New Neutrosophic Crisp Topological Concepts

Transcription

1 50 New Neutrosohic Cris Toological Concets A A Salama, Flentin Smarandache and S A Alblowi Deartment of Mathematics and Comuter Science, Faculty of Sciences, Pt Said University, Egyt E mail:drsalama44@gmailcom Deartment of Mathematics, University of New Mexico Gallu, NM, USA E mail:smarand@unmedu Deartment of Mathematics, King Abdulaziz University, Jeddah, Saudia ArabiaE mail: salwaalblowi@hotmailcom Abstract In this aer, we introduce the concet of ""neutrosohic cris neighbhoods system f the neutrosohic cris oint " Added to, we introduce and study the concet of neutrosohic cris local function, and construct a new tye of neutrosohic cris toological sace via neutrosohic cris ideals Possible alication to GIS toology rules are touched uon Keywds: Neutrosohic Cris Point, Neutrosohic Cris Ideal; Neutrosohic Cris Toology; Neutrosohic Cris Neighbhoods INTRODUCTION The idea of "neutrosohic set" was first given by Smarandache [4, 5] In 0 neutrosohic oerations have been investigated by Salama et al [4-] The fuzzy set was introduced by Zadeh [7] The intuitionstic fuzzy set was introduced by Atanassov [,, ] Salama et al []defined intuitionistic fuzzy ideal f a set and generalized the concet of fuzzy ideal concets, first initiated by Sarker [6] Neutrosohy has laid the foundation f a whole family of new mathematical theies, generalizing both their cris and fuzzy counterarts Here we shall resent the neutrosohic cris version of these concets In this aer, we introduce the concet of "neutrosohic cris oints "and "neutrosohic cris neigbourhoods systems" Added to we define the new concet of neutrosohic cris local function, and construct new tye of neutrosohic cris toological sace via neutrosohic cris ideals TERMINOLOGIES We recollect some relevant basic reliminaries, and in articular the wk of Smarandache in [4, 5], and Salama et al [4 -] Definition [] X be a non-emty fixed set A neutrosohic cris set (NCS f sht) A is an object having the fm A A, A, A where A, A and A are subsets of X satisfying A, A and A A Definition [] A A X be a nonemty set and X Then the neutrosohic cris oint N defined by N, c is called a neutrosohic cris oint (NCP f sht) in X, where NCP is a trile ({only one element in X}, the emty set,{the comlement of the same element in X}) Definition [] X be a nonemty set, and X a fixed element in X Then the neutrosohic cris set c NN is called vanishing neutrosohic cris oint (VNCP f sht) in X, where VNCP is a trile (the emty set,{only one element in X},{the comlement of the same element in X}) 4 Definition [] c, N be a NCP in X and A A, A, A a neutrosohic cris set in X (a) N is said to be contained in A ( N A iff A (b) N N be a VNCP in X, and neutrosohic cris set in X Then f sht) A A, A, A a N N is said to be AA Salama, Flentin Smarandache and S A Alblowi, New Neutrosohic Cris Toological Concets

2 5 (b) N N be a VNCP in X, and neutrosohic cris set in X Then A A, A, A a N N is said to be contained in A ( N N A f sht ) iff A 5 Definition [] X be non-emty set, and L a non emty family of NCSs We call L a neutrosohic cris ideal (NCL f sht) on X if i A L and B A BL [heredity], ii A L and B L A B L [Finite additivity] A neutrosohic cris ideal L is called a - neutrosohic cris ideal if M L M jl (countable additivity) j j, imlies The smallest and largest neutrosohic cris ideals on a non-emty set X are and the NSs on X Also, N NC Lf, NCL c are denoting the neutrosohic cris ideals (NCL f sht) of neutrosohic subsets having finite and countable sut of X resectively Meover, if A is a nonemty NS in X, then B NCS : B A is an NCL on X This is called the rincial NCL of all NCSs, denoted by NCL A Proosition [] L j : j J be any non - emty family of neutrosohic cris ideals on a set X Then L j L j are neutrosohic cris ideals on X, where L j j A j j A j and j A j and j A j, Remark [] The neutrosohic cris ideal defined by the single neutrosohic set N is the smallest element of the dered set of all neutrosohic cris ideals on X Proosition [] A neutrosohic cris set A A, A, A in the neutrosohic cris ideal L on X is a base of L iff every member of L is contained in A Neutrosohic Cris Neigbhoods System Definition A A, A, A set X, then,, be a neutrosohic cris set on a, X is called a neutrosohic cris oint An NCP, is said to be belong to a, neutrosohic cris set A A, A, A, of X, denoted by A, if may be defined by two tyes i) Tye : { } A,{ } A and { } A ii) Tye : { } A,{ } A and{ } A, A, B Theem A A A,, and B B B,, be neutrosohic cris subsets of X Then A B iff A imlies B f any neutrosohic cris oint in X A B and A Then two tyes Tye : { } A,{ } A and { } A Tye : { } A,{ } A and{ } A Thus B Conversely, take any x in X A and A and A Then is a neutrosohic cris oint in X and A By the hyothesis B Thus B, Tye : { } B,{ } B and { } B Tye : { } B,{ } B and{ } B Hence A B Theem A A, A, A, be a neutrosohic cris subset of X Then A : A Since : A may be two tyes A A Salama, Flentin Smarandache and S A Alblowi, New Neutrosohic Cris Toological Concets

3 5 Tye : { : A }, : A, A : Tye : { : A }, : A, A A : A, A, A Proosition, A j Hence A j : j J is a family of NCSs in X Then ( a ) iff A j f each j J ( a ) A iff j J such that A j Proosition j A A, A, A and neutrosohic cris sets in X Then a) A B iff f each we have B B, B, B be two A B and f each we have A B b) A B iff f each we have A B and f each we have Proosition Then A B A A, A, A be a neutrosohic cris set in X : A : A A A : Definition f : X Y be a function and be a nutrosohic cris oint in X Then the image of under f, denoted by f ( ), is defined by f ( ) q q q,where q f ( ), q f ( ) q f ( ), and It is easy to see that f ( ) is indeed a NCP in Y, namely f ( ) q, where q f ( ), and it is exactly the same meaning of the image of a NCP under the function f 4 4 Neutrosohic Cris Local functions 4 Definition be a neutrosohic cris oint of a neutrosohic cris toological sace X, A neutrosohic cris neighbourhood ( NCNBD f sht) of a neutrosohic cris oint if there is a neutrosohic cris oen set( NCOS f sht) B in X such that B A 4 Theem X, be a neutrosohic cris toological sace (NCTS f sht) of X Then the neutrosohic cris set A of X is NCOS iff A is a NCNBD of f every neutrosohic cris set A A be NCOS of X Clearly A is a NCBD of any A Conversely, let A Since A is a NCBD of, there is a NCOS B in X such that B A So we have A : A B : A A and A B : A Since each B is NCOS hence 4 Definition X, be a neutrosohic cris toological saces (NCTS f sht) and L be neutrsohic cris ideal (NCL, f sht) on X A be any NCS of X Then the neutrosohic cris local function L, of A is the union of all neutrosohic cris oints( NCP, f sht) P, such that if U N( ) and NA ) X : A U L f every U nbd of N(P), ) is called a neutrosohic cris local function of A with resect to which it will be denoted by and L ), simly L 4 Examle One may easily verify that If L= { N}, then N CA ) NCcl( A), f any neutrosohic cris set A NCSs on X any If L all NCSson X then ) N A NCSs on X, f 4 Theem X, be a NCTS and L, L be two toological neutrosohic cris ideals on X Then f any neutrosohic cris sets A, B of X then the following statements are verified A B ) NCB ), i) ii) L L L, ) ( L, ) ( iii) NCcl( A ) NCcl( A) A A Salama, Flentin Smarandache and S A Alblowi, New Neutrosohic Cris Toological Concets

4 5 iv) NC, v) A B NCB vi) NC( A B) NCB vii) L NC A viii) ) is neutrosohic cris closed set i) Since B A B A, let L U L f every U N U L, then NB L, then By hyothesis we get ii) Clearly L L imlies ) ( L, ) as there may be other IFSs which belong to L so that f GIFP L, but P may not be contained in L iii) Since N L f any NCL on X, therefe by (ii) and Examle, L O N NCcl( A) f any NCS A on X Suose P NCcl ( A L ) So f every U NCP NC( A ) U N, there exists P q q q L U such that f every V NCNBD of P N P, AU L Since U V N then A U V L, f every which leads to A U L U N( P ) therefe P NC A ) and so ( L NCcl NA While, the other inclusion follows directly Hence NCcl ( ) But the inequality Ncl ( ) iv) The inclusion NCB NC A B follows directly by (i) To show the other imlication, let B then f every U NC( ), A B U L, i e, A U B U L then, we have two cases A U L and B U L the converse, this means that exist U, U NP such that A U L, B U L, A U L and B U L Then A U U L and B U U L this gives A B U U L, U U NC( P) which contradicts the hyothesis Hence the equality holds in various cases vi) By (iii), we have NCcl( ) NCcl ( ) X, be a NCTS and L be NCL on X us define the neutrosohic cris closure oerat NCcl ( A) A NC( A ) f any NCS A of X Clearly, let NCcl (A) is a neutrosohic cris oerat NC (L) be NCT generated by NCcl f eve- ie NC L A : NCcl ( A ) A now L N NCcl A A A NCclA ry neutrosohic cris set A So, N ) c c ( N L all NCSs on X NCcl A A, Again because N, f every neutrosohic cris set A so NC L is the neutrosohic cris discrete toology on X So we can conclude by Theem 4(ii) NC ( N ) NC L ie NC NC, f any neutrosohic ideal L on X In articular, we have f two toological neutrosohic ideals L, and L on X, L L NC L NC L 4 Theem, be two neutrosohic cris toologies on X Then f any toological neutrosohic cris ideal L on X, imlies NA L, ) NA ), f every A L then NC NC Clear A basis L, ( NC f NC (L) can be described as follows: NC L, A B : A, B L Then we have the following theem 44 Theem NC L, A B : A, B L Fms a basis f the generated NT of the NCT X, with toological neutrosohic cris ideal L on X Straight fward The relationshi between NC and NC (L) established throughout the following result which have an immediately roof 45 Theem, be two neutrosohic cris toologies on X Then f any toological neutrosohic ideal L on X, imlies NC NC 46 Theem, be a NCTS and L, L be two neutrosohic cris ideals on X Then f any neutrosohic cris set A in X, we have A A Salama, Flentin Smarandache and S A Alblowi, New Neutrosohic Cris Toological Concets

5 54 i) L L, L, NC ( L ) L, NC ( L ) NC L L ) NC ( L ) ( L ) NC ( L ( ) ii) ( L L L, this means that there exists U NCP such that A U L L ie There exists L and L such that A U because of have A U and the heredity of L, and assuming ON Thus we A U therefe U A L and U A L Hence L, NC L P L, NC L because must belong to either but not to both This gives L L L, NC ( L ) L, NC ( ), L To show the second inclusion, let us assume P L, NC L This imlies that there exist U N P and L such that U A L By the heredity of L, if we assume that A and define U A Then we A U L Thus, have L L L L, NC ( L ) L, NC ( ), L and similarly, we can get L L L, ( ), L This gives the other inclusion, which comlete the roof 4 Collary cris ideal L on X Then, be a NCTS with toological neutrosohic i) ) ) and NC NC( NC ) ii) NC L L ) NC ( L ) NC ( ) ( L Follows by alying the revious statement [6] IM Hanafy, AA Salama and KM Mahfouz,," Neutrosohic Classical Events and Its Probability" International Journal of Mathematics and Comuter Alications Research(IJMCAR) Vol(),Issue,Mar (0) 7-78 [7] AA Salama and SA Alblowi, "Generalized Neutrosohic Set and Generalized Neutrosohic Toological Saces ", Journal Comuter Sci Engineering, Vol () No (7) (0)9- [8] AA Salama and SA Alblowi, Neutrosohic Set and Neutrosohic Toological Saces, ISORJ Mathematics, Vol(), Issue(), (0) --5 [9] A A Salama, "Neutrosohic Cris Points & Neutrosohic Cris Ideals", Neutrosohic Sets and Systems, Vol, No, (0) [0] A A Salama and F Smarandache, "Filters via Neutrosohic Cris Sets", Neutrosohic Sets and Systems, Vol, No, (0) 4-8 [] AA Salama and SA Alblowi, Intuitionistic Fuzzy Ideals Saces, Advances in Fuzzy Mathematics, Vol(7), Number, (0) 5-60 [] AA Salama, and HElagamy, "Neutrosohic Filters" International Journal of Comuter Science Engineering and Infmation Technology Reseearch (IJCSEITR), Vol, Issue(),Mar 0,(0) 07- [] A A Salama, FSmarandache and Valeri Kroumov "Neutrosohic Cris Sets & Neutrosohic Cris Toological Saces" Bulletin of the Research Institute of Technology (Okayama University of Science, Jaan), in January-February (04) (Acceted) [4] Flentin Smarandache, Neutrosohy and Neutrosohic Logic, First International Conference on Neutrosohy, Neutrosohic Logic, Set, Probability, and Statistics University of New Mexico, Gallu, NM 870, USA(00) [5] F Smarandache A Unifying Field in Logics: Neutrosohic Logic Neutrosohy, Neutrosohic Set, Neutrosohic Probability American Research Press, Rehoboth, NM, (999) [6] Debasis Sarker, Fuzzy ideal they, Fuzzy local function and generated fuzzy toology, Fuzzy Sets and Systems 87, 7 (997) [7] LA Zadeh, Fuzzy Sets, Infm and Control 8, 8-5(965) References [] K Atanassov, intuitionistic fuzzy sets, in VSgurev, ed, Vii ITKRS Session, Sofia (June 98 central Sci and Techn Library, Bulg Academy of Sciences (984) [] K Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 0, 87-96,(986) [] K Atanassov, Review and new result on intuitionistic fuzzy sets, rerint IM-MFAIS--88, Sofia, (988) [4] S A Alblowi, A A Salama & Mohmed Eisa, New Concets of Neutrosohic Sets, International Journal of Mathematics and Comuter Alications Research (IJMCAR),Vol, Issue, Oct (0) 95-0 [5] I Hanafy, AA Salama and K Mahfouz, Crelation of Neutrosohic Data, International Refereed Journal of Engineering and Science (IRJES), Vol(), Issue (0) PP9- Received: June 0, 04 Acceted: June 5, 04 A A Salama, Flentin Smarandache and S A Alblowi, New Neutrosohic Cris Toological Concets