int G y definition, int int F F, int G G F int G int G int F is the largest IFNS open set contained in F G F int G int F G 2 from 1 & 2int G THORM: 4.3 F = int F int G Let U,, be an IFNSTS. Let i[ l F ] int F ii[int F ] l F F be IFNSS over U. Then i l F = { G : G is intuitionistic fuzzy neutrosophic soft closed and { G : G [ l F ] =[ = { G : G ii int F F is intuitionistic fuzzy neutrosophic soft closed and F = int is intuitionistic fuzzy neutrosophic soft open and { G : G is intuitionistic fuzzy neutrosophic soft open and { G : G [int F ] =[ is intuitionistic fuzzy neutrosophicsoft open and { G : G = l F is intuitionistic fuzzy neutrosophic soft closed and G } F G }] G F G } } F G F } F G ] } opyright to IJIRST DOI: /IJIRST

7 ISSNOnlin : ISSN Print : DFINITION:4.5 n ISO 3297: 2007 ertified Organization Let U,, be an intuitionistic fuzzy neutrosophic soft topological space. Let F neutrosophic soft set over U. Then an intuitionistic fuzzy neutrosophic soft exterior of defined as ext F =int F be an intuitionistic fuzzy F denoted by ext F and is THORM :4.4 Let U,, be an intuitionistic fuzzy neutrosophic soft topological space. Let intuitionistic fuzzy neutrosophic soft sets over U.Then i ext G ext F ext G F ii ext F ext G ext F G iext F G int F G int F = ii F ext G G int F int G = ext F ext G ext = int F int G int F G int F G extf G ext F ext G ext F G DFINITION:4.6 F and G are two Let U,, be an intuitionistic fuzzy neutrosophic soft topological space. Let neutrosophic soft set over U. Then an intuitionistic fuzzy neutrosophic soft boundary of F and is defined as bou F l F l bou F F be an intuitionistic fuzzy F denoted by opyright to IJIRST DOI: /IJIRST

8 ISSNOnlin : ISSN Print : n ISO 3297: 2007 ertified Organization THORM:4.5 Let U,, be an intuitionistic fuzzy neutrosophic soft topological space. Let neutrosophic soft set over U.Then bou F ] int F int F int F ext F [ F be an intuitionistic fuzzy F l F l bou F [ bou F] [ l F] [ l F ] NOT: =int F int F = int F ext F ibou F int F 0 F = 0 iibou ext F V.ONLUSION In the present work, a new space called intuitionistic fuzzy neutrosophic soft topological space is introduced and some of its properties studied. RFRNS [1]rockiarani, I., and MartinaJency, J., More on fuzzy neutrosophic sets and fuzzy neutrosophic topological spaces, Inter.J.Innov.Research& Studies,Vol.3,Issue 5,PP , [2]rockiarani, I., Sumathi, I.R., and MartinaJency, J., Fuzzy neutrosophic soft topological spaces,inter.j.math.rchive, 410,PP , 2013 [3]. tanassov, K.T., Intuitionistic fuzzy sets, fuzzy sets and systems,vol.20,no.1,pp.87-96,1986. [4]hang,., fuzzy topological spaces, J.Math.nal.ppl.,24,PP ,1968 [5]Maji, P.K., Roy,.R., and iswas, R., Fuzzy soft sets, J.Fuzzy Math.93,PP , 2001 [6] Maji, P.K., Roy,.R., and iswas, R., Intuitionistic Fuzzy soft sets, J.FuzzyMath.93,PP ,2001 [7]Maji, P.K., Neutrosophic soft set,nn.fuzzy.math.inform,vol.5,no.1,pp ,2012 [8]Sadiayramovand Gigdemgunduz., On intuitionistic fuzzy soft topological spaces,inter.j.pureppl.math,vol.5,no.1,pp.66-79,2014 [9] Salama,.., and lblowi, S.., Neutrosophic set and neutrosophic topological spaces,iosrj.mathematics,vol.3,issue 4,PP.31-35,2012 [10] Shabir, M., and Naz, M., On soft topological spaces, ompute.math.ppl.vol.6,no.1,pp ,2001 [11] Smarandache, F., Neutrosophic set, generalization of the intuitionistic fuzzy sets, Inter.J.Pure ppl.math,24,pp ,2005 [12].TridivJyotiNeog,Dusmanta Kumar Sut and Hazarika., Fuzzy soft topological spaces, Inter.J.Latest Trend Math,Vol.21,PP.87-96,2012. opyright to IJIRST DOI: /IJIRST