Intuitionistic Neutrosophic Soft Set

Transcription

1 ISSN , England, UK Journal of Information and Computing Siene Vol. 8, No. 2, 2013, pp Intuitionisti Neutrosophi Soft Set Broumi Said 1 and Florentin Smarandahe 2 1 Administrator of Faulty of Arts and Humanities, Hay El Baraka Ben M'sik Casablana B.P. 7951, Hassan II University Mohammedia-Casablana, Moroo 2 Department of Mathematis, University of New Mexio, 705 Gurley Avenue,Gallup, NM 87301, USA (Reeived Deember11, 2012, aepted February 24, 2013) Abstrat. In this paper we study the onept of intuitionisti neutrosophi set of Bhowmik and Pal. We have introdued this onept in soft sets and defined intuitionisti neutrosophi soft set. Some definitions and operations have been introdued on intuitionisti neutrosophi soft set. Some properties of this onept have been established. Keywords: Soft sets, Neutrosophi set,intuitionisti neutrosophi set, Intuitionisti neutrosophi soft set. 1. Introdution In wide varities of real problems like, engineering problems, soial, eonomi, omputer siene, medial siene et. The data assoiated are often unertain or impreise, all real data are not neessarily risp, preise, and deterministi beause of their fuzzy nature. Most of these problem were solved by different theories, firstly by fuzzy set theory provided by Lotfi, Zadeh [1],Later several researhes present a number of results using different diretion of fuzzy set suh as : interval fuzzy set [13], intuitionisti fuzzy set by Atanassov[2], all these are suessful to some extent in dealing with the problems arising due to the vagueness present in the real world,but there are also ases where these theories failed to give satisfatory results, possibly due to indeterminate and inonsistent information whih exist in belif system, then in 1995, Smarandahe [3] intiated the theory of neutrosophi as new mathematial tool for handling problems involving impreise, indeterminay,and inonsistent data. Later on authors like Bhowmik and Pal [7] have further studied the intuitionisti neutrosophi set and presented various properties of it. In 1999 Molodtsov [4] introdued the onept of soft set whih was ompletely a new approhe for dealing with vagueness and unertainties,this onept an be seen free from the inadequay of parameterization tool. After Molodtsovs work, there have been many researhes in ombining fuzzy set with soft set, whih inorporates the benefiial properties of both fuzzy set and soft set tehniques ( see [12] [6] [8]). Reently, by the onept of neutrosophi set and soft set, first time, Maji [11] introdued neutrosophi soft set, established its appliation in deision making, and thus opened a new diretion, new path of thinking to engineers, mathematiians, omputer sientists and many others in various tests. This paper is an attempt to ombine the onepts: intuitionisti neutrosophi set and soft set together by introduing a new onept alled intuitionisti neutrosophi sof set, thus we introdue its operations namely equal,subset, union,and intersetion, We also present an appliation of intuitionisti neutrosophi soft set in deision making problem. The organization of this paper is as follow : in setion 2, we briefly present some basi definitions and preliminary results are given whih will be used in the rest of the paper. In setion 3, Intuitionisti neutrosophi soft set. In setion 4 an appliation of intuitionisti neutrosophi soft set in a deision making problem. Conlusions are there in the onluding setion Preliminaries Throughout this paper, let U be a universal set and E be the set of all possible parameters under 1 Corresponding author. Tel.: address: broumisaid78@gmail.om Published by World Aademi Press, World Aademi Union

2 Journal of Information and Computing Siene, Vol. 8 (2013) No. 2, pp onsideration with respet to U, usually, parameters are attributes, harateristis, or properties of objets in U. We now reall some basi notions of neutrosophi set, intuitionisti neutrosophi set and soft set. Definition 2.1 (see[3]). Let U be an universe of disourse then the neutrosophi set A is an objet having the form A = {< x: T A(x), I A(x),F A(x) >,x U}, where the funtions T,I,F : U ] 0,1 + [ define respetively the degree of membership, the degree of indeterminay, and the degree of non-membership of the element x X to the set A with the ondition. 0 T A(x) + I A(x) + F A(x) 3 +. From philosophial point of view, the neutrosophi 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 tehnial appliations, beause ] 0,1 + [will be diffiult to apply in the real appliations suh as in sientifi and engineering problems. Definition 2.2 (see [3]). A neutrosophi set A is ontained in another neutrosophi set B i.e. A B if x U, T A (x) T B (x), I A (x) I B (x), F A (x) F B (x). A omplete aount of the operations and appliation of neutrsophi set an be seen in [3 ] [10 ]. Definition 2.3(see[7]). intuitionisti neutrosophi set An element x of U is alled signifiant with respet to neutrsophi set A of U if the degree of truthmembership or falsity-membership or indeterminany-membership value, i.e.,t A(x) or F A(x) or I A(x) 0.5. Otherwise, we all it insignifiant. Also, for neutrosophi set the truth-membership, indeterminaymembership and falsity-membership all an not be signifiant. We define an intuitionisti neutrosophi set by A = {< x: T A(x), I A(x), F A(x) >,x U},where min { T A( x ), F A( x ) } 0.5, min { T A( x ), I A( x ) } 0.5, min { F A( x ), I A( x ) } 0.5, for all x U, with the ondition 0 T A(x) + I A(x) + F A(x) 2. As an illustration,let us onsider the following example. Example2.4.Assume that the universe of disourse U={x 1,x 2,x 3 },where x 1 haraterizes the apability, x 2 haraterizes the trustworthiness and x 3 indiates the pries of the objets. It may be further assumed that the values of x 1, x 2 and x 3 are in [0,1] and they are obtained from some questionnaires of some experts. The experts may impose their opinion in three omponents viz. the degree of goodness, the degree of indeterminay and that of poorness to explain the harateristis of the objets. Suppose A is an intuitionisti neutrosophi set ( IN S ) of U, suh that, A = {< x 1,0.3,0.5,0.4 >,< x 2,0.4,0.2,0.6 >,< x 3,0.7,0.3,0.5 >}, where the degree of goodness of apability is 0.3, degree of indeterminay of apability is 0.5 and degree of falsity of apability is 0.4 et. Definition 2.5 (see[4]). Let U be an initial universe set and E be a set of parameters. Let P(U) denotes the power set of U. Consider a nonempty set A, A E. A pair ( F, A ) is alled a soft set over U, where F is a mapping given by F : A P(U). As an illustration,let us onsider the following example. Example 2.6. Suppose that U is the set of houses under onsideration, say U = {h 1, h 2,..., h 5 }. Let E be the set of some attributes of suh houses, say E = {e 1, e 2,..., e 8 }, where e 1, e 2,..., e 8 stand for the attributes expensive, beautiful, wooden, heap, modern, and in bad repair, respetively. In this ase, to define a soft set means to point out expensive houses, beautiful houses, and so on. For example, the soft set (F, A) that desribes the attrativeness of the houses in the opinion of a buyer, say Thomas, may be defined like this: A={e 1,e 2,e 3,e 4,e 5 }; F(e 1 ) = {h 2, h 3, h 5 }, F(e 2 ) = {h 2, h 4 }, F(e 3 ) = {h 1 }, F(e 4 ) = U, F(e 5 ) = {h 3, h 5 }. For more details on the algebra and operations on intuitionisti neutrosophi set and soft set, the reader may refer to [ 5,6,8,9,12]. 3. Intuitionisti Neutrosophi Soft Set In this setion,we will initiate the study on hybrid struture involving both intuitionsti neutrosophi set and soft set theory. JIC for subsription: publishing@wau.org.uk

3 132 Broumi Said et.al.: Intuitionisti Neutrosophi Soft Set Definition 3.1. Let U be an initial universe set and A E be a set of parameters. Let N( U ) denotes the set of all intuitionisti neutrosophi sets of U. The olletion (F,A) is termed to be the soft intuitionisti neutrosophi set over U, where F is a mapping given by F : A N(U). Remark 3.2. we will denote the intuitionisti neutrosophi soft set defined over an universe by INSS. Let us onsider the following example. Example 3.3. Let U be the set of blouses under onsideration and E is the set of parameters (or qualities). Eah parameter is a intuitionisti neutrosophi word or sentene involving intuitionisti neutrosophi words. Consider E = { Bright, Cheap, Costly, very ostly, Colorful, Cotton, Polystyrene, long sleeve, expensive }. In this ase, to define a intuitionisti neutrosophi soft set means to point out Bright blouses, Cheap blouses, Blouses in Cotton and so on. Suppose that, there are five blouses in the universe U given by, U = {b 1,b 2,b 3,b 4,b 5 } and the set of parameters A = {e 1,e 2,e 3,e 4 }, where eah e i is a speifi riterion for blouses: e 1 stands for Bright, e 2 stands for Cheap, e 3 stands for ostly, e 4 stands for Colorful, Suppose that, F(Bright)={<b 1,0.5,0.6,0.3>,<b 2,0.4,0.7,0.2>,<b 3,0.6,0.2,0.3>,<b 4,0.7,0.3,0.2>,<b 5,0.8,0.2,0.3>}. F(Cheap)={<b 1,0.6,0.3,0.5>,<b 2,0.7,0.4,0.3>,<b 3,0.8,0.1,0.2>,<b 4,0.7,0.1,0.3>,<b 5,0.8,0.3,0.4}. F(Costly)={<b 1,0.7,0.4,0.3>,<b 2,0.6,0.1,0.2>,<b 3,0.7,0.2,0.5>,< b 4,0.5,0.2,0.6 >,< b 5,0.7,0.3,0.2 >}. F(Colorful)={<b 1,0.8,0.1,0.4>,<b 2,0.4,0.2,0.6>,<b 3,0.3,0.6,0.4>,<b 4,0.4,0.8,0.5>,< b 5,0.3,0.5,0.7 >}. The intuitionisti neutrosophi soft set ( INSS ) ( F, E ) is a parameterized family {F(e i ),i = 1,,10} of all intuitionisti neutrosophi sets of U and desribes a olletion of approximation of an objet. The mapping F here is blouses (.), where dot(.) is to be filled up by a parameter e i E. Therefore, F(e 1 ) means blouses (Bright) whose funtional-value is the intuitionisti neutrosophi set {< b 1,0.5,0.6,0.3 >,< b 2,0.4,0.7,0.2 >, < b 3,0.6,0.2,0.3 >,< b 4,0.7,0.3,0.2 >,< b 5,0.8,0.2,0.3 >}. Thus we an view the intuitionisti neutrosophi soft set ( INSS ) ( F, A ) as a olletion of approximation as below: ( F, A ) = { Bright blouses= {< b 1,0.5,0.6,0.3 >,< b 2,0.4,0.7,0.2 >, < b 3,0.6,0.2,0.3 >,< b 4,0.7,0.3,0.2 >,< b 5,0.8,0.2,0.3 >}, Cheap blouses= {< b 1,0.6,0.3,0.5 >,< b 2,0.7,0.4,0.3 >,< b 3,0.8,0.1,0.2 >, < b 4,0.7,0.1,0.3 >,< b 5,0.8,0.3,0.4 >}, ostly blouses= {< b 1,0.7,0.4,0.3 >,< b 2,0.6,0.1,0.2 >,< b 3,0.7,0.2,0.5 >,< b 4,0.5,0.2,0.6 >,< b 5,0.7,0.3,0.2 >}, Colorful blouses= {< b 1,0.8,0.1,0.4 >,< b 2,0.4,0.2,0.6 >,< b 3,0.3,0.6,0.4 >, < b 4,0.4,0.8,0.5>,< b 5,0.3,0.5,0.7 >}}. where eah approximation has two parts: (i) a prediate p, and (ii) an approximate value-set v ( or simply to be alled value-set v ). For example, for the approximation Bright blouses= {< b 1,0.5,0.6,0.3 >, < b 2,0.4,0.7,0.2 >,<b 3,0.6,0.2,0.3>,<b 4,0.7,0.3,0.2>,<b 5,0.8,0.2,0.3>}. we have (i) the prediate name Bright blouses, and (ii) the approximate value-set is{<b 1,0.5,0.6,0.3>,<b 2,0.4,0.7,0.2>,<b 3,0.6,0.2,0.3>,<b 4,0.7,0.3,0.2>,< b 5,0.8,0.2,0.3 >}. Thus, an intuitionisti neutrosophi soft set ( F, E ) an be viewed as a olletion of approximation like ( F, E ) = {p 1 = v 1,p 2 = v 2,,p 10 = v 10 }. In order to store an intuitionisti neutrosophi soft set in a omputer, we ould represent it in the form of a table as shown below ( orresponding to the intuitionisti neutrosophi soft set in the above example ). In this table, the entries are ij orresponding to the blouse b i and the parameter e j, where ij = (true-membership value of b i, indeterminay-membership value of b i, falsity membership value of b i ) in F(e j ). The table 1 represent the intuitionisti neutrosophi soft set ( F, A ) desribed above. JIC for ontribution: editor@ji.org.uk

4 Journal of Information and Computing Siene, Vol. 8 (2013) No. 2, pp U bright heap ostly olorful b 1 ( 0.5,0.6, 0.3 ) ( 0.6,0.3, 0.5 ) ( 0.7,0.4, 0.3 ) ( 0.8,0.1, 0.4 ) b 2 ( 0.4,0.7, 0.2 ) ( 0.7,0.4, 0.3 ) ( 0.6,0.1, 0.2 ) ( 0.4,0.2, 0.6 ) b 3 ( 0.6,0.2, 0.3 ) ( 0.8,0.1, 0.2 ) ( 0.7,0.2, 0.5 ) ( 0.3,0.6, 0.4 ) b 4 ( 0.7,0.3, 0.2 ) ( 0.7,0.1, 0.3 ) ( 0.5,0.2, 0.6 ) ( 0.4,0.8, 0.5 ) b 5 ( 0.8,0.2, 0.3 ) ( 0.8,0.3, 0.4 ) ( 0.7,0.3, 0.2 ) ( 0.3,0.5, 0.7 ) Table 1: Tabular form of the INSS ( F, A ). Remark 3.4.An intuitionisti neutrosophi soft set is not an intuituionisti neutrosophi set but a parametrized family of an intuitionisti neutrosophi subsets. Definition 3.5. Containment of two intuitionisti neutrosophi soft sets. For two intuitionisti neutrosophi soft sets ( F, A ) and ( G, B ) over the ommon universe U. We say that ( F, A ) is an intuitionisti neutrosophi soft subset of ( G, B ) if and only if (i) A B. (ii) F(e) is an intuitionisti neutrosophi subset of G(e). Or T F(e) (x) T G(e) (x), I F(e) (x) I G(e) (x), F F(e) (x) F G(e) (x), e A, x U. We denote this relationship by ( F, A ) ( G, B ). ( F, A ) is said to be intuitionisti neutrosophi soft super set of ( G, B ) if ( G, B ) is an intuitionisti neutrosophi soft subset of ( F, A ). We denote it by ( F, A ) ( G, B ). Example 3.6. Let (F,A) and (G,B) be two INSSs over the same universe U = {o 1,o 2,o 3,o 4,o 5 }. The INSS (F,A) desribes the sizes of the objets whereas the INSS ( G, B ) desribes its surfae textures. Consider the tabular representation of the INSS ( F, A ) is as follows. U small large olorful O 1 ( 0.4,0.3, 0.6 ) ( 0.3,0.1, 0.7 ) ( 0.4,0.1, 0.5 ) O 2 ( 0.3,0.1, 0.4 ) ( 0.4,0.2, 0.8 ) ( 0.6,0.3, 0.4 ) O 3 ( 0.6,0.2, 0.5 ) ( 0.3,0.1, 0.6 ) ( 0.4,0.3, 0.8 ) O 4 ( 0.5,0.1, 0.6 ) ( 0.1,0.5, 0.7 ) ( 0.3,0.3, 0.8 ) O 5 ( 0.3,0.2, 0.4 ) ( 0.3,0.1, 0.6 ) ( 0.5,0.2, 0.4 ) Table 2: Tabular form of the INSS ( F, A ). The tabular representation of the INSS ( G, B ) is given by table 3. U small large olorful very smooth O1 (0.6,0.4, 0.3 ) ( 0.7,0.2, 0.5 ) ( 0.5,0.7, 0.4 ) ( 0.1,0.8, 0.4 ) O2 ( 0.7,0.5, 0.2 ) ( 0.4,0.7, 0.3 ) ( 0.7,0.3, 0.2 ) ( 0.5,0.7, 0.3 ) O3 ( 0.6,0.3, 0.5 ) ( 0.7,0.2, 0.4 ) ( 0.6,0.4, 0.3 ) ( 0.2,0.9, 0.4 ) O4 ( 0.8,0.1, 0.4 ) ( 0.3,0.6, 0.4 ) ( 0.4,0.5, 0.7 ) ( 0.4,0.4, 0.5 ) O5 ( 0.5,0.4, 0.2 ) ( 0.4,0.1, 0.5 ) ( 0.6,0.4, 0.3 ) ( 0.5,0.8, 0.3 ) Table 3: Tabular form of the INSS ( G, B ). Clearly, by definition 3.5 we have ( F, A ) ( G, B ). Definition 3.7. Equality of two intuitionisti neutrosophi soft sets. Two INSSs ( F, A ) and ( G, B ) over the ommon universe U are said to be intuitionisti neutrosophi soft equal if ( F, A ) is an intuitionisti neutrosophi soft subset of ( G, B ) and ( G, B ) is an intuitionisti neutrosophi soft subset of ( F, A ) whih an be denoted by ( F, A )= ( G, B ). Definition 3.8. NOT set of a set of parameters. JIC for subsription: publishing@wau.org.uk

5 134 Broumi Said et.al.: Intuitionisti Neutrosophi Soft Set Let E = {e 1,e 2,,e n } be a set of parameters. The NOT set of E is denoted by E is defined by E ={ e 1, e 2,, e n }, where e i = not e i, i ( it may be noted that and are different operators ). Example 3.9. Consider the example 3.3. Here E = { not bright, not heap, not ostly, not olorful }. Definition Complement of an intuitionisti neutrosophi soft set. The omplement of an intuitionisti neutrosophi soft set ( F, A ) is denoted by (F,A) and is defined by (F,A) = (F, A), where F : A N(U) is a mapping given by F (α) = intutionisti neutrosophi soft omplement with T F (x) = F F(x), I F (x) = I F(x) and F F (x) = T F(x). Example As an illustration onsider the example presented in the example 3.2. the omplement (F,A) desribes the not attrativeness of the blouses. Is given below. F( not bright) = {< b 1,0.3,0.6,0.5 >,< b 2,0.2,0.7,0.4 >,< b 3,0.3,0.2,0.6 >, < b4,0.2,0.3,0.7 >< b 5,0.3,0.2,0.8 >}. F( not heap ) = {< b 1,0.5,0.3,0.6 >,< b 2,0.3,0.4,0.7 >,< b 3,0.2,0.1,0.8 >, < b 4,0.3,0.1,0.7 >,< b 5,0.4,0.3,0.8 >}. F( not ostly ) = {< b 1,0.3,0.4,0.7 >,< b 2,0.2,0.1,0.6 >,< b 3,0.5,0.2,0.7 >, < b 4,0.6,0.2,0.5 >,< b 5,0.2,0.3,0.7 >}. F( not olorful ) = {< b 1,0.4,0.1,0.8 >,< b 2,0.6,0.2,0.4 >,< b 3,0.4,0.6,0.3 >, < b 4,0.5,0.8,0.4 >< b 5,0.7,0.5,0.3 >}. Definition 3.12:Empty or Null intuitionisti neutrosopphi soft set. An intuitionisti neutrosophi soft set (F,A) over U is said to be empty or null intuitionisti neutrosophi soft (with respet to the set of parameters) denoted by Φ A or (Φ,A) if T F(e) (m) = 0,F F(e) (m) = 0 and I F(e) (m) = 0, m U, e A. Example Let U = {b 1,b 2,b 3,b 4,b 5 }, the set of five blouses be onsidered as the universal set and A = { Bright, Cheap, Colorful } be the set of parameters that haraterizes the blouses. Consider the intuitionisti neutrosophi soft set ( F, A) whih desribes the ost of the blouses and F(bright)={< b 1,0,0,0 >,< b 2,0,0,0 >,< b 3,0,0,0 >,< b 4,0,0,0 >, < b 5,0,0,0 >}, F(heap)={< b 1,0,0,0 >,< b 2,0,0,0 >,< b 3,0,0,0 >,< b 4, 0,0,0 >, < b 5,0,0,0 >}, F(olorful)={< b 1,0,0,0 >,< b 2,0,0,0 >,< b 3,0,0,0 >, < b 4,0,0,0 >,< b 5,0,0,0 >}. Here the NINSS ( F, A ) is the null intuitionisti neutrosophi soft set. Definition Union of two intuitionisti neutrosophi soft sets. Let (F,A) and (G,B) be two INSSs over the same universe U.Then the union of (F,A) and (G,B) is denoted by (F,A) (G,B) and is defined by (F,A) (G,B)=(K,C), where C=A B and the truth-membership, indeterminay-membership and falsity-membership of ( K,C) are as follows: T K(e)( m) = T F(e) (m), if e A B, = T G(e) (m), if e B A, = max (T F(e) (m),t G(e) (m)), if e A B. I K(e)( m) = I F(e) (m), if e A B, = I G(e) (m), if e B A, = min (I F(e) (m),i G(e) (m)), if e A B. F K(e)( m) = F F(e) (m), if e A B, = F G(e) (m), if e B A, = min (F F(e) (m),f G(e) (m)), if e A B. Example Let ( F, A ) and ( G, B ) be two INSSs over the ommon universe U. Consider the tabular representation of the INSS ( F, A ) is as follow: Bright Cheap Colorful b 1 ( 0.6,0.3, 0.5 ) ( 0.7,0.3, 0.4 ) ( 0.4,0.2, 0.6 ) b 2 ( 0.5,0.1, 0.8 ) ( 0.6,0.1, 0.3 ) ( 0.6,0.4, 0.4 ) b 3 ( 0.7,0.4, 0.3 ) ( 0.8,0.3, 0.5 ) ( 0.5,0.7, 0.2 ) b 4 ( 0.8,0.4, 0.1 ) ( 0.6,0.3, 0.2 ) ( 0.8,0.2, 0.3 JIC for ontribution: editor@ji.org.uk

6 Journal of Information and Computing Siene, Vol. 8 (2013) No. 2, pp b 5 ( 0.6,0.3, 0.2 ) ( 0.7,0.3, 0.5 ) ( 0.3,0.6, 0.5 Table 4: Tabular form of the INSS ( F, A ). The tabular representation of the INSS ( G, B ) is as follow: U Costly Colorful b 1 ( 0.6,0.2, 0.3) ( 0.4,0.6, 0.2 ) b 2 ( 0.2,0.7, 0.2 ) ( 0.2,0.8, 0.3 ) b 3 ( 0.3,0.6, 0.5 ) ( 0.6,0.3, 0.4 ) b 4 ( 0.8,0.4, 0.1 ) ( 0.2,0.8, 0.3 ) b 5 ( 0.7,0.1, 0.4 ) ( 0.5,0.6, 0.4 ) Table 5: Tabular form of the INSS ( G, B ). Using definition 3.12 the union of two INSS (F, A ) and ( G, B ) is ( K, C ) an be represented into the following Table. U Bright Cheap Colorful Costly b 1 ( 0.6,0.3, 0.5 ) ( 0.7,0.3, 0.4 ) ( 0.4,0.2, 0.2 ) ( 0.6,0.2, 0.3 ) b 2 ( 0.5,0.1, 0.8 ) ( 0.6,0.1, 0.3 ) ( 0.6,0.4, 0.3 ) ( 0.2,0.7, 0.2 ) b 3 ( 0.7,0.4, 0.3 ) ( 0.8,0.3, 0.5 ) ( 0.6,0.3, 0.2 ) ( 0.3,0.6, 0.5 ) b 4 ( 0.8,0.4, ( 0.6,0.3, ( 0.8,0.2, ( 0.8,0.4, 0.1 ) b 5 ( 0.6,0.3, 0.2 ) 0.2 ) ( 0.7,0.3, 0.5 ) 0.3 ) ( 0.5,0.6, 0.4 ) Table 6: Tabular form of the INSS ( K, C ). 0.1 ) ( 0.7,0.1, 0.4 ) Definition Intersetion of two intuitionisti neutrosophi soft sets. Let (F,A) and (G,B) be two INSSs over the same universe U suh that A B 0. Then the intersetion of (F,A) and ( G,B) is denoted by ( F,A) (G, B) and is defined by ( F, A ) ( G, B ) = ( K,C),where C =A B and the truth-membership, indeterminay membership and falsity-membership of ( K, C ) are related to those of (F,A) and (G,B) by: T K(e)( m) = min (T F(e) (m),t G(e) (m)), I K(e)( m) = min (I F(e) (m),i G(e) (m)), F K(e)( m) = max (F F(e) (m),f G(e) (m)), for all e C. Example Consider the above example The intersetion of ( F, A ) and ( G, B ) an be represented into the following table : U Colorful b 1 ( 0.4,0.2,0.6) b 2 ( 0.2,0.4,0.4) b 3 ( 0.6,0.3,0.4) b 4 ( 0.8,0.2,0.3) b 5 ( 0.3,0.6,0.5) Table 7: Tabular form of the INSS ( K, C ). Proposition If (F, A) and (G, B) are two INSSs over U and on the basis of the operations defined above,then: JIC for subsription: publishing@wau.org.uk

7 136 Broumi Said et.al.: Intuitionisti Neutrosophi Soft Set (1) idempoteny laws: (F,A) (F,A) = (F,A). (F,A) (F,A) = (F,A). (2) Commutative laws : (F,A) (G,B) = (G,B) (F,A). (F,A) (G,B) = (G,B) (F,A). (3) (F,A) Φ = (F,A). (4) (F,A) Φ = Φ. (5) [(F,A) ] = (F,A). Proof. The proof of the propositions 1 to 5 are obvious. Proposition If ( F, A ), ( G, B ) and ( K, C ) are three INSSs over U,then: (1) (F,A) [(G,B) (K,C)] = [(F,A) (G,B)] (K,C). (2) (F,A) [(G,B) (K,C)] = [(F,A) (G,B)] (K,C). (3) Distributive laws: (F,A) [(G,B) (K,C)] = [(F,A) (G,B)] [(F,A) (K,C)]. (4) (F,A) [(G,B) (K,C)] = [(H,A) (G,B)] [(F,A) (K,C)]. Exemple Let (F,A) ={ b 1,0.6,0.3,0. 1, b 2,0.4,0.7,0. 5),(b 3,0.4,0.1,0.8) }, (G,B) ={ (b 1,0.2,0.2,0.6), (b 2 0.7,0.2,0.4), (b 3,0.1,0.6,0.7) } and (K,C) ={ (b 1, 0.3,0.8,0.2), b 2, 0.4,0.1,0.6), b 3,0.9,0.1,0.2)} be three INSSs of U, Then: (F,A) (G,B) = { b 1, 0.6,0.2,0.1, b 2, 0.7,0.2,0.4, b 3,0.4,0.1,0.7 }. (F,A) (K,C) = { b 1,0.6,0.3,0.1, b 2, 0.4,0.1,0.5, b 3,0.9,0.1,0.2 }. (G,B) (K,C)] = { b 1,0.2,0.2,0.6, b 2,0.4,0.1,0.6, b 3, 0.1,0.1,0.7 }. (F,A) [(G,B) (K,C)] = { b 1,0.6,0.2,0.1, b 2,0.4,0.1,0.5, b 3,0.4,0.1,0.7 }. [(F,A) (G,B)] [(F,A) (K,C)] = { b 1,0.6,0.2,0.1, b 2,0.4,0.1,0.5, b 3,0.4,0.1,0.7 }. Hene distributive (3) proposition verified. Proof, an be easily proved from definition 3.14.and Definition AND operation on two intuitionisti neutrosophi soft sets. Let ( F, A ) and ( G, B ) be two INSSs over the same universe U. then ( F, A ) AND ( G, B) denoted by ( F, A ) ( G, B )and is defined by ( F, A ) ( G, B ) = ( K, A B ), where K(α, β)=f(α) B(β) and the truth-membership, indeterminay-membership and falsity-membership of ( K, A B ) are as follows: T K(α,β) (m) = min(t F(α) (m),t G(β) (m)), I K(α,β) (m) = min(i F(α) (m),i G(β) (m)) F K(α,β) (m) = max(f F(α) (m),f G(β) (m)), α A, β B. Example Consider the same example 3.15 above. Then the tabular representation of (F,A) AND ( G, B ) is as follow: u (bright, ostly) (bright, Colorful) (heap, ostly) b 1 ( 0.6,0.2, 0.5 ) ( 0.4,0.3, 0.5 ) ( 0.6,0.2, 0.4 ) b 2 ( 0.2,0.1, 0.8 ) ( 0.2,0.1, 0.8 ) ( 0.2,0.1, 0.3 ) b 3 ( 0.3,0.4, 0.5 ) ( 0.6,0.3, 0.4 ) ( 0.3,0.3, 0.5 ) b 4 ( 0.8,0.4, 0.1 ) ( 0.2,0.4,0.3 ) ( 0.6,0.3, 0.2 ) b 5 ( 0.6,0.1, 0.4 ) ( 0.5,0.3, 0.4 ) ( 0.7,0.1, 0.5) u (heap, olorful) (olorful, ostly) (olorful, olorful) b 1 ( 0.4,0.3, 0.4 ) ( 0.4,0.2, 0.6 ) ( 0.4,0.2, 0.6 ) b 2 ( 0.2,0.1, 0.3 ) ( 0.2,0.4, 0.4 ) ( 0.2,0.4, 0.4 ) b 3 ( 0.6,0.3, 0.5 ) ( 0.3,0.6, 0.5 ) ( 0.5,0.3, 0.4 ) b 4 ( 0.2,0.3, 0.3 ) ( 0.8,0.2, 0.3 ) ( 0.2,0.2, 0.3 ) b 5 ( 0.5,0.3, 0.5 ) ( 0.3,0.1, 0.5 ) ( 0.3,0.6, 0.5 ) Table 8: Tabular representation of the INSS ( K, A B). Definition If (F,A) and (G,B) be two INSSs over the ommon universe U then (F,A) OR(G,B) denoted by (F,A) (G,B) is defined by ( F, A) (G, B ) = (O,A B), where, the truth-membership, indeterminay membership and falsity-membership of O( α, β) are given as follows: JIC for ontribution: editor@ji.org.uk

8 Journal of Information and Computing Siene, Vol. 8 (2013) No. 2, pp TO(α,β) (m) = max(t F(α) (m),t G(β) (m)), I O(α,β) (m) = min(i F(α) (m),i G(β) (m)), (m)), α A, β B. FO(α,β) (m) = min(f F(α) (m),f G(β) Example 3.24 Consider the same example 3.14 above. Then the tabular representation of ( F, A ) OR ( G, B ) is as follow: u (bright, ostly) (bright, olorful) (heap, ostly) b 1 ( 0.6,0.2, 0.3 ) ( 0.6,0.3, 0.2 ) ( 0.7,0.2, 0.3 ) b 2 ( 0.5,0.1, 0.2 ) ( 0.5,0.1, 0.3 ) ( 0.6,0.1, 0.2 ) b 3 ( 0.7,0.4, 0.3 ) ( 0.7,0.3, 0.3 ) ( 0.8,0.3, 0.5 ) b 4 ( 0.8,0.4, 0.1 ) ( 0.8,0.4, 0.1 ) ( 0.8,0.3, 0.1 ) b 5 ( 0.7,0.1, 0.2 ) ( 0.6,0.3, 0.4 ) ( 0.7,0.1, 0.4 ) u (heap, olorful) (olorful, ostly) (olorful, olorful) b 1 ( 0.7,0.3, 0.2 ) ( 0.6,0.2, 0.3 ) ( 0.4,0.2, 0.2 ) b 2 ( 0.6,0.1, 0.3 ) ( 0.6,0.4, 0.2 ) ( 0.6,0.4, 0.3 ) b 3 ( 0.8,0.3, 0.4 ) ( 0.5,0.6, 0.2 ) ( 0.5,0.7, 0.2 ) b 4 ( 0.6,0.3, 0.2 ) ( 0.8,0.2, 0.1 ) ( 0.8,0.2, 0.3 ) b 5 ( 0.7,0.3, 0.4 ) ( 0.7,0.1, 0.4 ) ( 0.5,0.6, 0.4) Table 9: Tabular representation of the INSS ( O, A B). Proposition if ( F, A ) and ( G, B ) are two INSSs over U, then : (1) [(F,A) (G,B)] = (F,A) (G,B) (2) [(F,A) (G,B)] = (F,A) (G,B) Proof1. Let (F,A)={<b, T F(x) (b), I F(x) (b), F F(x) (b)> b U} and (G,B) = {< b, T G(x) (b), I G(x) (b), F G(x) (b) > b U} be two INSSs over the ommon universe U. Also let (K,A B) = (F,A) (G,B), where, K(α, β) = F(α) G(β) for all (α, β) A B then K(α, β) = {< b, min(t F(α) (b),t G(β) (b)), min(i F(α) (b),i G(β) (b)), max(f F(α) (b),f G(β) (b)) > b U}. Therefore, [(F,A) (G,B)] = (K,A B) = {< b, max(f F(α) (b),f G(β) (b)), min(i F(α) (b),i G(β) (b)), min(t F(α) (b),t G(β) (b)) > b U}. Again (F,A) (G,B) = {< b, max(f F (α) (b)),f G (β) (b)), min(i F (α) (b),i G (β) (b)), min(t F (α) (b), T G (β) (b)) > b U}. = {< b, min(t F(α) (b),t G(β) (b)), min(i F(α) (b),i G(β) (b)), max(f F(α) (b),f G(β) (b)) > b U}. = {< b, max(f F(α) (b), F G(β) (b)), min(i F(α) (b),i G(β) (b)), min(t F(α) (b),t G(β) (b)) > b U}. It follows that [(F,A) (G,B)] = (F,A) (G,B). Proof 2. Let ( F, A ) = {< b, T F(x) (b), I F(x) (b), F F(x) (b) > b U} and (G,B) = {< b, T G(x) (b),i G(x) (b),f G(x) (b) > b U} be two INSSs over the ommon universe U. Also let (O,A B) = (F,A) (G,B), where, O (α,β) = F(α) G(β) for all (α,β) A B. then O(α,β) = {< b, max(t F(α) (b),t G(β) (b)), min(i F(α) (b),i G(β) (b)), min(f F(α) (b),f G(β) (b)) > b U}. [(F,A) (G,B)] = (O,A B) ={< b, min(f F(α) (b),f G(β) (b)), min(i F(α) (b),i G(β) (b)), max(t F(α) (b),t G(β) (b)) > b U}. Again (H,A) (G,B) = {< b,min(f F (α) (b),f G (β) (b)),min(i F (α) (b),i G (β) (b)), max(t F (α) (b),t G (β) (b)),> b U}. JIC for subsription: publishing@wau.org.uk

9 138 Broumi Said et.al.: Intuitionisti Neutrosophi Soft Set = {< b,max(t F(α) (b),t G(β) (b)),min(i F (α) (b),i G (β) (b)),min(f F(α) (b),f G(β) (b))> b U}. = {< b, min(f F(α) (b),f G(β) (b)),min(i F(α) (b),i G(β) (b)), max(t F(α) (b),t G(β) (b)) > b U}. It follows that [(F,A) (G,B)] = (F,A) (G,B). 4. An appliation of intuitionisti neutrosophi soft set in a deision making problem For a onrete example of the onept desribed above, we revisit the blouse purhase problem in Example 3.3. So let us onsider the intuitionisti neutrosophi soft set S = (F,P) (see also Table 10 for its tabular representation), whih desribes the "attrativeness of the blouses" that Mrs. X is going to buy. on the basis of her m number of parameters (e 1,e 2,,e m ) out of n number of blouses(b 1,b 2,,b n ). We also assume that orresponding to the parameter e j (j =1,2,,m) the performane value of the blouse b i (i = 1,2,,n) is a tuple p ij = (T F(ej) (b i ),I F(ej) (b i ),T F(ej) (b i )), suh that for a fixed i that values p ij (j = 1,2,,m) represents an intuitionisti neutrosophi soft set of all the n objets. Thus the performane values ould be arranged in the form of a matrix alled the riteria matrix. The more are the riteria values, the more preferability of the orresponding objet is. Our problem is to selet the most suitable objet i.e. the objet whih dominates eah of the objets of the spetrum of the parameters e j. Sine the data are not risp but intuitionisti neutrosophi soft the seletion is not straightforward. Our aim is to find out the most suitable blouse with the hoie parameters for Mrs. X. The blouse whih is suitable for Mrs. X need not be suitable for Mrs. Y or Mrs. Z, as the seletion is dependent on the hoie parameters of eah buyer. We use the tehnique to alulate the sore for the objets Definition: Comparison matrix The Comparison matrix is a matrix whose rows are labelled by the objet names of the universe suh as b 1,b 2,,b n and the olumns are labelled by the parameters e 1,e 2,,e m. The entries are ij, where ij, is the number of parameters for whih the value of b i exeeds or is equal to the value b j. The entries are alulated by ij =a + d -, where a is the integer alulated as how many times T bi (e j ) exeeds or equal to T bk (e j ), for b i b k, b k U, d is the integer alulated as how many times I bi(ej) exeeds or equal to I bk(ej), for b i b k, b k U and is the integer how many times F bi(ej) exeeds or equal to F bk (e j ), for b i b k, b k U. Definition 4.2. Sore of an objet. The sore of an objet b i is S i and is alulated as : S i = j ij Now the algorithm for most appropriate seletion of an objet will be as follows. Algorithm (1) input the intuitionisti Neutrosophi Soft Set ( F, A). (2) input P, the hoie parameters of Mrs. X whih is a subset of A. (3) onsider the INSS ( F, P) and write it in tabular form. (4) ompute the omparison matrix of the INSS ( F, P). (5) ompute the sore S i of b i, i. (6) find S k = maxi S i (7) if k has more than one value then any one of b i may be hosen. To illustrate the basi idea of the algorithm, now we apply it to the intuitionisti neutrosophi soft set based deision making problem. Suppose the wishing parameters for Mrs. X where P={Bright,Costly, Polystyreneing,Colorful,Cheap}. Consider the INSS ( F, P ) presented into the following table. U Bright ostly Polystyreneing Colorful Cheap b 1 ( 0.6,0.3, 0.4 ) ( 0.5,0.2, 0.6 ) ( 0.5,0.3, 0.4 ) ( 0.8,0.2, 0.3 ) ( 0.6,0.3, 0.2 ) b 2 ( 0.7,0.2, 0.5 ) ( 0.6,0.3, 0.4 ) ( 0.4,0.2, 0.6 ) ( 0.4,0.8, 0.3 ) ( 0.8,0.1, 0.2 ) b 3 ( 0.8,0.3, 0.4 ) ( 0.8,0.5, 0.1 ) ( 0.3,0.5, 0.6 ) ( 0.7,0.2, 0.1 ) ( 0.7,0.2, 0.5 ) b 4 ( 0.7,0.5, 0.2 ) ( 0.4,0.8, 0.3 ) ( 0.8,0.2, 0.4 ) ( 0.8,0.3, 0.4 ) ( 0.8,0.3, 0.4 ) JIC for ontribution: editor@ji.org.uk

10 Journal of Information and Computing Siene, Vol. 8 (2013) No. 2, pp b 5 ( 0.3,0.8, 0.4 ) ( 0.3,0.6, 0.1 ) ( 0.7,0.3, 0.2 ) ( 0.6,0.2, 0.4 ) ( 0.6,0.4, 0,2 ) Table 10: Tabular form of the INSS (F, P). The omparison-matrix of the above INSS ( F, P) is represented into the following table. U Bright Costly Polystyreneing Colorful Cheap b b b b b Table 11: Comparison matrix of the INSS ( F, P ). Next we ompute the sore for eah b i as shown below: U Sore (S i ) b 1 3 b 2 2 b 3 11 b 4 19 b 5 17 Clearly, the maximum sore is the sore 19, shown in the table above for the blouse b 4. Hene the best deision for Mrs. X is to selet b 4, followed by b Conlusions In this paper we study the notion of intuitionisti neutrosophi set initiated by Bhowmik and Pal. We use this onept in soft sets onsidering the fat that the parameters ( whih are words or sentenes ) are mostly intutionisti neutrosophi set; but both the onepts deal with impreision, We have also defined some operations on INSS and prove some propositions. Finally, we present an appliation of INSS in a deision making problem. Aknowledgements. The authors are thankful to the anonymous referee for his valuable and onstrutive remarks that helped to improve the larity and the ompleteness of this paper. 6 Referenes [1]. Zadeh, L. (1965). Fuzzy sets, Inform and Control [2]. Atanassov, K. (1986). Intuitionisti fuzzy sets.fuzzy Sets and Systems [3]. Smarandahe, F. (1999). A Unifying Field in Logis. Neutrosophy: Neutrosophi Probability, Set and Logi. Rehoboth: Amerian Researh Press. [4]. D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) [5]. P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) [6]. M. Irfan Ali, Feng Feng, Xiaoyan Liu, Won Keun Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl. 57 (2009) [7]. M. Bhowmik and M.Pal. Intuitionisti neutrosophi set, Journal of Information and Computing Siene Vol. 4, No. 2, 2009, pp [8]. P. K. Maji, R. Biswas, and A. R. Roy, Soft Set Theory, Comput. Math. Appl. 45 (2003) [9]. P.K.Maji,R.Biswas,andA.R.Roy,An appliation of soft sets in a deision making problem, Comput. Math. Appl. 44 (2002) [10]. F. Smarandahe, Neutrosophi set, a generalisation of the intuitionisti fuzzy sets, Inter. J.Pure Appl. Math. 24 (2005) [11]. P. K. Maji, Neutrosophi soft set,volume x, No. x, (Month 201y), pp. 1- xx ISSN ,Annals of Fuzzy JIC for subsription: publishing@wau.org.uk

11 140 Broumi Said et.al.: Intuitionisti Neutrosophi Soft Set Mathematis and Informatis. [12]. R. Roy and P. K. Maji, A fuzzy soft set theoreti approah appliation of soft set in a deision making problem, Comput. Appl. Math. 203 (2007) [13]. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, vol. 20, pp , JIC for ontribution: editor@ji.org.uk