Book Series, Vol. 16, 2017 Florentin Smarandache and Mohamed Abdel-Baset ISBN 978-1-59973-526-9 ISBN 978-1-59973-526-9 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering Quarterly Editor-in-Chief: Associate Editors: Prof. FLORENTIN SMARANDACHE W. B. Vasantha Kandasamy, Indian Institute of Technology, Chennai, Tamil Nadu, India. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. A. A. Salama, Faculty of Science, Port Said University, Egypt. Address: Yanhui Guo, School of Science, St. Thomas University, Miami, USA. Neutrosophic Sets and Systems Francisco Gallego Lupiaňez, Universidad Complutense, Madrid, Spain. Department of Mathematics and Science Peide Liu, Shandong University of Finance and Economics, China. University of New Mexico Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. 705 Gurley Avenue Jun Ye, Shaoxing University, China. Gallup, NM 87301, USA Ştefan Vlăduţescu, University of Craiova, Romania. E-mail: [email protected] Valeri Kroumov, Okayama University of Science, Japan. Home page: http://fs.gallup.unm.edu/NSS Dmitri Rabounski and Larissa Borissova, independent researchers. Surapati Pramanik, Nandalal Ghosh B.T. College, Panpur, West Bengal, India. Irfan Deli, Kilis 7 Aralık University, 79000 Kilis, Turkey. Associate Editor-in-Chief: Rıdvan Şahin, Faculty of Science, Ataturk University, Erzurum, Turkey. Dr. Mohamed Abdel-Basset Luige Vladareanu, Romanian Academy, Bucharest, Romania. Faculty of Computers and Informatics A. A. A. Agboola, Federal University of Agriculture, Abeokuta, Nigeria. Le Hoang Son, VNU Univ. of Science, Vietnam National Univ. Hanoi, Vietnam. Operations Research Dept. Huda E. Khalid, University of Telafer, College of Basic Education, Telafer - Mosul, Iraq. Zagazig University, Egypt Maikel Leyva-Vázquez, Universidad de Guayaquil, Guayaquil, Ecuador. Muhammad Akram, University of the Punjab, New Campus, Lahore, Pakistan. Paul Wang, Pratt School of Engineering, Duke University, Durham, USA. Darjan Karabasevic, University Business Academy, Novi Sad, Serbia. Dragisa Stanujkic, John Naisbitt University, Belgrade, Serbia. Edmundas K. Zavadskas, Vilnius Gediminas Technical University, Vilnius, Lithuania. Contents Volume 16 2017 Ferhat Taș, Selçuk Topal. Bèzier Curve Modeling for Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Neutrosophic Data Problem ………………....………. 3 Tapan Kumar Roy, F. Smarandache. Neutrosophic Cu- 44 bic MCGDM Method Based on Similarity Measure ….. P. Iswarya, Dr. K. Bageerathi. A Study on Neutrosoph- ic Frontier and Neutrosophic Semi-frontier in Neutro- 6 Eman.M.El-Nakeeb, Hewayda ElGhawalby, A.A. sophic Topological Spaces ....………………………..... Salama, S.A.El-Hafeez. Neutrosophic Crisp Mathemat- 57 ical Morphology .........…………………………………. I. Arokiarani, R. Dhavaseelan, S. Jafari, M. Parimala. On Some New Notions and Functions in Neutrosophic 16 Kanika Bhutani, Swati Aggarwal. Neutrosophic Rough Topological Spaces …………………………………..... Soft Set - A Decision Making Approach to Appendici- 70 tis Problem .....................………………………………. R. Dhavaseelan, S. Jafari, R. Narmada Devi, Md. Hanif Page. Neutrosophic Baire Spaces ..……………………. 20 F. Smarandache, N. Abbas, Y. Chibani, B. Hadjadji, Z. A. Omar. PCR5 and Neutrosophic Probability in Target 76 R. Cabezas Padilla, J. González Ruiz, M. Villegas Ala- Identification (revisited) ……................………………. va, M. Leyva Vázquez. A Knowledge-based Recom- 24 Suriana Alias, Daud Mohamad, Adibah Shuib. Rough mendation Framework using SVN Numbers .............…. Neutrosophic Multisets ……......………………………. 80 Okpako Abugor Ejaita, Asagba P.O. An Improved Framework for Diagnosing Confusable Diseases 28 E. J. Henríquez Antepara, J. E. Arízaga Gamboa, M. R. Using Neutrosophic Based Neural Network ………...... Campoverde Méndez, M. E. Peña González. Compe- 89 tencies Interdepencies Analysis based on Neutrosophic R. Dhavaseelan, S. Jafari, F. Smarandache. Compact Cognitive Mapping ……....……………………………. Open Topology and Evaluation Map via Neutrosophic 35 Sets ………………………………….............................. Nguyen Xuan Thao, Florentin Smarandache, Nguyen Van Dinh. Support-Neutrosophic Set: A New Concept 93 R. Dhavaseelan, M. Parimala, S. Jafari, F. Smaran- in Soft Computing …………………….....……………. dache. On Neutrosophic Semi-Supra Open Set and 39 Neutrosophic Semi-Supra Continuous Functions ....…... The Educational Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA. Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, Vol. 16, 2017 Neutrosophic Sets and Systems An International Book Series in Information Science and Engineering Copyright Notice Copyright @ Neutrosophics Sets and Systems ademic or individual use can be made by any user without All rights reserved. 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Any part of Neutro- tive: this means that electronic copying, distribution and sophic Sets and Systems howsoever used in other publica- printing of both full-size version of the book and the in- tions must include an appropriate citation of this book. dividual papers published therein for non-commercial, ac- Information for Authors and Subscribers “Neutrosophic Sets and Systems” has been created for pub- Neutrosophic Probability is a generalization of the classical lications on advanced studies in neutrosophy, neutrosophic set, probability and imprecise probability. neutrosophic logic, neutrosophic probability, neutrosophic statis- Neutrosophic Statistics is a generalization of the classical tics that started in 1995 and their applications in any field, such statistics. as the neutrosophic structures developed in algebra, geometry, What distinguishes the neutrosophics from other fields is the topology, etc. <neutA>, which means neither <A> nor <antiA>. The submitted papers should be professional, in good Eng- <neutA>, which of course depends on <A>, can be indeter- lish, containing a brief review of a problem and obtained results. minacy, neutrality, tie game, unknown, contradiction, ignorance, Neutrosophy is a new branch of philosophy that studies the imprecision, etc. origin, nature, and scope of neutralities, as well as their interac- tions with different ideational spectra. This theory considers every notion or idea <A> together with All submissions should be designed in MS Word format using its opposite or negation <antiA> and with their spectrum of neu- our template file: tralities <neutA> in between them (i.e. notions or ideas support- http://fs.gallup.unm.edu/NSS/NSS-paper-template.doc. ing neither <A> nor <antiA>). The <neutA> and <antiA> ideas together are referred to as <nonA>. A variety of scientific books in many languages can be down- Neutrosophy is a generalization of Hegel's dialectics (the last one loaded freely from the Digital Library of Science: is based on <A> and <antiA> only). http://fs.gallup.unm.edu/eBooks-otherformats.htm. According to this theory every idea <A> tends to be neutralized and balanced by <antiA> and <nonA> ideas - as a state of equi- To submit a paper, mail the file to the Editor-in-Chief. To order librium. printed issues, contact the Editor-in-Chief. This book series is a In a classical way <A>, <neutA>, <antiA> are disjoint two by non-commercial, academic edition. It is printed from private two. But, since in many cases the borders between notions are donations. vague, imprecise, Sorites, it is possible that <A>, <neutA>, <an- tiA> (and <nonA> of course) have common parts two by two, or Information about the neutrosophics you get from the UNM even all three of them as well. website: Neutrosophic Set and Neutrosophic Logic are generalizations http://fs.gallup.unm.edu/neutrosophy.htm. of the fuzzy set and respectively fuzzy logic (especially of intui- tionistic fuzzy set and respectively intuitionistic fuzzy logic). In The home page of the book series can be accessed on neutrosophic logic a proposition has a degree of truth (T), a de- http://fs.gallup.unm.edu/NSS. gree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ]-0, 1+[. Copyright © Neutrosophic Sets and Systems Neutrosophic Sets and Systems, Vol. 16, 2017 3 University of New Mexico Be`zier Curve Modeling for Neutrosophic Data Problem FerhatTas¸1,Selc¸ukTopal2 1DepartmentofMathematics,IstanbulUniversity,Istanbul,Turkey.E-mail:[email protected] 2DepartmentofMathematics,BitlisErenUniversity,Bitlis,Turkey.E-mail:[email protected] Abstract: Neutrosophic set concept is defined with membership, lemforthefirsttime. Thismodelisbasedonneutrosophicsetsand non-membershipandindeterminacydegrees.Thisconceptistheso- neutrosophicrelations. Neutrosophiccontrolpointsaredefinedac- lutionandrepresentationoftheproblemswithvariousfields.Inthis cordingtothesepoints,resultinginneutrosophicBe`ziercurves. paper,ageometricmodelisintroducedforNeutrosophicdataprob- Keywords:NeutrosophicSets,NeutrosophicLogic,Be`zierCurve 1 Introduction entirelyonsubjectspace(discourseuniverse). Inthissense, the conceptofneutrosophicsetisthesolutionandrepresentationof Whiletoday’stechnologiesarerapidlydeveloping,thecontribu- theproblemswithvariousfields. tion of mathematics is fundamental and leading the science. In Recently,geometricinterpretationsofdatathatuncertaintruth particular, the developments in geometry are not only modeling were presented by Wahab and friends [4, 5, 6, 7]. They studied themathematicsoftheobjectsbutalsobeinggeometricallymod- geometricmodelsoffuzzyandintuitionisticfuzzydataandgave eledinmostabstractconcepts. Whatistheuseoftheseabstract fuzzyinterpolationandBe`ziercurvemodeling. Inthispaper,we conceptsinmodeling? Inthefutureofscience,therewillbearti- considerageometricmodelingofneutrosophicdata. ficialintelligence. Forthedevelopmentofthistechnology,many branchesofscienceworktogetherandespeciallythetopicssuch 2 Preliminaries aslogic, datamining, quantumphysics, machinelearningcome to the forefront. Of course, the place where these areas can co- operateisthecomputerenvironment. Datacanbetransferredin In this section, we will first give some fundamental definitions variousways. Oneofthemistotransferthedataasageometric dealing with Bzier curve and Neutrosophic sets (elements). We model. The first method that comes to mind in terms of a ge- willthenintroducethenewdefinitionsneededtoformaNeutro- ometric model is the Bzier technique. Although this method is sophicBe`ziercurve. generallyusedforcurveandsurfacedesigns, itisusedinmany Definition1 Let P ,(i = 0,1,2,...,n),P ∈ E3 be the set of disciplinesrangingfromthesolutionofdifferentialequationsto i i points. ABe´ziercurvewithdegreenisdefinedby robotmotionplanning. The embodied state of the adventure of obtaining meaning B(t)=Bn(t)P ,t∈[0,1] (1) andmathematicalresultsfromuncertaintystates(fuzzy)wasbe- i i gun by Zadeh [1]. Fuzzy sets proposed by Zadeh provided a whereBn(t) = (cid:80)n (cid:0)n(cid:1)(1−t)n−iti andP aretheBernstein newdimensiontotheconceptofclassicalsets. Atanassovintro- i i=0 i i polynomialfunctionandthecontrolpoints, respectively. Notice ducedintuitionisticfuzzysetsdealingwithmembershipandnon- thatthereare(n+1)-controlpointsforaBe`ziercurvewithde- membershipdegrees[2]. NeutrosophywasproposedbySmaran- gree n. Because n−interpolation is done with (n+1)-control dache as a mathematical application of the concept neutrality points[8,9,10,11]. [3]. Neutrosophicsetconceptisdefinedwithmembership,non- membership and indeterminacy degrees. Neutrosophic set con- Definition2 Let E be a universe and A ⊆ E. N = ceptisseparatedfromintuitionisticfuzzysetbythedifferenceas {(x,T(x),I(x),F(x)) : x ∈ A} is a neutrosophic element follow: intuitionistic fuzzy sets are defined by degree of mem- where T = N → [0,1] (membership function), I = N → p p bershipandnon-membershipdegreeand,uncertaintydegreesby [0,1] (indeterminacy function) and F = N → [0,1] (non- p the1-(membershipdegreeplusnon-membershipdegree),while membershipfunction). degree of uncertainty are considered independently of the de- gree of membership and non-membership in neutrosophic sets. Definition3 Let A∗ = {(x,T(x),I(x),F(x)) : x ∈ A} and Here,membership,non-membershipanduncertainty(indetermi- B∗ = {(y,T(y),I(y),F(y)) : y ∈ B} be neutrosophic ele- nacy)degreescanbejudgedaccordingtotheinterpretationinthe ments. NR = {((x,y),T(x,y),I(x,y),F(x,y)) : (x,y) ∈ spaces to be used, such as truth and falsity degrees. It depends A×B}isaneutrosophicrelationonA∗andB∗. Ferhat Taș, Selçuk Topal. Bèzier Curve Modeling for Neutrosophic Data Problem 4 Neutrosophic Sets and Systems, Vol. 16, 2017 3 Neutrosophic Be`zier Model Point Truthdegree Indeterminacydegree Falsitydegree (2,3) 0.6 0.4 0.7 Definition4 NS of P∗ in space N is NCP and P∗ = {Pi∗} (1,3) 0.5 0.6 0.2 where i = 0,...,n is a set of NCPs where there exists (4,6) 0.7 0.5 0.3 Tp = N → [0,1] as membership function, Ip = N → [0,1] (3,5) 0.3 0.2 0.7 as indeterminacy function and F = N → [0,1] as non- p membershipfunctionwith Table1: Aneutrosophicdataexample  0 ifP ∈/ N  i T (P∗)= a∈(0,1) ifP ∈∼N p i 1 ifP ∈N i  0 ifP ∈/ N  i F (P∗)= c∈(0,1) ifP ∈∼N p i 1 ifP ∈N i  0 ifP ∈/ N  i I (P∗)= e∈(0,1) ifP ∈∼N p i 1 ifP− ∈N. i Be`zier Neutrosophic curves are generated based on the con- trol points from one of TC = {(x,y,T(x,y))}, IC = {(x,y,I(x,y))} and FC = {(x,y,F(x,y))} sets. Thus, there will be three different neutrosophic Be`zier curve models for a Figure1: NeutrosophicBe´ziercurvesfordatainTable1. neutrosophicrelationandvariablesxandy. Aneutrosophiccon- trolpointrelationcanbedefinedasasetofn+1pointsthatshows a position and coordinate of a location and is used to described stochastic processes. In this article, we used the Be´zier tech- threecurvewhicharedenotedby nique for visualizing neutrosophic data. This model is suitable NR ={NR ,NR ,...,NR } pi p0 p1 pn for statisticians, data scientists, economists and engineers. Fur- andcanbewrittenas thermore,thedifferentialgeometricpropertiesofthismodelcan {((x ,y ),T(x ,y ),I(x ,y ),F(x ,y )) 0 0 0 0 0 0 0 0 be investigated as in [8] for classification of neutrosophic data. ,...,((x ,y ),T(x ,y ),I(x ,y ),F(x ,y ))} n n n n n n n n Ontheotherhand, transformingtheimagesofobjectsintoneu- inordertocontroltheshapeofacurvefromaneutrosophicdata. trosophic data is an important problem [12]. In our model, the curve and the data can be transformed into each other by the Definition5 A neutrosophic Be´zier curve with degree n is de- blossoming method, which can be used in neutrosophic image finedby processing. This and similar applications can be studied in the NB(t)=Bn(t)NR ,t∈[0,1] (2) future. i pi EverysetofTC = {(x,y,T(x,y))},IC = {(x,y,I(x,y))} and FC = {(x,y,F(x,y))} determines a Be´zier curve. Thus References wegetthreeBe´ziercurves. ANeutrosophicBe´ziercurveisde- finedbythesethreecurves. Soitisasetofcurvesjustlikeinits [1] Zadeh,L.A.(1965).Fuzzysets.Informationandcontrol,8(3),338-353. definition. [2] Atanassov,K.T.(1986).Intuitionisticfuzzysets.FuzzysetsandSystems, As an illustrative example, we can consider a neutrosophic 20(1),87-96. datainTable1. OnecanseetherearethreequbicBe´ziercurves. [3] Smarandache,F.(2005).AUnifyingFieldinLogics:NeutrosophicLogic. Neutrosophy,NeutrosophicSet,NeutrosophicProbability: Neutrosophic 4 Conclusion and Future Work Logic.Neutrosophy,NeutrosophicSet,NeutrosophicProbability.Infinite Study. Visualization or geometric modeling of data plays an impor- [4] Wahab,A.F.,Ali,J.M.,Majid,A.A.andTap,A.O.M.,(2004),July.Fuzzy tantroleindatamining, databases, stockmarket, economy, and setingeometricmodeling.InComputerGraphics, ImagingandVisual- Ferhat Taș, Selçuk Topal. Bèzier Curve Modeling for Neutrosophic Data Problem Neutrosophic Sets and Systems, Vol. 16, 2017 5 ization,2004.CGIV2004.Proceedings.InternationalConferenceon(pp. [8] Tantay, B., Tas¸, F., (2011).TheCurvatureofaBe´zierControlPolyline, 227-232).IEEE. Math.Comput.Appl.16,no.2:350-358. [5] Wahab, Abd Fatah, Jamaludin Md Ali, and Ahmad Abd Majid. (2009) [9] Gallier,J.H.(2000).Curvesandsurfacesingeometricmodeling: theory ”FuzzyGeometricModeling.”InComputerGraphics,ImagingandVisu- andalgorithms.MorganKaufmann. alization,2009.CGIV’09.SixthInternationalConferenceon,pp.276-280. IEEE. [10] Farin, G. E. (2002). Curves and surfaces for CAGD: a practical guide. MorganKaufmann. [6] Wahab, Abd Fatah, Rozaimi Zakaria, and Jamaludin Md Ali. (2010) ”FuzzyinterpolationrationalBe´ziercurve.”InComputerGraphics,Imag- [11] Marsh, D. (2006). Applied geometry for computer graphics and CAD. ingandVisualization(CGIV),2010SeventhInternationalConferenceon, SpringerScience&BusinessMedia. pp.63-67.IEEE,2010. [12] Cheng, H. D., Guo, Y. (2008). A new neutrosophic approach to image [7] Wahab,AbdFatah,MohammadIzatEmirZulkifly,andMohdSallehuddin thresholding.NewMathematicsandNaturalComputation,4(03),291-308. Husain.,(2016)”Be´ziercurvemodelingforintuitionisticfuzzydataprob- lem.”AIPConferenceProceedings.Eds.ShaharuddinSalleh, etal.Vol. 1750.No.1.AIPPublishing. Received: April 3, 2017. Accepted: April 24, 2017. Ferhat Taș, Selçuk Topal. Bèzier Curve Modeling for Neutrosophic Data Problem 6 Neutrosophic Sets and Systems, Vol. 16, 2017 University of New Mexico A Study on Neutrosophic Frontier and Neutrosophic Semi-frontier in Neutrosophic Topological Spaces P. Iswarya1 and Dr. K. Bageerathi2 1Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur, India E mail ID : [email protected] 2Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur, India E mail ID : [email protected] ABSTRACT. In this paper neutrosophic frontier and Kharal [4] in 2014. In this paper, we are extending neutrosophic semi-frontier in neutrosophic topology are the above concepts to neutrosophic topological introduced and several of their properties, characterizations spaces. We study some of the basic properties of and examples are established. neutrosophic frontier and neutrosophic semi-frontier in neutrosophic topological spaces with examples. MATHEMATICS SUBJECT CLASSIFICATION Properties of neutrosophic semi-interior, neutros- (2010) : 03E72 ophic semi-closure, neutrosophic frontier and neut- KEYWORDS : Neutrosophic frontier and Neutrosophic rosophic semi-frontier have been obtained in neutros- semi-frontier. ophic product related spaces. I. INTRODUCTION II.NEUTROSOPHIC FRONTIER Theory of Fuzzy sets [21], Theory of Intuitionistic fuzzy sets [2], Theory of Neutrosophic In this section, the concepts of the sets [10] and the theory of Interval Neutrosophic sets neutrosophic frontier in neutrosophic topological [13] can be considered as tools for dealing with space are introduced and also discussed their uncertainties. However, all of these theories have characterizations with some related examples. their own difficulties which are pointed out in [10]. In 1965, Zadeh [21] introduced fuzzy set theory as a Definition 2.1 Let , ,   [0, 1] and  +  +   1. mathematical tool for dealing with uncertainties A neutrosophic point [ NP for short ] x of X is a where each element had a degree of membership. The (,,) NS of X which is defined by Intuitionistic fuzzy set was introduced by Atanassov [2] in 1983 as a generalization of fuzzy set, where besides the degree of membership and the degree of In this case, x is called the support of x non-membership of each element. The neutrosophic (,,) and ,  and  are called the value, intermediate set was introduced by Smarandache [10] and value and the non-value of x , respectively. A NP explained, neutrosophic set is a generalization of (,,) x is said to belong to a NS A =   ,  ,   in X, Intuitionistic fuzzy set. In 2012, Salama, Alblowi (,,) A A A denoted by x  A if    (x),    (x) and [18], introduced the concept of Neutrosophic (,,) A A topological spaces. They introduced neutrosophic   A(x). Clearly a neutrosophic point can be topological space as a generalization of Intuitionistic represented by an ordered triple of neutrosophic fuzzy topological space and a Neutrosophic set points as follows : x(,,) = ( x , x , C ( x C ()) ). besides the degree of membership, the degree of A class of all NPs in X is denoted as NP (X). indeterminacy and the degree of non-membership of each element. Definition 2.2 Let X be a NTS and let A  NS (X). The concepts of neutrosophic semi-open Then x  NP (X) is called a neutrosophic (,,) sets, neutrosophic semi-closed sets, neutrosophic frontier point [ NFP for short ] of A if x  (,,) semi-interior and neutrosophic semi-closure in NCl (A)  NCl (C (A)). The intersection of all the neutrosophic topological spaces were introduced by NFPs of A is called a neutrosophic frontier of A and P. Iswarya and Dr. K. Bageerathi [12] in 2016. is denoted by NFr (A). That is, Frontier and semifrontier in intuitionistic fuzzy NFr (A) = NCl (A)  NCl (C (A)). topological spaces were introduced by Athar P. Iswarya, Dr. K. Bageerathi. A Study on Neutrosophic Frontier and Neutrosophic Semi-frontier in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 16, 2017 7 Proposition 2.3 For each A  NS(X), A  NFr (A)  Example 2.7 From Example 2.4, NFr (C) = G  C. NCl (A). But C  C (). Proof : Let A be the NS in the neutrosophic topological space X. Then by Definition 2.2, Theorem 2.8 If a NS A is NOS, then NFr (A)  A  NFr (A) = A  ( NCl (A)  NCl (C (A)) ) C (A). = (A  NCl (A) )  ( A  NCl (C (A)) ) Proof : Let A be the NS in the neutrosophic  NCl (A)  NCl (C (A)) topological space X. Then by Definition 4.3 [18] ,  NCl (A) A is NOS implies C (A) is NCS in X. By Theorem 2.6, Hence A  NFr (A)  NCl (A). NFr (C (A))  C (A) and by Theorem 2.5, we get NFr (A)  C (A). From the above proposition, the inclusion cannot be replaced by an equality as shown by the following The converse of the above theorem is not true as example. shown by the following example. Example 2.4 Let X = { a, b } and  = { 0 , A, B, C, Example 2.9 From Example 2.4, NFr (G) = G  N D, 1 }. Then (X, ) is a neutrosophic topological C (G) = C. But G  . N space. The neutrosophic closed sets are C () = { 1 , N E, F, G, H, 0 } where Theorem 2.10 For a NS A in the NTS X, C (NFr (A)) N A =  ( 0.5, 1, 0.1), (0.9, 0.2, 0.5) , = NInt (A)  NInt (C (A)). B =  ( 0.2, 0.5, 0.9), (0, 0.5, 1) , Proof : Let A be the NS in the neutrosophic C =  ( 0.5, 1, 0.1), (0.9, 0.5, 0.5) , topological space X. Then by Definition 2.2, C (NFr (A)) = C (NCl (A)  NCl (C (A))) D =  ( 0.2, 0.5, 0.9), (0, 0.2, 1) , By Proposition 3.2 (1) [18] , E =  ( 0.1, 0, 0.5), (0.5, 0.8, 0.9) , = C (NCl (A))  C (NCl (C (A))) F =  ( 0.9, 0.5, 0.2), (1, 0.5, 0) , By Proposition 4.2 (b) [18] , G =  ( 0.1, 0, 0.5), (0.5, 0.5, 0.9)  and = NInt (C (A))  NInt (A) H =  ( 0.9, 0.5, 0.2), (1, 0.8, 0) . Hence C (NFr (A)) = NInt (A)  NInt (C (A)). Here NCl (A) = 1 and NCl (C (A)) = NCl (E) = E. N Then by Definition 2.2, NFr (A) = E. Theorem 2.11 Let A  B and B  NC (X) ( resp., Also A  NFr (A) =  (0.5, 1, 0.1), (0.9, 0.8, 0.5)   B  NO (X) ). Then NFr (A)  B ( resp., NFr (A)  1 . Therefore NCl (A) = 1 ⊈  (0.5, 1, 0.1), (0.9, 0.8, N N C (B) ), where NC (X) ( resp., NO (X) ) denotes the 0.5) . class of neutrosophic closed ( resp., neutrosophic open) sets in X. Theorem 2.5 For a NS A in the NTS X, NFr (A) = Proof : By Proposition 1.18 (d) [12] , A  B , NFr (C (A)). Proof : Let A be the NS in the neutrosophic NCl (A)  NCl (B) -------------------- (1). topological space X. Then by Definition 2.2, By Definition 2.2, NFr (A) = NCl (A)  NCl (C (A)) NFr (A) = NCl (A)  NCl (C (A)) = NCl (C (A))  NCl (A)  NCl (B)  NCl (C (A)) by (1) = NCl (C (A))  NCl (C (C (A)))  NCl (B) By Definition 4.4 (b) [18] , Again by Definition 2.2, = NFr (C (A)) = B Hence NFr (A) = NFr (C (A)). Hence NFr (A)  B. Theorem 2.12 Let A be the NS in the NTS X. Then Theorem 2.6 If a NS A is a NCS, then NFr (A)  A. Proof : Let A be the NS in the neutrosophic NFr (A) = NCl (A) – NInt (A). Proof : Let A be the NS in the neutrosophic topological space X. Then by Definition 2.2, topological space X. By Proposition 4.2 (b) [18] , NFr (A) = NCl (A)  NCl (C (A)) C (NCl (C (A))) = NInt (A) and by Definition 2.2,  NCl (A) NFr (A) = NCl (A)  NCl (C (A)) By Definition 4.4 (a) [18] , = NCl (A) – C (NCl (C (A))) = A by using A – B = A  C (B) Hence NFr (A)  A, if A is NCS in X. By Proposition 4.2 (b) [18] , = NCl (A) – NInt (A) The converse of the above theorem needs not be Hence NFr (A) = NCl (A) – NInt (A). true as shown by the following example. P. Iswarya, Dr. K. Bageerathi. A Study on Neutrosophic Frontier and Neutrosophic Semi-frontier in Neutrosophic Topological Spaces 8 Neutrosophic Sets and Systems, Vol. 16, 2017 Theorem 2.13 For a NS A in the NTS X, Then C (A ) =  ( 0.2, 0.1, 0.7), (0.3, 0.1, 0.5) . Then 2 NFr (NInt (A))  NFr (A). by Definition 2.2, NFr (A ) = G. 2 Proof : Let A be the NS in the neutrosophic Therefore NFr (A ) = G ⊈ 0 = NFr (NCl (A )). 2 N 2 topological space X. Then by Definition 2.2, NFr (NInt (A)) = NCl (NInt (A))  NCl (C (NInt (A))) Theorem 2.17 Let A be the NS in the NTS X. Then By Proposition 4.2 (a) [18] , NInt (A)  A – NFr (A). = NCl (NInt (A))  NCl (NCl (C (A))) Proof : Let A be the NS in the neutrosophic By Definition 4.4 (b) [18] , topological space X. Now by Definition 2.2, = NCl (NInt (A))  NCl (C (A)) A – NFr (A) = A − (NCl (A)  NCl (C (A))) By Definition 4.4 (a) [18] , = ( A – NCl (A))  (A – NCl (C (A)))  NCl (A)  NCl (C (A)) = A – NCl (C (A)) Again by Definition 2.2,  NInt (A). = NFr (A) Hence NInt (A)  A – NFr (A). Hence NFr (NInt (A))  NFr (A). Example 2.18 From Example 2.14, A – NFr (A ) = 1 1 The converse of the above theorem is not true as  ( 0.3, 0.2, 0.8), (0.1, 0.1, 0.6) . shown by the following example. Therefore A – NFr (A ) =  ( 0.3, 0.2, 0.8), (0.1, 0.1, 1 1 0.6)  ⊈ 0 = NInt (A ). N 1 Example 2.14 Let X = { a, b } and  = { 0 , A, B, C, N D, 1 }. Then (X, ) is a neutrosophic topological Remark 2.19 In general topology, the following N space. The neutrosophic closed sets are C () = { 1 , conditions are hold : N E, F, G, H, 0 } where NFr (A)  NInt (A) = 0 , N N A =  ( 0.5, 0.6, 0.7), (0.1, 0.9, 0.4) , NInt (A)  NFr (A) = NCl (A), B =  ( 0.3, 0.9, 0.2), (0.4, 0.1, 0.6) , NInt (A)  NInt (C (A))  NFr (A) = 1 . N C =  ( 0.5, 0.9, 0.2), (0.4, 0.9, 0.4) , But the neutrosophic topology, we give D =  ( 0.3, 0.6, 0.7), (0.1, 0.1, 0.6) , counter-examples to show that the conditions of the E =  ( 0.7, 0.4, 0.5), (0.4, 0.1, 0.1) , above remark may not be hold in general. F =  ( 0.2, 0.1, 0.3), (0.6, 0.9, 0.4) , G =  ( 0.2, 0.1, 0.5), (0.4, 0.1, 0.4)  and Example 2.20 From Example 2.14, H =  ( 0.7, 0.4, 0.3), (0.6, 0.9, 0.1) . NFr (A2)  NInt (A2) = G  C = G  0N. Define A =  ( 0.4, 0.2, 0.8), (0.4, 0.5, 0.1) . Then 1 NInt (A )  NFr (A ) = C  G = C  1 = NCl (A ). C (A ) =  ( 0.8, 0.8, 0.4), (0.1, 0.5, 0.4) . 2 2 N 2 1 Therefore by Definition 2.2, NFr (A ) = H ⊈ 0 = 1 N NFr (NInt (A1)). NInt (A2)  NInt (C (A2))  NFr (A2) = C  0N  G = C  1 . N Theorem 2.15 For a NS A in the NTS X, Theorem 2.21 Let A and B be the two NSs in the NTS NFr (NCl (A))  NFr (A). Proof : Let A be the NS in the neutrosophic X. Then NFr (A  B)  NFr (A)  NFr (B). topological space X. Then by Definition 2.2, Proof : Let A and B be the two NSs in the NTS X. NFr (NCl (A)) = NCl (NCl (A))  NCl (C (NCl (A))) Then by Definition 2.2, By Proposition 1.18 (f) [12] and 4.2 (b) [18] , NFr (A  B) = NCl (A  B)  NCl (C (A  B)) = NCl (A)  NCl (NInt (C (A))) By Proposition 3.2 (2) [18] , By Proposition 1.18 (a) [12] , = NCl (A  B)  NCl ( C (A)  C (B) )  NCl (A)  NCl (C (A)) by Proposition 1.18 (h) and (o) [12] , Again by Definition 2.2,  (NCl (A)  NCl (B))  (NCl (C (A))  NCl(C (B))) = NFr (A) = [(NCl (A)  NCl (B) )  NCl (C (A)) ] Hence NFr (NCl (A))  NFr (A).  [ ( NCl (A)  NCl (B) )  NCl (C (B)) ] = [(NCl (A)  NCl (C (A)))(NCl (B)  NCl(C (A)))] The converse of the above theorem is not true as  [(NCl (A)  NCl (C (B)))(NCl(B)  NCl(C (B)))] shown by the following example. Again by Definition 2.2, = [NFr (A)  ( NCl (B)  NCl (C (A))) ] Example 2.16 From Example 2.14, let A =  ( 0.7,  [ ( NCl (A)  NCl (C (B)) )  NFr(B) ] 2 0.9, 0.2), (0.5, 0.9, 0.3) . = ( NFr (A)  NFr (B))  [ ( NCl (B)  NCl (C (A)) )  ( NCl (A)  NCl (C (B)) ) ] P. Iswarya, Dr. K. Bageerathi. A Study on Neutrosophic Frontier and Neutrosophic Semi-frontier in Neutrosophic Topological Spaces Neutrosophic Sets and Systems, Vol. 16, 2017 9  NFr (A)  NFr (B). The converse of the above theorem needs not be Hence NFr (A  B)  NFr (A)  NFr (B). true as shown by the following example. The converse of the above theorem needs not be Example 2.26 From Example 2.24, true as shown by the following example. ( NFr (B )  NCl (B ) )  ( NFr (B )  NCl (B ) ) = 1 2 2 1 (1  1 )  (1  1 ) = 1  1 = 1 ⊈ H = N N N N N N N Example 2.22 By Example 2.14, we define NFr (B  B ). 1 2 A =  ( 0.2, 0, 0.5), (0.4, 0.1, 0.1) , 1 A =  ( 0.7, 0.9, 0.2), (0.5, 0.9, 0.3) , Corollary 2.27 For any NSs A and B in the NTS X, 2 A  A = A =  ( 0.7, 0.9, 0.2), (0.5, 0.9, 0.1)  and NFr (A  B)  NFr (A)  NFr (B). 1 2 3 A  A = A =  ( 0.2, 0, 0.5), (0.4, 0.1, 0.3) . Then Proof : Let A and B be the two NSs in the NTS X. 1 2 4 C (A ) =  ( 0.5, 1, 0.2), (0.1, 0.9, 0.4) , Then by Definition 2.2, 1 C (A ) =  ( 0.2, 0.1, 0.7), (0.3, 0.1, 0.5) , NFr (A  B) = NCl (A  B)  NCl (C (A  B)) 2 By Proposition 3.2 (1) [18] , C (A ) =  ( 0.2, 0.1, 0.7), (0.1, 0.1, 0.5)  and 3 C (A ) =  ( 0.5, 1, 0.2), (0.3, 0.9, 0.4) . = NCl (A  B)  NCl ( C (A)  C (B) ) 4 By Proposition 1.18 (n) and (h) [12] , Therefore NFr (A )  NFr (A ) = E  G = E ⊈ G = 1 2  (NCl (A)  NCl (B))  (NCl (C(A))  NCl (C (B))) NFr (A ) = NFr (A  A ). 3 1 2 = ( NCl (A)  NCl (B)  NCl (C (A)) ) Note 2.23 The following example shows that  ( NCl (A)  NCl (B)  NCl (C (B)) ) Again by Definition 2.2, NFr (A  B) ⊈ NFr (A)  NFr (B) and NFr (A)  NFr (B) ⊈ NFr (A  B). = ( NFr (A)  NCl (B) )  ( NCl (A)  NFr (B) )  NFr (A)  NFr (B) Example 2.24 From Example 2.22, NFr (A  A ) = Hence NFr (A  B)  NFr (A)  NFr (B). 1 2 NFr (A ) = E ⊈ G = NFr (A )  NFr (A ). 4 1 2 The equality in the above corollary may not hold as From Example 2.14, We define B =  ( 0.4, 0.5, 0.1), 1 seen in the following example. (0.2, 0.9, 0.5) , B =  ( 0.5, 0.2, 0.9), (0.8, 0.4, 0.7) , 2 Example 2.28 From Example 2.24, B  B = B =  ( 0.5, 0.5, 0.1), (0.8, 0.9, 0.5)  and 1 2 3 NFr (B )  NFr (B ) = 1  1 = 1 ⊈ H = NFr (B ) B  B = B =  ( 0.4, 0.2, 0.9), (0.2, 0.4, 0.7) . 1 2 N N N 4 1 2 4 = NFr (B  B ). Then 1 2 C (B ) =  ( 0.1, 0.5, 0.4), (0.5, 0.1, 0.2) , 1 Theorem 2.29 For any NS A in the NTS X, C (B ) =  ( 0.9, 0.8, 0.5), (0.7, 0.6, 0.8) , 2 (1) NFr (NFr (A))  NFr (A), C (B ) =  ( 0.1, 0.5, 0.5), (0.5, 0.1, 0.8)  and 3 (2) NFr (NFr (NFr (A)))  NFr (NFr (A)). C (B ) =  ( 0.9, 0.8, 0.4), (0.7, 0.6, 0.2) . 4 Proof : (1) Let A be the NS in the neutrosophic Therefore NFr (B )  NFr (B ) = 1  1 = 1 ⊈ H 1 2 N N N topological space X. Then by Definition 2.2, = NFr (B ) = NFr (B  B ). 4 1 2 NFr (NFr (A)) = NCl (NFr (A))  NCl (C (NFr (A))) Again by Definition 2.2, Theorem 2.25 For any NSs A and B in the NTS X, = NCl ( NCl (A)  NCl (C (A)) )  NFr (A  B)  ( NFr (A)  NCl (B) )  ( NFr (B)  NCl ( C ( NCl (A)  NCl (C (A)) ) ) NCl (A) ). By Proposition 1.18 (f) [12] and by 4.2 (b) [18] , Proof : Let A and B be the two NSs in the NTS X.  ( NCl (NCl (A))  NCl (NCl (C (A))) ) Then by Definition 2.2,  NCl ( NInt (C (A))  NInt (A) ) NFr (A  B) = NCl (A  B)  NCl (C (A  B)) By Proposition 1.18 (f) [12] , By Proposition 3.2 (1) [18] , = ( NCl (A)  NCl (C (A)) )  ( NCl (NInt (C (A))) = NCl (A  B)  NCl ( C (A)  C (B) )  NCl (NInt (A)) By Proposition 1.18 (n) and (h) [12] ,  NCl (A)  NCl (C (A))  (NCl (A)  NCl (B))  (NCl (C(A))  NCl (C (B))) By Definition 2.2, = [ ( NCl (A)  NCl (B) )  NCl (C(A)) ] = NFr (A)  [ ( NCl (A)  NCl (B) )  NCl (C(B)) ] Therefore NFr (NFr (A))  NFr (A). Again by Definition 2.2, = ( NFr (A)  NCl (B) )  ( NFr (B)  NCl (A) ) (2) By Definition 2.2, Hence NFr (A  B)  ( NFr (A)  NCl (B) )  NFr (NFr (NFr (A))) = NCl (NFr (NFr (A)))  ( NFr (B)  NCl (A) ). NCl (C (NFr (NFr (A)))) P. Iswarya, Dr. K. Bageerathi. A Study on Neutrosophic Frontier and Neutrosophic Semi-frontier in Neutrosophic Topological Spaces