Neutrosophic Sets and Systems

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2 ISSN prt ISSN X ole Neutrosophc Sets ad Systems Iteratoal Joural Iformato Scece ad Egeerg Edtor--Chef: Prof Floret Smaradache Departmet of Mathematcs ad Scece Uversty of New Meco 705 Gurley veue Gallup NM 8730 US E-mal: Home page: ssocate Edtors: Dmtr Rabousk ad Larssa orssova depedet researchers W Vasatha Kadasamy IIT Chea Taml Nadu Ida Sad roum Uv of Hassa II Mohammeda Casablaca Morocco Salama Faculty of Scece Port Sad Uversty Egypt Yahu Guo School of Scece St Thomas Uversty Mam US Fracsco Gallego Lupaňez Uversdad Complutese Madrd Spa Pede Lu Shadog Uverstuy of Face ad Ecoomcs Cha Pabtra Kumar Ma Math Departmet K N Uversty W Ida S lbolw Kg bdulazz Uv Jeddah Saud raba Ju Ye Shaog Uversty Cha Ştefa Vlăduţescu Uversty of Craova Romaa Volume Cotets 03 Floret Smaradache Neutrosophc Measure ad Neutrosophc Itegral 3 Ju Ye other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso-Makg Method 8 Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group 3 Fu Yuhua Neutrosophc Eamples Physcs 6 Salama ad Floret Smaradache Flters va Neutrosophc Crsp Sets 34 Yahu Guo ad bdulkadr Şegür Novel Image Segmetato lgorthm ased o Neutrosophc Flterg ad Level Set 46 Salama Neutrosophc Crsp Pots & Neutrosophc Crsp Ideals 50 Sad roum ad Floret Smaradache Several Smlarty Measures of Neutrosophc Sets 54 Pgpg Ch ad Pede Lu Eteded TOPSIS Method for the Multple ttrbute Decso Makg Problems ased o Iterval Neutrosophc Set 63 Floret Smaradache ad Ştefa Vlăduțescu Commucato vs Iformato a omatc Neutrosophc Soluto 38 Copyrght Neutrosophc Sets ad Systems

3 Neutrosophc Sets ad Systems Vol 03 Neutrosophc Sets ad Systems Iteratoal Joural Iformato Scece ad Egeerg Copyrght Notce Neutrosophcs Sets ad Systems ll rghts reserved The authors of the artcles do hereby grat Neutrosophc Sets ad Systems o-eclusve worldwde royalty-free lcese to publsh ad dstrbute the artcles accordace wth the udapest Ope Itatve: ths meas that electroc copyg dstrbuto ad prtg of both full-sze verso of the oural ad the dvdual papers publshed there for o-commercal academc or dvdual use ca be made by ay user wthout permsso or charge The authors of the artcles publshed Neutrosophc Sets ad Systems reta ther rghts to use ths oural as a whole or ay part of t ay other publcatos ad ay way they see ft y part of Neutrosophc Sets ad Systems howsoever used other publcatos must clude a approprate ctato of ths oural Iformato for uthors ad Subscrbers Neutrosophc Sets ad Systems has bee created for publcatos o advaced studes eutrosophy eutrosophc set eutrosophc logc eutrosophc probablty eutrosophc statstcs that started 995 ad ther applcatos ay feld such as the eutrosophc structures developed algebra geometry topology etc The submtted papers should be professoal good Eglsh cotag a bref revew of a problem ad obtaed results Neutrosophy s a ew brach of phlosophy that studes the org ature ad scope of eutraltes as well as ther teractos wth dfferet deatoal spectra Ths theory cosders every oto or dea <> together wth ts opposte or egato <at> ad wth ther spectrum of eutraltes <eut> betwee them e otos or deas supportg ether <> or <at> The <eut> ad <at> deas together are referred to as <o> Neutrosophy s a geeralzato of Hegel's dalectcs the last oe s based o <> ad <at> oly ccordg to ths theory every dea <> teds to be eutralzed ad balaced by <at> ad <o> deas - as a state of equlbrum I a classcal way <> <eut> <at> are dsot two by two ut sce may cases the borders betwee otos are vague mprecse Sortes t s possble that <> <eut> <at> ad <o> of course have commo parts two by two or eve all three of them as well Neutrosophc Set ad Neutrosophc Logc are geeralzatos of the fuzzy set ad respectvely fuzzy logc especally of tutostc fuzzy set ad respectvely tutostc fuzzy logc I eutrosophc logc a proposto has a degree of truth T a degree of determacy I ad a degree of falsty F where T I F are stadard or o-stadard subsets of ] - 0 [ Neutrosophc Probablty s a geeralzato of the classcal probablty ad mprecse probablty Neutrosophc Statstcs s a geeralzato of the classcal statstcs What dstgushes the eutrosophcs from other felds s the <eut> whch meas ether <> or <at> <eut> whch of course depeds o <> ca be determacy eutralty te game ukow cotradcto gorace mprecso etc ll submssos should be desged MS Word format usg our template fle: varety of scetfc books may laguages ca be dowloaded freely from the Dgtal Lbrary of Scece: To submt a paper mal the fle to the Edtor--Chef To order prted ssues cotact the Edtor--Chef Ths oural s ocommercal academc edto It s prted from prvate doatos Iformato about the eutrosophcs you get from the UNM webste: Copyrght Neutrosophc Sets ad Systems

4 Neutrosophc Sets ad Systems Vol 03 Neutrosophc Measure ad Neutrosophc Itegral Floret Smaradache Uversty of New Meco Math & Scece Dvso 705 Gurley ve Gallup NM 8730 US E-mal: bstract Sce the world s full of determacy the eutrosophcs foud ther place to cotemporary research We ow troduce for the frst tme the otos of eutrosophc measure ad eutrosophc tegral Neutrosophc Scece meas developmet ad applcatos of eutrosophc logc/set/measure/tegral/ probablty etc ad ther applcatos ay feld It s possble to defe the eutrosophc measure ad cosequetly the eutrosophc tegral ad eutrosophc probablty may ways because there are varous types of determaces depedg o the problem we eed to solve Idetermacy s dfferet from radomess Idetermacy ca be caused by physcal space materals ad type of costructo by tems volved the space or by other factors Neutrosophc measure s a geeralzato of the classcal measure for the case whe the space cotas some determacy Neutrosophc Itegral s defed o eutrosophc measure Smple eamples of eutrosophc tegrals are gve Keywords: eutrosophy eutrosophc measure eutrosophc tegral determacy radomess probablty Itroducto to Neutrosophc Measure Itroducto Let <> be a tem <> ca be a oto a attrbute a dea a proposto a theorem a theory etc d let <at> be the opposte of <>; whle <eut> be ether <> or <at> but the eutral or determacy ukow related to <> For eample f <> = vctory the <at> = defeat whle <eut> = te game If <> s the degree of truth value of a proposto the <at> s the degree of falsehood of the proposto whle <eut> s the degree of determacy e ether true or false of the proposto lso f <> = votg for a caddate <at> = votg agast that caddate whle <eut> = ot votg at all or castg a blak vote or castg a black vote I the case whe <at> does ot est we cosder ts measure be ull {mat=0} d smlarly whe <eut> does ot est ts measure s ull { meut = 0} Defto of Neutrosophc Measure We troduce for the frst tme the scetfc oto of eutrosophc measure Let X be a eutrosophc space ad Σ a σ -eutrosophc algebra over X eutrosophc measure ν s defed by for eutrosophc set Σ by 3 ν :X R = m meutmat ν wth at = the opposte of ad eut = the eutral determacy ether or at as defed above; for ay X ad Σ m meas measure of the determate part of ; meut meas measure of determate part of ; ad mat meas measure of the determate part of at; where ν s a fucto that satsfes the followg two propertes: a Null empty set: ν Φ = 000 b Coutable addtvty or σ -addtvty: For all coutable collectos { } of dsot eutrosophc L sets Σ oe has: ν = m m eut m at m X L L L L where X s the whole eutrosophc space ad mat mx = mx m = m at L L L 3 Neutrosophc Measure Space eutrosophc measure space s a trplet X Σ ν 4 Normalzed Neutrosophc Measure eutrosophc measure s called ormalzed f 4ν X = m X m eutx m atx = 3 wth 3 = 4 ad Where of course X s the whole eutrosophc measure space 5 Fte Neutrosophc Measure Space Floret Smaradache Neutrosophc Measure ad Neutrosophc Itegral

5 4 Neutrosophc Sets ad Systems Vol 03 Let X We say that ν = aaa 3 s fte f all a a ad a 3 are fte real umbers X Σ ν s called fte eutrosophc measure space f ν X = abc such that all a b ad c are fte rather tha fte 6 σ-fte Neutrosophc Measure eutrosophc measure s called σ-fte f X ca be decomposed to a coutable uo of eutrosophcally measurable sets of fe eutrosophc measure alogously a set X s sad to have a σ-fte eutrosophc measure f t s a coutable uo of sets wth fte eutrosophc measure 7 Neutrosophc om of No-Negatvty We say that the eutrosophc measure ν satsfes the aom of o-egatvty f: Σ ν = a a a3 0 f a 0a 0 ad a3 0 4 Whle a eutrosophc measure ν that satsfes oly the ull empty set ad coutable addtvty aoms hece ot the o-egatvty aom takes o at most oe of the ± values 8 Measurable Neutrosophc Set ad Measurable Neutrosophc Space The members of Σ are called measurable eutrosophc sets whle X Σ s called a measurable eutrosophc space 9 Neutrosophc Measurable Fucto fucto f : X ΣX Y Σ Y mappg two measurable eutrosophc spaces s called eutrosophc measurable fucto f Σ Y f Σ the X verse mage of a eutrosophc Y -measurable set s a eutrosophc X -measurable set 0 Neutrosophc Probablty Measure s a partcular case of eutrosophc measure ν s th eutrosophc probablty measure e a eutrosophc measure that measures probable/possble propostos 0 ν X 3 5 where X s the whole eutrosophc probablty sample space We use ostadard umbers such for eample to deomate the absolute measure measure all possble worlds ad stadard umbers such as to deomate the relatve measure measure at least oe world Etc We deote the eutrosophc probablty measure by NP for a closer coecto wth the classcal probablty P Neutrosophc Category Theory The eutrosophc measurable fuctos ad ther eutrosophc measurable spaces form a eutrosophc category where the fuctos are arrows ad the spaces obects We troduce the eutrosophc category theory whch meas the study of the eutrosophc structures ad of the eutrosophc mappgs that preserve these structures The classcal category theory was troduced about 940 by Eleberg ad Mac Lae eutrosophc category s formed by a class of eutrosophc obects XYZ ad a class of eutrosophc morphsms arrows νξω such that: a If Hom X Y represet the eutrosophc morphsms from X to Y the Hom X Y ad Hom X 'Y ' are dsot ecept whe X = X' ad Y = Y' ; b The composto of the eutrosophc morphsms verfy the aoms of ssocatvty: ν ξ ω = ν ξ ω Idetty ut: for each eutrosophc obect X there ests a eutrosophc morphsm deoted d X called eutrosophc detty of X such that d X ν = ν ad ξ d X = ξ Fg Propertes of Neutrosophc Measure a Mootocty If ad are eutrosophcally measurable wth where ν = m m eut m at ad m m eut m at ν = the m m meut meut mat mat 6 Let ν X = 3 ad ν Y = y y y3 We say that ν X ν Y f y y ad 3 y 3 b ddtvty Floret Smaradache Neutrosophc Measure ad Neutrosophc Itegral

6 Neutrosophc Sets ad Systems Vol 03 5 the ν = ν ν If =Φ 7 where we defe a b c a b c = a a b b a b m X where X s the whole eutrosophc space ad a b mx = mx m m = mx a a 3 3 = m at at 3 Neutrosophc Measure Cotuous from elow or bove eutrosophc measure ν s cotuous from below f for eutrosophcally measurable sets wth for all the uo of the sets s eutrosophcally measurable ad ν = lmν 0 = d a eutrosophc measure ν s cotuous from above f for eutrosophcally measurable sets wth for all ad at least oe has fte eutrosophc measure the tersecto of the sets ad eutrosophcally measurable ad ν = lmν = 4 Geeralzatos Neutrosophc measure s a geeralzato of the fuzzy measure because whe m eut = 0 ad mat s gored we get ν = m 00 m ad the two fuzzy measure aoms are verfed: ν = a If =Φ the b If the ν ν The eutrosophc measure s practcally a trple classcal measure: a classcal measure of the determate part of a eutrosophc obect a classcal part of the determate part of the eutrosophc obect ad aother classcal measure of the determate part of the opposte eutrosophc obect Of course f the determate part does ot est ts measure s zero ad the measure of the opposte obect s gored the eutrosophc measure s reduced to the classcal measure 5 Eamples Let s see some eamples of eutrosophc obects ad eutrosophc measures a If a book of 00 sheets covers cluded has 3 mssg sheets the 9 ν book = where ν s the eutrosophc measure of the book umber of pages b If a surface of 5 5 square meters has cracks of 0 0 square meters the ν surface = where ν s the eutrosophc measure of the surface c If a de has two erased faces the ν de = 40 4 where ν s the eutrosophc measure of the de s umber of correct faces d appromate umber N ca be terpreted as a eutrosophc measure N = d where d s ts determate part ad ts determate part Its at part s cosdered 0 For eample f we do t kow eactly a quatty q but oly that t s betwee let s say q [ 0809 ] the q = 08 where 08 s the determate part of q ad ts determate part [ 00 ] We get a egatve eutrosophc measure f we appromate a quatty measured a verse drecto o the -as to a equvalet postve quatty For eample f r [ 6 4] the r = 6 where -6 s the determate part of r ad [ 0 ] s ts determate part Its at part s also 0 e Let s measure the truth-value of the proposto G = through a pot eteror to a le oe ca draw oly oe parallel to the gve le The proposto s complete sce t does ot specfy the type of geometrcal space t belogs to I a Eucldea geometrc space the proposto G s true; a Remaa geometrc space the proposto G s false sce there s o parallel passg through a eteror pot to a gve le; a Smaradache geometrc space costructed from med spaces for eample from a part of Eucldea subspace together wth aother part of Remaa space the proposto G s determate true ad false the same tme ν G = 5 f I geeral ot well determed obects otos deas etc ca become subect to the eutrosophc theory Itroducto to Neutrosophc Itegral Defto of Neutrosophc Itegral Usg the eutrosophc measure we ca defe a eutrosophc tegral The eutrosophc tegral of a fucto f s wrtte as: fdν 6 X where X s the a eutrosophc measure space Floret Smaradache Neutrosophc Measure ad Neutrosophc Itegral

7 6 Neutrosophc Sets ad Systems Vol 03 ad the tegral s take wth respect to the eutrosophc measure ν Idetermacy related to tegrato ca occur multple ways: wth respect to value of the fucto to be tegrated or wth respect to the lower or upper lmt of tegrato or wth respect to the space ad ts measure Frst Eample of Neutrosophc Itegral: Idetermacy Related to Fucto s Values Let fn: [a b] R 7 where the eutrosophc fucto s defed as: fn = g 8 wth g the determate part of fn ad the determate part of fnwhere for all [a b] oe has: [0 h ] h 0 9 determat part a ad a determate part ε e a = a ε 3 where ε [00] 4 Therefore b b X fdν = f d 5 a a where the determacy belogs to the terval: a 0 [0 f d] 6 a Or a dfferet way: b X a b fdν = f d a 0 7 where smlarly the determacy belogs to the terval: Therefore the values of the fucto fn are appromate e f [ g g h ] 0 N Smlarly the eutrosophc tegral s a appromato: b b b f dν = g d d N a a a 0 Secod Eample of Neutrosophc Itegral: Idetermacy Related to the Lower Lmt Suppose we eed to tegrate the fucto f: X R o the terval [a b] from X but we are usure about the lower lmt a Let s suppose that the lower lmt a has a a 0 8 [0 f d] a Refereces [] uthor C D uthor ad E F uthor Joural volume year page [] Floret Smaradache Itroducto to Neutrosophc Probablty ppled Quatum Physcs SO/NS DS Physcs bstract Servce [] Floret Smaradache Itroducto to Neutrosophc Probablty ppled Quatum Physcs ullet of the merca Physcal Socety 009 PS prl Meetg Volume 54 Number 4 Saturday Tuesday May 5 009; Dever Colorado US [3] Floret Smaradache Itroducto to Neutrosophc Probablty ppled Quatum Physcs ullet of Pure ad ppled Sceces Physcs 3-5 Vol D No 003 [4] Floret Smaradache "Neutrosophy / Neutrosophc Probablty Set ad Logc" merca Research Press Rehoboth US 05 p 998 [5] Floret Smaradache Ufyg Feld Logcs: Neutrosophc Logc Neutrosophc Set Neutrosophc Probablty ad Statstcs [6] Floret Smaradache Neutrosophc Physcs as a ew feld of research ullet of the merca Physcal Socety PS Floret Smaradache Neutrosophc Measure ad Neutrosophc Itegral

8 Neutrosophc Sets ad Systems Vol 03 7 March Meetg 0 Volume 57 Number Moday Frday February 7 March 0; osto Massachusetts [7] Floret Smaradache V Chrstato Neutrosophc Logc Vew to Schrodger's Cat Parado ullet of the merca Physcal Socety 008 Jot Fall Meetg of the Teas ad Four Corers Sectos of PS PT ad Zoes 3 ad 6 of SPS ad the Socetes of Hspac & lack Physcsts Volume 53 Number Frday Saturday October ; El Paso Teas [8] Floret Smaradache Vc Chrstato The Neutrosophc Logc Vew to Schrodger Cat Parado Revsted ullet of the merca Physcal Socety PS March Meetg 00 Volume 55 Number Moday Frday March ; Portlad Orego [9] Floret Smaradache Neutrosophc Degree of Paradocty of a Scetfc Statemet ullet of the merca Physcal Socety 0 ual Meetg of the Four Corers Secto of the PS Volume 56 Number Frday Saturday October 0; Tusco rzoa [0] Floret Smaradache -Valued Refed Neutrosophc Logc ad Its pplcatos to Physcs ullet of the merca Physcal Socety 03 ual Fall Meetg of the PS Oho-Rego Secto Volume 58 Number 9 Frday Saturday October ; Ccat Oho [] Floret Smaradache Neutrosophc Dagram ad Classes of Neutrosophc Paradoes or To the Outer-Lmts of Scece ullet of the merca Physcal Socety 7th eal Iteratoal Coferece of the PS Topcal Group o Shock Compresso of Codesed Matter Volume 56 Number 6 Suday Frday Jue 6 July 0; Chcago Illos Receved: November 03 ccepted: December 5 03 Floret Smaradache Neutrosophc Measure ad Neutrosophc Itegral

9 Neutrosophc Sets ad Systems Vol 03 8 other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso- Makg Method Ju Ye Departmet of Electrcal ad Iformato Egeerg Shaog Uversty 508 Huacheg West Road Shaog Zheag 3000 PR Cha E-mal: yehu@alyucom bstract sgle valued eutrosophc set SVNS whch s the subclass of a eutrosophc set ca be cosdered as a powerful tool to epress the determate ad cosstet formato the process of decso makg The correlato s oe of the most broadly appled dces may felds ad also a mportat measure data aalyss ad classfcato patter recogto decso makg ad so o Therefore we propose aother form of correlato coeffcet betwee SVNSs ad establsh a multple attrbute decso makg method usg the correlato coeffcet of SVNSs uder sgle valued eutrosophc evromet Through the weghted correlato coeffcet betwee each alteratve ad the deal alteratve the rakg order of all alteratves ca be determed ad the best alteratve ca be easly detfed as well Fally two llustratve eamples are employed to llustrate the actual applcatos of the proposed decso-makg approach Keywords: Correlato coeffcet; Sgle valued eutrosophc set; Decso makg Itroducto To hadle the determate formato ad cosstet formato whch est commoly real stuatos Smaradache [] frstly preseted a eutrosophc set from phlosophcal pot of vew whch s a powerful geeral formal framework ad geeralzed the cocept of the classc set fuzzy set terval-valued fuzzy set tutostc fuzzy set terval-valued tutostc fuzzy set paracosstet set dalethest set paradost set ad tautologcal set [ ] I the eutrosophc set a truthmembershp a determacy-membershp ad a falstymembershp are represeted depedetly Its fuctos T I ad F are real stadard or ostadard subsets of ] 0 [ e T : X ] 0 [ I : X ] 0 [ ad F : X ] 0 [ Obvously t wll be dffcult to apply real scetfc ad egeerg areas Therefore Wag et al [3] proposed the cocept of a sgle valued eutrosophc set SVNS whch s the subclass of a eutrosophc set ad provded the set-theoretc operators ad varous propertes of SVNSs Thus SVNSs ca be appled real scetfc ad egeerg felds ad gve us a addtoal possblty to represet ucertaty mprecse complete ad cosstet formato whch est real world However the correlato coeffcet s oe of the most frequetly used tools egeerg applcatos Therefore Haafy et al [4] troduced the correlato of eutrosophc data The Ye [5] preseted the correlato coeffcet of SVNSs based o the eteso of the correlato coeffcet of tutostc fuzzy sets ad proved that the cose smlarty measure of SVNSs s a specal case of the correlato coeffcet of SVNSs ad the appled t to sgle valued eutrosophc multcrtera decso-makg problems Haafy et al [6] preseted the cetrod-based correlato coeffcet of eutrosophc sets ad vestgated ts propertes Recetly S roum ad F Smaradache [8] Correlato coeffcet of terval eutrosophc set ad vestgated ts propertes I ths paper we propose aother form of correlato coeffcet betwee SVNSs ad vestgate ts propertes The a multple attrbute decso-makg method usg the correlato coeffcet of SVNSs s establshed uder sgle valued eutrosophc evromet To do so the rest of the paper s orgazed as follows Secto brefly descrbes some cocepts of SVNSs I Secto 3 we develop aother form of correlato coeffcet betwee SVNSs ad vestgate ts propertes Secto 4 establshes a multple attrbute decso-makg method usg the correlato coeffcet of SVNSs uder sgle valued eutrosophc evromet I Secto 5 two llustratve eamples are preseted to demostrate the applcatos of the developed approach Secto 6 cotas a cocluso ad future research Ju Ye other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso-Makg Method

10 Neutrosophc Sets ad Systems Vol 03 Ju Ye other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso-Makg Method Some cocepts of SVNSs Smaradache [] frstly preseted the cocept of a eutrosophc set from phlosophcal pot of vew ad gave the followg defto of a eutrosophc set Defto [] Let X be a space of pots obects wth a geerc elemet X deoted by eutrosophc set X s characterzed by a truth-membershp fucto T a determacy-membershp fucto I ad a falstymembershp fucto F The fuctos T I ad F are real stadard or ostadard subsets of ] 0 [ e T : X ] 0 [ I : X ] 0 [ ad F : X ] 0 [ There s o restrcto o the sum of T I ad F so 0 sup T sup I sup F 3 Obvously t s dffcult to apply practcal problems Therefore Wag et al [3] troduced the cocept of a SVNS whch s a stace of a eutrosophc set to apply real scetfc ad egeerg applcatos I the followg we troduce the defto of a SVNS [3] Defto [3] Let X be a space of pots obects wth geerc elemets X deoted by SVNS X s characterzed by a truth-membershp fucto T a determacy-membershp fucto I ad a falstymembershp fucto F for each pot X T I F [0 ] Thus SVNS ca be epressed as { } X F I T = The the sum of T I ad F satsfes the codto 0 T I F 3 Defto 3 [3] The complemet of a SVNS s deoted by c ad s defed as { } X T I F c = Defto 4 [3] SVNS s cotaed the other SVNS f ad oly f T T I I ad F F for every X Defto 5 [3] Two SVNSs ad are equal wrtte as = f ad oly f ad 3 Correlato coeffcet of SVNSs Motvated by aother correlato coeffcet betwee tutostc fuzzy sets [7] ths secto proposes aother form of correlato coeffcet betwee SVNSs as a geeralzato of the correlato coeffcet of tutostc fuzzy sets [7] Defto 6 For ay two SVNSs ad the uverse of dscourse X = { } aother form of correlato coeffcet betwee two SVNSs ad s defed by { } [ ] [ ] [ ] = = = = = F I T F I T F F I I T T C C C N ma ma Theorem The correlato coeffcet N satsfes the followg propertes: N = N ; 0 N ; 3 N = f = Proof It s straghtforward The equalty N 0 s obvous Thus we oly prove the equalty N [ ] F F F F F F I I I I I I T T T T T T F F I I T T N = = = ccordg to the Cauchy Schwarz equalty: y y y y y y where R ad y y y R we ca obta [ ] [ ] [ ] N N F I T F I T N = = = Thus [ ] [ ] / / N N N The { } ma N N N Therefore N 3 If = there are T = T I = I ad F = F for ay X ad = Thus there are N = I practcal applcatos the dffereces of mportace are cosdered the elemets the uverse Therefore we eed to take the weghts of the elemets = to accout Let w be the weght for each elemet = w [0 ] ad = = w the we have the followg weghted correlato coeffcet betwee the SVNSs ad : [ ] [ ] [ ] = = = = F I T w F I T w F F I I T T w W ma If w = / / / T the Eq reduce to Eq Note that W also satsfy the three propertes of Theorem 9

11 0 Neutrosophc Sets ad Systems Vol 03 Theorem Let w be the weght for each elemet = w [0 ] ad w = the the weghted = correlato coeffcet W defed Eq also satsfes the followg propertes: W = W ; 0 W ; 3 W = f = Sce the process to prove these propertes s smlar to that Theorem we do ot repeat t here 4 Decso-makg method usg the correlato coeffcet of SVNSs Ths secto proposes a sgle valued eutrosophc multple attrbute decso-makg method usg the proposed correlato coeffcet of SVNSs Let = { m} be a set of alteratves ad C = {C C C } be a set of attrbutes ssume that the weght of a attrbute C = etered by the decso-maker s w w [0 ] ad = = I ths case the characterstc of a alteratve = m wth respect to a attrbute C = s represeted by a SVNS form: = { C T C I C F C C C = } where T C I C F C [0 ] ad 0 T C I C F C 3 for C C = ad = m For coveece the values of the three fuctos T C I C F C are deoted by a sgle valued eutrosophc value SVNV a = t f = m; = whch s usually derved from the evaluato of a alteratve wth respect to a attrbute C by the epert or decso maker Thus we ca establsh a sgle valued eutrosophc decso matr D = a m : D a = m t f = t f tm m f m t t f t m m f f m t f t t m m f f m I the decso-makg method the cocept of deal pot has bee used to help detfy the best alteratve the decso set The deal alteratve provdes a useful theoretcal costruct agast whch to evaluate alteratves Geerally the evaluato attrbutes ca be categorzed to two kds beeft attrbutes ad cost attrbutes Let H be a collecto of beeft attrbutes ad L be a collecto of cost attrbutes deal SVNV ca be defed by a deal elemet for a beeft attrbute the deal alteratve * as * * * * a = t f = ma t m m f for H Ju Ye other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso-Makg Method whle a deal SVNV ca be defed by a deal elemet for a cost attrbute the deal alteratve * as * * * * a = t f = m t ma ma f for L The by applyg Eq the weghted correlato coeffcet betwee a alteratve = m ad the deal alteratve * s gve by * W = ma = w = * * * [ t t f f ] * * * [ t f ] w t f w = [ ] 3 The the bgger the measure value W * = m s the better the alteratve s because the alteratve s close to the deal alteratve * Through the weghted correlato coeffcet betwee each alteratve ad the deal alteratve the rakg order of all alteratves ca be determed ad the best oe ca be easly detfed as well 5 Illustratve eamples I ths secto two llustratve eamples for the multple attrbute decso-makg problems are provded to demostrate the applcato of the proposed decsomakg method 5 Eample Now we dscuss the decso-makg problem adapted from the lterature [5] There s a vestmet compay whch wats to vest a sum of moey the best opto There s a pael wth four possble alteratves to vest the moey: s a car compay; s a food compay; 3 3 s a computer compay; 4 4 s a arms compay The vestmet compay must take a decso accordg to the three attrbutes: C s the rsk; C s the growth; 3 C 3 s the evrometal mpact where C ad C are beeft attrbutes ad C 3 s a cost attrbute The weght vector of the three attrbutes s gve by w = T The four possble alteratves are to be evaluated uder the above three attrbutes by the form of SVNVs For the evaluato of a alteratve wth respect to a attrbute C = 3 4; = 3 t s obtaed from the questoare of a doma epert For eample whe we ask the opo of a epert about a alteratve wth respect to a attrbute C he or she may say that the possblty whch the statemet s good s 04 ad the statemet s poor s 03 ad the degree whch he or she s ot sure s 0 For the eutrosophc otato t ca be epressed as a = Thus whe the four

12 Neutrosophc Sets ad Systems Vol 03 possble alteratves wth respect to the above three attrbutes are evaluated by the epert we ca obta the followg sgle valued eutrosophc decso matr D: D = The we utlze the developed approach to obta the most desrable alteratves From the sgle valued eutrosophc decso matr we ca obta the followg deal alteratve: * = C 0700 C 0600 C } { 3 y usg Eq 3 we ca obta the values of the weghted correlato coeffcet W * = 3 4: W * = 0806 W * = 0950 W 3 * = ad W 4 * = Thus the rakg order of the four alteratves s 4 3 Therefore the alteratve 4 s the best choce amog the four alteratves 5 Eample mult-crtera decso makg problem s cocered wth a maufacturg compay whch wats to select the best global suppler accordg to the core competeces of supplers Now suppose that there are a set of four supplers = { 3 4} whose core competeces are evaluated by meas of the four attrbutes: C s the level of techology ovato; C s the cotrol ablty of flow; 3 C 3 s the ablty of maagemet; 4 C 4 s the level of servce where C C ad C are all beeft attrbutes ssume that the weght vector for the four attrbutes s w = T The proposed decso makg method s appled to solve ths problem for selectg supplers For the evaluato of a alteratve = 3 4 wth respect to a crtero C = 3 4 t s obtaed from the questoare of a doma epert For eample whe we ask the opo of a epert about a alteratve wth respect to a crtero C he or she may say that the possblty whch the statemet s good s 05 ad the statemet s poor s 03 ad the degree whch he or she s ot sure s 0 For the eutrosophc otato t ca be epressed as a = Thus whe the four possble alteratves wth respect to the above four attrbutes are evaluated by the smlar method from the epert we ca obta the followg sgle valued eutrosophc decso matr D: D = The we employ the developed approach to obta the most desrable alteratves From the sgle valued eutrosophc decso matr we ca obta the followg deal alteratve: { * = C0600 C05003 C C y applyg Eq 3 we ca obta the values of the weghted correlato coeffcet W * = 3 4: W * = W * = W 3 * = ad W 4 * = 0753 Thus the rakg order of the four alteratves s 3 4 Therefore the alteratve s the best choce amog the four alteratves From the two eamples we ca see that the proposed sgle valued eutrosophc multple attrbute decsomakg method s more sutable for real scetfc ad egeerg applcatos because t ca hadle ot oly complete formato but also the determate formato ad cosstet formato whch est commoly real stuatos 6 Cocluso I ths paper we proposed aother form of the correlato coeffcet betwee SVNSs The a multple attrbute decso-makg method has bee establshed sgle valued eutrosophc settg by meas of the weghted correlato coeffcet betwee each alteratve ad the deal alteratve Through the correlato coeffcet the rakg order of all alteratves ca be determed ad the best alteratve ca be easly detfed as well Fally two llustratve eamples llustrated the applcatos of the developed approach The the techque proposed ths paper s sutable for hadlg decso-makg problems wth sgle value eutrosophc formato ad ca provde a useful way for decsomakers I the future we shall cotue workg the applcatos of the correlato coeffcet betwee SVNSs to other domas such as data aalyss ad classfcato patter recogto ad medcal dagoss Refereces [] F Smaradache ufyg feld logcs eutrosophy: Neutrosophc probablty set ad logc merca Research Press Rehoboth999 [] F Smaradache Neutrosophc set geeralzato of the tutostc fuzzy set Iteratoal Joural of Pure ad ppled Mathematcs [3] H Wag F Smaradache YQ Zhag ad R Suderrama Sgle valued eutrosophc sets Multspace ad Multstructure [4] IM Haafy Salama ad K Mahfouz Correlato of eutrosophc Data Iteratoal Refereed Joural of Egeerg ad Scece } Ju Ye other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso-Makg Method

13 Neutrosophc Sets ad Systems Vol 03 [5] J Ye Multcrtera decso-makg method usg the correlato coeffcet uder sgle-valued eutrosophc evromet Iteratoal Joural of Geeral Systems [6] M Haafy Salama ad K M Mahfouz Correlato Coeffcets of Neutrosophc Sets by Cetrod Method Iteratoal Joural of Probablty ad Statstcs [7] ZS Xu J Che ad JJ Wu Clusterg algorthm for tutostc fuzzy sets Iformato Sceces [8] S roum F Smaradache Correlato Coeffcet of Iterval Neutrosophc set Perodcal of ppled Mechacs ad Materals Vol wth the ttle Egeerg Decsos ad Scetfc Research erospace Robotcs omechacs Mechacal Egeerg ad Maufacturg; Proceedgs of the Iteratoal Coferece ICMER ucharest October 03 Receved: Nov st 03 ccepted: Nov 0 03 Ju Ye other Form of Correlato Coeffcet betwee Sgle Valued Neutrosophc Sets ad Its Multple ttrbute Decso-Makg Method

14 Neutrosophc Sets ad Systems Vol 03 3 Soft Neutrosophc Group Muhammad Shabr Mumtaz l Muazza Naz 3 ad Floret Smaradache 4 Departmet of Mathematcs Quad--zam Uversty Islamabad 44000Paksta E-mal: mshabrbhatt@yahoocouk mumtazal770@yahoocom 3 Departmet of Mathematcal Sceces Fatma Jah Wome Uversty The Mall Rawalpd Paksta E-mal: muazzaaz@yahoocom 4 Uversty of New Meco 705 Gurley ve Gallup New Meco 8730 US E-mal: fsmaradache@gmalcom bstracti ths paper we eted the eutrosophc group ad subgroup to soft eutrosophc group ad soft eutrosophc subgroup respectvely Propertes ad theorems related to them are proved ad may eamples are gve Keywords:Neutrosophc groupeutrosophc subgroupsoft setsoft subsetsoft groupsoft subgroupsoft eutrosophc group soft eutrosophc subgroup Itroducto The cocept of eutrosophc set was frst troduced by Smaradache [36] whch s a geeralzato of the classcal sets fuzzy set [8] tutostc fuzzy set [4] ad terval valued fuzzy set [7] Soft Set theory was tated by Molodstov as a ew mathematcal tool whch s free from the problems of parameterzato adequacy I hs paper [] he preseted the fudametal results of ew theory ad successfully appled t to several drectos such as smoothess of fuctos game theory operatos research Rema-tegrato Perro tegrato theory of probablty Later o may researchers followed hm ad worked o soft set theory as well as applcatos of soft sets decso makg problems ad artfcal tellgece Now ths dea has a wde rage of research may felds such as databases [5 6] medcal dagoss problem [7] decso makg problem [8] topology [9] algebra ad so oma gave the cocept of eutrosophc soft set [8] ad later o roum ad Smaradache defed tutostc eutrosophc soft set We have worked wth eutrosophc soft set ad ts applcatos group theory Prelmares Nuetrosophc Groups Defto [4] Let G * be ay group ad let ág ÈIñ { a bi : a b G} { } = Î The eutrosophc group s geerated by I ad G uder * deoted by N G = ág ÈIñ * I s called the eutrosoph- c elemet wth the property I = I For a teger I ad I are eutrosophc elemets ad 0 I = 0 I - the verse of I s ot defed ad hece does ot est Theorem [ 4 ] Let N G be a eutrosophc group The N G geeral s ot a group; N G always cotas a group Defto pseudo eutrosophc group s defed as a eutrosophc group whch does ot cota a proper subset whch s a group Defto 3 Let N G be a eutrosophc group The N G s sad to be a N H s a proper subset N H of eutrosophc subgroup of N G f eutrosophc group that s N H cotas a proper subset whch s a group N H s sad to be a pseudo eutrosophc subgroup f t does ot cota a proper subset whch s a group Eample NZ NQ NR ad NC are eutrosophc groups of teger ratoal real ad comple umbers respectvely Eample Let Z7 = { o 6} be a group uder addto modulo 7 N G = { áz7 È Iñ ' 'modulo7} s a eutrosophc group whch s fact a group For N G = { a bi : a b Î Z7 } s a group uder ` ' modulo 7 Defto 4 Let N G be a fte eutrosophc group Let P be a proper subset of N G whch uder the Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

15 4 Neutrosophc Sets ad Systems Vol 03 operatos of N G s a eutrosophc group If / o P o N G the we call P to be a Lagrage Defto 5 eutrosophc subgroup N G s called weakly Lagrage eutrosophc group f N G has at least oe Lagrage eutrosophc subgroup Defto 6 N G s called Lagrage free eutrosophc group f N G has o Lagrage eutrosophc subgroup Defto7 Let N G be a fte eutrosophc group Suppose L s a pseudo eutrosophc subgroup of N G ad f o L / o N G the we call L to be a pseudo Lagrage eutrosophc subgroup N G has at least oe pseudo La- Defto 8 If grage eutrosophc subgroup the we call N G to be a weakly pseudo Lagrage eutrosophc group N G has o pseudo Lagrage eutro- Defto 9 If sophc subgroup the we call N G to be pseudo Lagrage free eutrosophc group Defto 0 Let N G be a eutrosophc group We say a eutrosophc subgroup H of N G s ormal f we ca fd ad y N G such that H = Hy for all y Î N G Note = y or y = - ca also occur Defto eutrosophc group N G whch has o otrval eutrosophc ormal subgroup s called a smple eutrosophc group Defto Let N G be a eutrosophc group proper pseudo eutrosophc subgroup P of N G s sad to be ormal f we have P = Py for all y Î N G eutrosophc group s sad to be pseudo smple eutrosophc group f N G has o otrval pseudo ormal subgroups Soft Sets Throughout ths subsecto U refers to a tal uverse E s a set of parameters P U s the power set of U ad Ì E Molodtsov [] defed the soft set the followg maer: Defto3 [ ] par F s called a soft set over U where F s a mappg gve by F : P U I other words a soft set over U s a parameterzed famly of subsets of the uverse U For e Î F e may be cosdered as the set of e -elemets of the soft set F or as the set of e-appromate elemets of the soft set Eample 3 Suppose that U s the set of shops E s the set of parameters ad each parameter s a word or setece Let ì hgh retormal ret ü E = ï í ï ý ï good codto bad codto î ïþ F whch descrbes the Let us cosder a soft set attractveess of shops that Mr Z s takg o ret Suppose that there are fve houses the uverse U = { h h h3 h4 h5} uder cosderato ad that { e e e } = 3 be the set of parameters where e stads for the parameter 'hgh ret e stads for the parameter 'ormal ret e 3 stads for the parameter ' good codto Suppose that F e = { h h 4} F e = { h h5} F e = { h h h } The soft set F s a appromated famly { Fe = 3} of subsets of the set U whch gves us a collecto of appromate descrpto of a obect Thus we have the soft set F as a collecto of appromatos as below: F = { hgh ret = { h h4} ormal ret { h h } = h h h } = 5 good codto { 3 4 5} Defto 4 [ 3 ] For two soft sets F ad Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

16 Neutrosophc Sets ad Systems Vol 03 5 H over U H f Í ad Fe Í He for all e Î F s called a soft subset of Ths relatoshp s deoted by F Ì H Smlarly F s called a soft superset of H f H s a soft subset of F whch s deoted by F É H Defto 5 [ 3 ] Two soft sets F ad H over U are called soft equal f F s a soft subset of H ad H s a soft subset of F Defto 6 Let [ ] 3 F ad G be two Ç = where HC soft sets over a commo uverse U such that Ç ¹ f The ther restrcted tersecto s deoted by F G HC R s defed as Hc = Fc Ç Gc for all c C = Ç Defto 7[ 3 ] The eteded tersecto of two soft sets F ad G over a commo uverse U s the soft set HC where C = È ad for all e Î C H e s defed as ìï Fe f e - He = ï í Ge f e - ïfe ÇGe f e Ç ïî We wrte F Ç e G = HC Defto 8 [ 3 ] The restrcted uo of two soft sets G over a commo uverse U s F ad the soft set HC where C = È ad for all e Î C H e s defed as the soft set HC = F È G where C = Ç ad R Î Hc = Fc È Gc for all c C Defto 9[ 3 ] The eteded uo of two soft sets F ad G over a commo uverse U s the soft set HC where C = È ad for all e Î C H e s defed as ìï Fe f e - He = ï í Ge f e - ïfe ÈGe f e Ç ïî We wrte F È G = HC 3 Soft Groups e Defto 0[ ] Let F be a soft set over G The F s sad to be a soft group over G f ad oly f F G forall Î Eample 4 Suppose that G = = S3 = { e3333} The F s a soft group over S 3 where F e = { e} F = { e } F 3 = { e 3 } F 3 = { e 3 } F = F = { e } Let Defto [ ] F be a soft group over G The F s sad to be a detty soft group over G f F { e} = for all Î where e s the detty elemet of G ad F s sad to be a absolute soft group f F = G for all Î Defto The restrcted product HC of two soft groups F ad K over G s deoted by the soft set HC = F K where = Ç ad H s a set valued fucto from C C Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

17 6 Neutrosophc Sets ad Systems Vol 03 P G ad s defed as Hc FcKc to all c Î C The soft set = for HC s called the restrcted soft product of F ad 3 Soft Neutrosophc Group K over G Defto 3 Let N G be a eutrosophc group ad F be soft set over N G The F s called soft eutrosophc group over N G f ad oly for all f F N G Î Eample 5 Let ì 0 3 I I 3 I I I 3 I ü NZ 4 =í ï ï ý ï I I 3 I3 I3 I3 3I î ïþ be a eutrosophc group uder addto modulo 4 Let { e e e3 e4} F s soft eutrosophc group over = be the set of parameters the N Z 4 where F e = { 0 3 } F e = { 0 I I 3 I} F e3 = { 0 I I} F e4 = { 0 I I3 I I I 3 I} Theorem Let F ad H be two soft eutrosophc groups over N G The ther tersecto F Ç H s aga a soft eutrosophc group over N G Theorem 3 Let F ad Proof The proof s straghtforward H be two soft eutrosophc groups over NG If Ç = f the F H N G È s a soft eutrosophc group over H be two soft eu- Theorem 4 Let F ad trosophc groups over N G If F e Í H e all e Î the for F s a soft eutrosophc subgroup of H Theorem 5 The eteded uo of two soft eutrosophc N G s ot a groups F ad K over soft eutrosophc group over N G Proof Let F ad K be two soft eutrosophc groups over N G Let C = È the for all e Î C F È e K = HC where = F e If e Î - H e = K e If e Î - = F e ÈK e If e Î Ç s uo of two subgroups may ot be aga a subgroup Clearly f e C H e may ot be = Ç the a subgroup of N G Hece the eteded uo HC s ot a soft eutrosophc group over N G Eample 6 Let F ad eutrosophc groups over N Z uder addto modulo where F e = { 0 } F e = { 0 I} d K be two soft = { } = { } K e 0 K e 0 I 3 The clearly ther eteded uo s ot a soft eutrosophc group as H e = F e È K e = { 0 I} s ot a subgroup of N Z Theorem 6 The eteded tersecto of two soft eutrosophc groups over N G s soft eutrosophc group over N G Theorem 7 The restrcted uo of two soft eutrosophc N G s ot a groups F ad K over soft eutrosophc group over N G Theorem 8 The restrcted tersecto of two soft eutrosophc groups over N G s soft eutrosophc group over N G Theorem 9 The restrcted product of two soft eutrosoph- Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

18 Neutrosophc Sets ad Systems Vol 03 7 ct groups F ad K over soft eutrosophc group over N G N G s a Theorem 0 The ND operato of two soft eutrosophc groups over N G s soft eutrosophc group over N G Theorem The OR operato of two soft eutrosophc groups over N G may ot be a soft eutrosophc group Defto 4 soft eutrosophc group whch does ot cota a proper soft group s called soft pseudo eutrosophc group Eample 7 Let NZ = Z È I = { 0 I I} be a eutrosophc group uder addto modulo Let = { e e e3} be the set of parameters the F s a soft pseudo eutrosophc group over N G where F e = { 0 } F e = { 0 I} F e = { 0 I} 3 Theorem The eteded uo of two soft pseudo eu- K over trosophc groups F ad N G s ot a soft pseudo eutrosophc group over N G Eample 8 Let NZ = Z È I = { 0 I I} be a eutrosophc group uder addto modulo Let F ad K be two soft pseudo eutrosophc groups over N G where Fe = { 0 } Fe = { 0 I} Fe = { 0 I} d 3 = { } = { } K e 0 I K e 0 Clearly ther restrcted uo s ot a soft pseudo eutrosophc group as uo of two subgroups s ot a subgroup Theorem 3 The eteded tersecto of two soft pseudo K over eutrosophc groups F ad N G s aga a soft pseudo eutrosophc group over N G Theorem 4 The restrcted uo of two soft pseudo eu- K over trosophc groups F ad N G s ot a soft pseudo eutrosophc group over N G Theorem 5 The restrcted tersecto of two soft pseudo K over eutrosophc groups F ad N G s aga a soft pseudo eutrosophc group over N G Theorem 6 The restrcted product of two soft pseudo K over eutrosophc groups F ad N G s a soft pseudo eutrosophc group over N G Theorem 7 The ND operato of two soft pseudo eutrosophc groups over N G soft pseudo eutrosophc soft group over N G Theorem 8 The OR operato of two soft pseudo eutrosophc groups over N G may ot be a soft pseudo eutrosophc group Theorem9 Every soft pseudo eutrosophc group s a soft eutrosophc group Proof The proof s straght forward Remark The coverse of above theorem does ot hold N Z be a eutrosophc group ad Eample 9 Let 4 F be a soft eutrosophc group over N Z 4 The F e = { 0 3 } F e = { 0 I I 3 I} F e3 = { 0 I I} ut F s ot a soft pseudo eutrosophc group as H s clearly a proper soft subgroup of F where H e = { 0 } H e = { 0 } N G s a soft pseudo Theorem 0 F over Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

19 8 Neutrosophc Sets ad Systems Vol 03 eutrosophc group f N G s a pseudo eutrosophc group Proof Suppose that N G be a pseudo eutrosophc group the t does ot cota a proper group ad for all e F over Î the soft eutrosophc group N G s such that F e N G Sce each F e s a pseudo eutrosophc subgroup whch does ot cota a proper group whch make F s soft pseudo eutrosophc group Eample 0 Let NZ = Z È I = { 0 I I} be a pseudo eutrosophc group uder addto modulo The F a soft pseudo eutrosophc soft group clearly over N Z where F e = { 0 } F e = { 0 I} F e3 = { 0 I} Defto 5 Let F ad eutrosophc groups over N G The H be two soft H s a soft eutrosophc subgroup of F deoted as H F f Ì ad H e F e for all e Î NZ = Z È I be a soft eu- Eample Let 4 4 trosophc group uder addto modulo 4 that s ì 0 3 I I 3 I I I 3 I ü N Z4 = ï í ï ý ï I I 3 I3 I3 I3 3I î ïþ F be a soft eutrosophc group over Let N Z 4 the = { 0 3 } = { 0 3 } F e3 = { 0 I I} = { } F e F e I I I F e 0 I I3 I I I 3 I 4 H s a soft eutrosophc subgroup of where F = { } = { } = { 0 3 } H e 0 H e 0 I H e I I I 4 Theorem soft group over G s always a soft eutrosophc subgroup of a soft eutrosophc group over N G f Ì Proof Let F be a soft eutrosophc group over N G ad H be a soft group over G s G Ì N G ad for all b Î H b G Ì N G Ths mples H e F e for all e Î as Ì Hece H F Eample Let F be a soft eutrosophc group over N Z 4 the F e = { 0 3 } F e = { 0 I I 3 I} F e3 = { 0 I I} Let = { e e } such that H F where Clearly e Î 3 = { } H 3 = { } Ì ad H e F e H e 0 0 for all Theorem soft eutrosophc group over N G always cotas a soft group over G Proof The proof s followed from above Theorem H be two soft Defto 6 Let F ad pseudo eutrosophc groups over N G The H s called soft pseudo eutrosophc subgroup of F deoted as H F f Ì H e F e for all e Î Eample 3 Let F be a soft pseudo eutrosophc group over N Z 4 where Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

20 Neutrosophc Sets ad Systems Vol 03 9 = { } = { } F e 0 I I3 I F e 0 I Hece H F where H e = { I} 0 Theorem 3 Every soft eutrosophc group F over N G has soft eutrosophc subgroup as well as soft pseudo eutrosophc subgroup Proof Straghtforward F be a soft eutrosophc Defto 7 Let group over N G the soft eutrosophc group over N G f F = { e} for all F s called the detty Î where e s the detty elemet of G H be a soft eutrosophc Defto 8 Let group over N G the eutrosophc group over N G f F = N G for all Î H s called Full-soft Eample 4 Let ì a bi : a b R ad ü NR = ï í ï ý ï I s determacy î ïþ s a eutrosophc real group where R s set of real umbers ad I = I therefore I = I for a postve teger The real group where F s a Full-soft eutrosophc Fe = NR for all e Theorem 4 Every Full-soft eutrosophc group cota absolute soft group Theorem 5 Every absolute soft group over G s a soft eutrosophc subgroup of Full-soft eutrosophc group over N G Theorem 6 Let N G be a eutrosophc group If order of N G s prme umber the the soft eutrosophc group F over N G s ether detty soft eutrosophc group or Full-soft eutrosophc group Proof Straghtforward Defto 9 Let F be a soft eutrosophc group over N G If for all e Î each F e s Lagrage eutrosophc subgroup of N G the F s called soft Lagrage eutrosophc group over N G Eample 5 Let N Z { } = { I I } s a 3 / 0 eutrosophc group uder multplcato modulo 3 Now N Z 3 / 0 { } { I } are subgroups of { } whch dvdes order of { } N Z 3 / 0 The the soft eutrosophc group F = { F e = { } F e = { I} } s a eample of soft Lagrage eutrosophc group Theorem 7 If N G s Lagrage eutrosophc group the F over N G s soft Lagrage eutrosophc group but the coverse s ot true geeral Theorem 8 Every soft Lagrage eutrosophc group s a soft eutrosophc group Proof Straghtforward Remark The coverse of the above theorem does ot hold Eample 6 Let NG = { 3 4 I I 3 I 4 I} be a eutrosophc group uder multplcato modulo 5 F be a soft eutrosophc group over ad N G where F e = { I I I I} F e = { } F e I I 3 I 4 I = { } 3 ut clearly t s ot soft Lagrage eutrosophc group as F e whch s a subgroup of N G does ot dvde order of N G Theorem 9 If N G s a eutrosophc group the the soft Lagrage eutrosophc group s a soft eutrosophc group Proof Suppose that N G be a eutrosophc group ad F be a soft Lagrage eutrosophc group over N G The by above theorem F s also soft eutrosophc group Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

21 0 Neutrosophc Sets ad Systems Vol 03 Eample 7 Let N Z 4 be a eutrosophc group ad F s a soft Lagrage eutrosophc group over N Z 4 uder addto modulo 4 where F e = { 0 3 } F e = { 0 I I 3 I} F e3 = { 0 I I} ut F has a proper soft group H where H e = { 0 } H e3 = { 0 } Hece F s soft eutrosophc group Theorem 30 Let F ad K be two soft Lagrage eutrosophc groups over N G The Ther eteded uo F È e K over N G s ot soft Lagrage eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s ot soft Lagrage eutrosophc group over N G 3 Ther restrcted uo F È R K over N G s ot soft Lagrage eutrosophc group over N G 4 Ther restrcted tersecto F ÇR K over N G s ot soft Lagrage eutrosophc group over N G 5 Ther restrcted product F K over N G s ot soft Lagrage eutrosophc group over N G Theorem 3 Let F ad H be two soft Lagrage eutrosophc groups over N G The Ther ND operato F K s ot soft Lagrage eutrosophc group over N G Ther OR operato F K s ot a soft Lagrage eutrosophc group over N G Defto 30 Let F be a soft eutrosophc group over N G The F s called soft weakly Lagrage eutrosophc group f atleast oe F e s a Lagrage eutrosophc subgroup of N G for some e Î Eample 8 Let N G = { 3 4 I I 3 I 4 I} be a eutrosophc group uder multplcato modulo 5 F s a soft weakly Lagrage eutrosophc the group over N G where F e = { I I I I} F e = { } F e3 = { I I 3 I 4 I} s F e ad 3 N G do ot dvde order of F e whch are subgroups of N G Theorem 3 Every soft weakly Lagrage eutrosophc F s soft eutrosophc group group Remark 3 The coverse of the above theorem does ot hold geeral N Z be a eutrosophc group u- Eample 9 Let 4 der addto modulo 4 ad { e e } = be the set of parameters the F s a soft eutrosophc group over N Z 4 where F e = { I I I} F e = { I} ut ot soft weakly Lagrage eutrosophc group over 4 N Z Defto 3 Let F be a soft eutrosophc group over N G The F s called soft Lagrage free eutrosophc group f F e s ot Lagrageeutrosophc subgroup of N G for all e Î Eample 0 Let NG = {34 I I3 I4 I} be a eutrosophc group uder multplcato modulo 5 Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

22 Neutrosophc Sets ad Systems Vol 03 ad the F be a soft Lagrage free eutrosophc group over N G where F e = { 4 I34 I I I} F e = { I34 I I I} s F e ad N G do ot dvde order of F e whch are subgroups of N G Theorem 33 Every soft Lagrage free eutrosophc N G s a soft eutrosophc group F over group but the coverse s ot true F be a soft eutrosophc group Defto 3 Let over N G If for all e Î each F e s a pseudo Lagrage eutrosophc subgroup of N G the F s called soft pseudo Lagrage eutrosophc group over N G Eample Let N Z 4 be a eutrosophc group uder addto modulo 4 ad = { e e} be the set of parameters the F s a soft pseudo Lagrage eutrosophc group over N Z 4 where F e = { 0 I I3 I} F e = { 0 I} Theorem 34 Every soft pseudo Lagrage eutrosophc group s a soft eutrosophc group but the coverse may ot be true Proof Straghtforward K be two soft Theorem 35 Let F ad pseudo Lagrage eutrosophc groups over N G The Ther eteded uo F È e K over N G s ot a soft pseudo Lagrage eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s ot pseudo Lagrage eutrosophc soft group over N G 3 Ther restrcted uo F È K over R N G s ot pseudo Lagrage eutrosophc soft group over N G 4 Ther restrcted tersecto F ÇR K over N G s also ot soft pseudo Lagrage eutrosophc group over N G 5 Ther restrcted product F K over N G s ot soft pseudo Lagrage eutrosophc group over N G Theorem 36 Let F ad H be two soft pseudo Lagrage eutrosophc groups over N G The Ther ND operato F K s ot soft pseudo Lagrage eutrosophc group over N G Ther OR operato F K s ot a soft pseudo Lagrage eutrosophc soft group over N G Defto 33 Let F be a soft eutrosophc group over N G The F s called soft weakly pseudo Lagrage eutrosophc group f atleast oe F e s a pseudo Lagrage eutrosophc subgroup of N G for some e Î Eample Let N G = { 3 4 I I 3 I 4 I} be a eutrosophc group uder multplcato modulo 5 F s a soft weakly pseudo Lagrage eutro- The sophc group over N G where F e = { I I I I} F e = { I} 3 4 N G does ot d- s F e whch s a subgroup of vde order of N G Theorem 37 Every soft weakly pseudo Lagrage eutrosophc group F s soft eutrosophc group Remark 4 The coverse of the above theorem s ot true geeral Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

23 Neutrosophc Sets ad Systems Vol 03 Eample 3 Let N Z 4 be a eutrosophc group uder addto modulo 4 ad = { e e} be the set of parameters the F s a soft eutrosophc group over N Z 4 where F e = { I I 3 I} F e = { 0 I} ut t s ot soft weakly pseudo Lagrage eutrosophc group K be two soft Theorem 38 Let F ad weakly pseudo Lagrage eutrosophc groups over N G The Ther eteded uo F È e K over N G s ot soft weakly pseudo Lagrage eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s ot soft weakly pseudo Lagrage eutrosophc group over N G 3 Ther restrcted uo F È R K over N G s ot soft weakly pseudo Lagrage eutrosophc group over N G 4 Ther restrcted tersecto F ÇR K over N G s ot soft weakly pseudo Lagrage eutrosophc group over N G 5 Ther restrcted product F K over N G s ot soft weakly pseudo Lagrage eutrosophc group over N G Defto 34 Let F be a soft eutrosophc group over N G The F s called soft pseudo Lagrage free eutrosophc group f F e s ot pseudo Lagrage eutrosophc subgroup of N G for all e Î Eample 4 Let N G = { 3 4 I I 3 I 4 I} be a eutrosophc group uder multplcato modulo 5 F s a soft pseudo Lagrage free eutro- The sophc group over N G where F e = { I I I I} F e = { I I I I} s F e ad N G do ot dvde order of F e whch are subgroups of N G Theorem 39 Every soft pseudo Lagrage free eutrosophc group F over N G s a soft eutrosophc group but the coverse s ot true K be two soft Theorem 40 Let F ad pseudo Lagrage free eutrosophc groups over N G The Ther eteded uo F È e K over N G s ot soft pseudo Lagrage free eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s ot soft pseudo Lagrage free eutrosophc group over N G 3 Ther restrcted uo F È R K over N G s ot pseudo Lagrage free eutrosophc soft group over N G 4 Ther restrcted tersecto F ÇR K over N G s ot soft pseudo Lagrage free eutrosophc group over N G 5 Ther restrcted product F K over N G s ot soft pseudo Lagrage free eutrosophc group over N G Defto 35 soft eutrosophc group F over N G s called soft ormal eutrosophc group over Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

24 Neutrosophc Sets ad Systems Vol 03 3 N G f F e s a ormal eutrosophc subgroup of N G for all e Î Eample 5 Let N G = { eabciaibici } be a eutrosophc group uder multplcatowherea = b = c = ebc = cb= aac = ca= bab = ba= c The F s a soft ormal eutrosophc group over N G where = { e a I ai} = { e b I bi} = { e c I ci} F e F e F e 3 Theorem 4 Every soft ormal eutrosophc group F over N G s a soft eutrosophc group but the coverse s ot true H be two soft Theorem 4 Let F ad ormal eutrosophc groups over N G The Ther eteded uo F È e K over N G s ot soft ormal eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s soft ormal eutrosophc group over N G 3 Ther restrcted uo F È R K over N G s ot soft ormal eutrosophc group over N G 4 Ther restrcted tersecto F ÇR K over N G s soft ormal eutrosophc group over N G 5 Ther restrcted product F K over N G Theorem 43 Let F ad over N G s ot soft ormal eutrosophc soft group H be two soft ormal eutrosophc groups over N G The Ther ND operato F K s soft ormal eutrosophc group over N G Ther OR operato F K s ot soft ormal eutrosophc group over N G Defto 36 Let F be a soft eutrosophc group over N G The F s called soft pseudo ormal eutrosophc group f F e s a pseudo ormal eutrosophc subgroup of N G for all e Î Eample 6 Let N Z = Z È I = { 0 I I} be a eutrosophc group uder addto modulo ad let = { e e} be the set of parameters the F s soft pseudo ormal eutrosophc group over N G where F e = { 0 I} F e = { 0 I} s F e ad of N G F e are pseudo ormal subgroup Theorem 44 Every soft pseudo ormal eutrosophc group F over N G s a soft eutrosophc group but the coverse s ot true K be two soft Theorem 45 Let F ad pseudo ormal eutrosophc groups over N G The Ther eteded uo F È K over N G s ot soft pseudo ormal eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s soft pseudo ormal eutrosophc group over N G 3 Ther restrcted uo F È R K over N G s ot soft pseudo ormal eutrosophc e Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

25 4 Neutrosophc Sets ad Systems Vol 03 group over N G 4 Ther restrcted tersecto F ÇR K over N G s soft pseudo ormal eutrosophc group over N G 5 Ther restrcted product F K over N G s ot soft pseudo ormal eutrosophc group over N G Theorem 46 Let F ad K be two soft pseudo ormal eutrosophc groups over N G The Ther ND operato F K s soft pseudo ormal eutrosophc group over N G Ther OR operato F K s ot soft pseudo ormal eutrosophc group over N G Defto 37 Let N G be a eutrosophc group The F s called soft cougate eutrosophc group over N G f ad oly f eutrosophc subgroup of N G for all e F e s cougate Î Eample 7 Let ì I I 3 I 4 I 5 I ü N G = ï í ï ý ï I I3 I5 5I î ïþ be a eutrosophc group uder addto modulo 6 ad = are cou- let P = { I 3 3I} ad K { 0 4 I 4 4 I I 4I} gate eutrosophc subgroups of N G The F s soft cougate eutrosophc group over N G where F e = { I 3 3 I} F e = { I I I I} Theorem 47 Let F ad K be two soft cougate eutrosophc groups over N G The Ther eteded uo F È e K over N G s ot soft cougate eutrosophc group over N G Ther eteded tersecto F Ç e K over N G s aga soft cougate eutrosophc group over N G 3 Ther restrcted uo F È R K over N G s ot soft cougate eutrosophc group over N G 4 Ther restrcted tersecto F ÇR K over N G s soft cougate eutrosophc group over N G 5 Ther restrcted product F K over N G s ot soft cougate eutrosophc group over N G Theorem 48 Let F ad K be two soft cougate eutrosophc groups over N G The Ther ND operato F K s aga soft cougate eutrosophc group over N G Ther OR operato F K s ot soft cougate eutrosophc group over N G Cocluso I ths paper we eted the eutrosophc group ad subgrouppseudo eutrosophc group ad subgroup to soft eutrosophc group ad soft eutrosophc subgroup ad respectvely soft pseudo eutrosophc group ad soft pseudo eutrosophc subgroup The ormal eutrosophc subgroup s eteded to soft ormal eutrosophc subgroup We showed all these by gvg varous eamples order Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

26 Neutrosophc Sets ad Systems Vol 03 5 to llustrate the soft part of the eutrosophc otos used ckowledgemets We would lke to thak Edtors ad referees for ther kd suggestos that helped to mprove ths paper Refereces [] gboola D kwu YTOyebo Neutrosophc groups ad subgroups It J Math Comb 30-9 [] ktas ad N Cagma Soft sets ad soft group If Sc [3] MI l F Feg XY Lu WK M ad M Shabr O some ew operatos soft set theory Comput Math ppl [4] K taassov Itutostc fuzzy sets Fuzzy Sets Syst [5] Sroum F Smaradache Itutostc Neutrosophc Soft Set J If & Comput Sc [6] D Che ECC Tsag DS Yeug X Wag The parameterzato reducto of soft sets ad ts applcatoscomput Math ppl [7] M Gorzalzay method of ferece appromate reasog based o tervalvaluedfuzzy sets Fuzzy Sets ad Systems [8] P K Ma Neutrosophc Soft Set als of Fuzzy Mathematcs ad Iformatcs Smaradache asc Neutrosophc lgebrac Structures ad ther pplcatos to Fuzzy ad Neutrosophc Models Hes 49 pp 004 [7] Sezg & O tagu Soft Groups ad Normalstc Soft Groups Comput Math ppl [8] L Zadeh Fuzzy sets Iform Cotrol Receved:November 5 03 ccepted:november 8 03 [9] PK Ma R swas ad R Roy Soft set theory Comput Math ppl [0] PK Ma R Roy ad R swas applcato of soft sets a decso makg problem Comput Math ppl [] D Molodtsov Soft set theory frst results Comput Math ppl [] Z Pawlak Rough sets It J If Compu Sc [3] Floret Smaradache Ufyg Feld Logcs Neutrosophy: Neutrosophc Probablty Set ad Logc Rehoboth: merca Research Press 999 [4] W Vasatha Kadasamy & Floret Smaradache Some Neutrosophc lgebrac Structures ad Neutrosophc N-lgebrac Structures 9 p Hes 006 [5] W Vasatha Kadasamy & Floret Smaradache N-lgebrac Structures ad S-N- lgebrac Structures 09 pp Hes Phoe 006 [6] W Vasatha Kadasamy & Floret Muhammad Shabr Mumtaz l Muazza Naz ad Floret Smaradache Soft Neutrosophc Group

27 Neutrosophc Sets ad Systems Vol 03 6 Neutrosophc Eamples Physcs Fu Yuhua CNOOC Research Isttute No6 Dogzhmewaaoe Street eg 0007 Cha E-mal: fuyh945@sacom bstract: Neutrosophy ca be wdely appled physcs ad the lke For eample oe of the reasos for 0 Nobel Prze for physcs s "for the dscovery of the acceleratg epaso of the uverse through observatos of dstat superovae" but accordg to eutrosophy there est seve or e states of acceleratg epaso ad cotracto ad the eutrosophc state the uverse other two eamples are "a revso to Gödel's completeess theorem by eutrosophy" ad "s eutral eutrosophc fudametal teractos" I addto the "partal ad temporary ufed theory so far" s dscussed cludg "partal ad temporary ufed electromagetc theory so far" "partal ad temporary ufed gravtatoal theory so far" "partal ad temporary ufed theory of four fudametal teractos so far" ad "partal ad temporary ufed theory of atural scece so far" Keywords: Neutrosophy applcato eutrosophc eample physcs partal ad temporary ufed theory so far Itroducto Neutrosophy s proposed by Prof Floret Smaradache 995 Neutrosophy s a ew brach of phlosophy that studes the org ature ad scope of eutraltes as well as ther teractos wth dfferet deatoal spectra Ths theory cosders every oto or dea <> together wth ts opposte or egato <t-> ad the spectrum of "eutraltes" <Neut-> e otos or deas located betwee the two etremes supportg ether <> or <t-> The <Neut-> ad <t-> deas together are referred to as <No-> Neutrosophy s the base of eutrosophc logc eutrosophc set eutrosophc probablty ad statstcs used egeerg applcatos especally for software ad formato fuso medce mltary cyberetcs ad physcs Neutrosophc Logc s a geeral framework for ufcato of may estg logcs such as fuzzy logc especally tutostc fuzzy logc paracosstet logc tutostc logc etc The ma dea of NL s to characterze each logcal statemet a 3D Neutrosophc Space where each dmeso of the space represets respectvely the truth T the falsehood F ad the determacy I of the statemet uder cosderato where T I F are stadard or o-stadard real subsets of ]- 0 [ wthout ecessarly coecto betwee them More formato about Neutrosophy may be foud refereces [-4] Now we dscuss the eutrosophc eamples physcs ad the lke Dscusso o "the acceleratg epaso of the uverse" Oe of the reasos for 0 Nobel Prze for physcs s "for the dscovery of the acceleratg epaso of the uverse through observatos of dstat superovae" ut accordg to eutrosophy "the acceleratg epaso of the uverse" s debatable Supposg that "the epaso of the uverse" s a dea <> ts opposte or egato <t-> should be "the cotracto of the uverse" ad the spectrum of "eutraltes" <Neut-> should be "the stable state of the uverse" e the state located betwee the two etremes supportg ether epaso or cotracto I fact the area earby a black hole s the state of cotracto because the mass of black hole or smlar black hole s mmese ad t produces a very strog gravtatoal feld so that all matters ad radatos cludg the electromagetc wave or lght wll be uable to escape f they eter to a crtcal rage aroud the black hole The vewpot of "the acceleratg epaso of the uverse" uepectedly turs a bld eye to the fact that partal uverse such as the area earby a black hole s the state of cotracto s for "the stable state of the uverse" t should be located at the trasto area betwee epaso area ad cotracto area ga rug the same program to the state of "the epaso of the uverse" supposg that "the Fu Yuhua Neutrosophc Eamples Physcs

28 Neutrosophc Sets ad Systems Vol 03 7 acceleratg epaso of the uverse" s a dea <> ts opposte or egato <t-> should be "the deceleratg epaso of the uverse" ad the spectrum of "eutraltes" <Neut-> should be "the uform epaso of the uverse" e the state located betwee the two etremes supportg ether acceleratg epaso or deceleratg epaso Smlarly rug the same program to the state of "the cotracto of the uverse" t ca be dvded to three cases: "the acceleratg cotracto of the uverse" "the deceleratg cotracto of the uverse" ad "the uform cotracto of the uverse" To sum up there est seve states the uverse: acceleratg epaso deceleratg epaso uform epaso acceleratg cotracto deceleratg cotracto uform cotracto ad stable state I addto accordg to eutrosophy aother kd of seve states s as follows: log-term epaso short-term epaso medum-term epaso log-term cotracto short-term cotracto medum-term cotracto ad stable state It should be oted that the stable state ca be also dvded to three cases such as: "log-term stable state" "short-term stable state" ad "medum-term stable state"; thus there est e states the uverse Cosderg all possble stuatos besdes these seve or e states due to the lmtatos of huma kowledge there may be other ukow states From ths eample we ca see that all of the absolute soltary ad oe-sded vewpots are completely wrog ut wth the help of eutrosophy may of these mstakes may be avoded 3 revso to Gödel's completeess theorem by eutrosophy ccordg to referece [4] the ma cotets of the revso are as follows s well-kow eutrosophy paves the way to cosder all possble stuatos ut we ca see that the proof of Gödel's completeess theorem all possble stuatos are ot cosdered Frst the proof the followg stuato s ot cosdered: wrog results ca be deduced from some aoms For eample from the aom of choce a parado the doublg ball theorem ca be deduced whch says that a ball of volume ca be decomposed to peces ad reassembled to two balls both of volume It follows that certa cases the proof of Gödel's completeess theorem may be faulty Secod the proof of Gödel's completeess theorem oly four stuatos are cosdered that s oe proposto ca be proved to be true caot be proved to be true ca be proved to be false caot be proved to be false ad ther combatos such as oe proposto ca ether be proved to be true or be proved to be false ut those are ot all possble stuatos I fact there may be may kds of determate stuatos cludg t ca be proved to be true some cases ad caot be proved to be true other cases; t ca be proved to be false some cases ad caot be proved to be false other cases; t ca be proved to be true some cases ad ca be proved to be false other cases; t caot be proved to be true some cases ad caot be proved to be false other cases; t ca be proved to be true some cases ad ca ether be proved to be true or be proved to be false other cases; ad so o ecause so may stuatos are ot cosdered we may say that the proof of Gödel's completeess theorem s faulty at least s ot oe wth all sded cosderatos I order to better uderstad the case we cosder a etreme stuato where oe proposto as show Gödel's completeess theorem ca ether be proved or dsproved It may be assumed that ths proposto ca be proved 9999 cases oly case t ca ether be proved or dsproved We wll see whether or ot ths stuato has bee cosdered the proof of Gödel's completeess theorem Some people may argue that ths stuato s equvalet to the oe where the proposto ca ether be proved or dsproved ut the dfferece les the dstcto betwee the part ad the whole If oe case may represet the whole stuato may mportat theores caot be appled For eample the geeral theory of relatvty volves sgular pots; the law of uversal gravtato does ot allow the case where the dstace r s equal to zero ccordgly whether or ot oe may say that the geeral theory of relatvty ad the law of uversal gravtato caot be appled as a whole? Smlarly the stuato also caot be cosdered as the oe that ca be proved ut ths problem may be easly solved wth the eutrosophc method Moreover f we apply the Gödel's completeess theorem to tself we may obta the followg possblty: oe of all formal mathematcal aom systems the Gödel's completeess theorem ca ether be proved or dsproved If all possble stuatos ca be cosdered the Gödel's completeess theorem ca be mproved prcple ut wth our boudless uverse beg ever chagg ad beg etremely comple t s mpossble "cosderg all possble stuatos" s far as "cosderg all possble stuatos" s cocered the Smaradache s eutrosophy s qute good possbly the best Therefore ths paper proposes to revse the Gödel's completeess theorem to the complete aom wth Smaradache s eutrosophy Cosderg all possble stuatos wth Smaradache s eutrosophy oe may revse the Gödel's Icompleteess theorem to the completeess aom: y proposto Fu Yuhua Neutrosophc Eamples Physcs

29 8 Neutrosophc Sets ad Systems Vol 03 ay formal mathematcal aom system wll represet respectvely the truth T the falsehood F ad the determacy I of the statemet uder cosderato where T I F are stadard or o-stadard real subsets of ]- 0 [ 4 S eutral eutrosophc fudametal teractos s well-kow accordg to the preset uderstadg there are four fudametal teractos or forces: gravtatoal electromagetc weak ad strog teracto Whle accordace wth the eutrosophy theory that betwee a etty ad ts opposte there est termedate ettes thus besdes the estg four fudametal teractos there must est s eutral eutrosophc fudametal teractos as s ew forms of fudametal teracto For eample betwee strog teracto ad weak teracto there ests termedate teracto amely eutral eutrosophc strog-weak fudametal teracto NSW fudametal teracto t ether strog teracto or weak teracto but somethg betwee Smlarly cosderg other fve pars of opposte teractos: strog ad electromagetc fudametal teracto strog ad gravtatoal fudametal teracto weak ad electromagetc fudametal teracto weak ad gravtatoal fudametal teracto ad electromagetc ad gravtatoal fudametal teracto respectvely other fve eutral eutrosophc fudametal teractos are as follows: eutral eutrosophc strog-electromagetc fudametal teracto NSE fudametal teracto eutral eutrosophc strog-gravtatoal fudametal teracto NSG fudametal teracto eutral eutrosophc weak-electromagetc fudametal teracto NWE fudametal teracto eutral eutrosophc weakgravtatoal fudametal teracto NWG fudametal teracto ad eutral eutrosophc electromagetcgravtatoal fudametal teracto NEG fudametal teracto Thus there may be te fudametal teractos all together 5 Several ufed theores Whether or ot the ufed theory ca be ested? ccordg to eutrosophy there are three cases as follows: the ufed theory ca be ested the ufed theory caot be ested ad the eutrosophc case such as the "partal ad temporary ufed theory so far" Now we dscuss the "partal ad temporary ufed theory so far" What s the "ufed theory"? I 980 Stephe Hawkg oce clamed physcsts have see the outle of "fal theory" ths theory of everythg ca epress all laws of ature wth a sgle ad beautful mathematcal model perhaps that t s so smple ad ca be wrtte o a T-shrt I other words for ay feld the strct "ufed theory" refers to that all the laws of ths feld ca be epressed a sgle mathematcal model If followg ths cocept to uderstad the strct "ufed theory" we have to say such a "ufed theory" s smply caot be ested I other words there s oly "partal ad temporary ufed theory so far" Now we dscuss that the strct "ufed electromagetc theory" caot be ested 5 Why the strct "ufed electromagetc theory" caot be ested ad applyg least square method to establsh "partal ad temporary ufed electromagetc theory so far" It mght be argued that Mawell's equatos are "ufed electromagetc theory" Facg wth ths argumet we ask three questos Frst whether or ot all the electromagetc laws ca be cluded or derved by Mawell's equatos? Secod whether or ot the later appeared hgh temperature supercoductvty problem ad the lke ca be solved by Mawell's equatos? Thrd whether or ot the faster-tha-lght FTL problems ca be solved by Mawell's equatos? If egatve aswers were gve to these three questos the t should be ackowledged that Mawell's equatos are ot strct "ufed electromagetc theory" but oly "partal ad temporary ufed electromagetc theory" ased o the same reaso the "theory of the ufed weak ad electromagetc teracto" caot be ested ad there s oly "partal ad temporary theory of the ufed weak ad electromagetc teracto so far" Now we establsh the "partal ad temporary ufed electromagetc theory so far" Frst of all for ay feld applyg least square method to establsh ths feld s "partal ad temporary ufed theory so far" the correspodg epresso s "partal ad temporary ufed varatoal prcple so far" Supposg that for a certa doma Ω we already establsh the followg geeral equatos F = 0 = O boudary V the boudary codtos are as follows = 0 = m pplyg least square method for ths feld ad the domas ad boudary codtos the "partal ad temporary ufed theory so far" ca be epressed the followg form of "partal ad temporary ufed varatoal prcple so far" Fu Yuhua Neutrosophc Eamples Physcs

30 Neutrosophc Sets ad Systems Vol 03 9 Π = W F dω W dv = Ω m ' m 3 where: m 0 was troduced referece [5] dcatg the mmum ad ts value should be equal to zero V 0 W ad W are sutable postve weghted costats; for the ' smplest cases all of these weghted costats ca be take as If oly a certa equato s cosdered we ca oly make ts correspodg weghted costat s equal to ad the other weghted costats are all equal to 0 y usg ths method we already establshed the "partal ad temporary ufed water gravty wave theory so far" ad the correspodg "partal ad temporary ufed water gravty wave varatoal prcple so far" referece [6]; ad establshed the "partal ad temporary ufed theory of flud mechacs so far" ad the correspodg "partal ad temporary ufed varatoal prcple of flud mechacs so far" referece [7] Some scholars may sad ths s smply the applcato of least square method our aswer s: the smplest way may be the most effectve way It should be oted that due to that tme we caot realze that the strct "ufed theory" caot be ested therefore refereces [6] ad [7] the wrog deas that "ufed water gravty wave theory" "ufed water gravty wave varatoal prcple" "ufed theory of flud mechacs" ad "ufed varatoal prcple of flud mechacs" were appeared Now we correct these mstakes ths paper It should also be oted that Eq ca be cluded Eq therefore we wll oly dscuss Eq rather tha dscuss Eq Now we wrte Mawell's equatos as follows F = 0 doma Ω where: where: where: where: F = D ρ F = 0 doma Ω F = E / t F 3 = 0 doma Ω3 F 3 = F = 0 4 doma Ω4 F = H D / t 4 I addto for sotropc medum the followg equatos should be added F 5 = 0 doma Ω5 where: F 5 = D ε 0 ε r E F 6 = 0 doma Ω6 where: F 6 = μ 0 μr H F 7 = 0 doma Ω7 where: F7 = γe esdes these equatos the Coulomb's law reads F 8 = 0 doma Ω8 kq q r where: F8 = f accordg to the epermetal data k = N m²/c² Due to the lmted space other equatos of electromagetsm are o loger lsted lso a umber of coservato equatos such as the equato of coservato of eergy ad a umber of laws such as the law of composto of veloctes are also o loger lsted ll of them wll be dscussed below I addto some soltary equatos establshed oly for the soltary pots or specal cases ca be wrtte as follows = m 4 S = 0 For eample the scale factor the Coulomb's law ca be wrtte as the followg soltary equato S = 0 where: S = k N m²/c² other eample s that plasma problem the sheldg dstace Debye dstace ca be wrtte as the followg soltary equato S = 0 where: S = D ε 0kT / e lso due to lmted space other electromagetc soltary equatos are o loger lsted For the reaso that some soltary equatos caot be ru the tegral process they wll be ru the square sum process pplyg least square method "partal ad temporary ufed electromagetc theory so far" ca be epressed the followg form of "partal ad temporary ufed electromagetc varatoal prcple so far" m Π EM = W F dω W ' S = m0 5 Ω where: the subscrpt EM deotes that the sutable scope s the electromagetsm all of the equatos F = 0 deote so far dscovered derved all of the equatos related to electromagetsm all of the equatos S = 0 deote so far dscovered derved all of the soltary equatos related to electromagetsm ad W ad W are sutable postve weghted costats Clearly here ad m are all very large tegers ' Fu Yuhua Neutrosophc Eamples Physcs

31 30 Neutrosophc Sets ad Systems Vol 03 5 pplyg least square method to establsh "partal ad temporary ufed gravtatoal theory so far" Frstly t should be oted that for dfferet gravtatoal problems the dfferet formulas or dfferet gravtatoal theores should be appled The "uversal gravtatoal formulas or equatos" actually caot be ested For ths cocluso may scholars do ot realze t I addto all of the dfferet gravtatoal formulas ca be wrtte as the form of Eq amely the form that the rght sde of the epresso s equal to zero The frst formula should be metoed s Newto's uversal gravtatoal formula F GMm r = 6 It ca be wrtte as the followg form F 6 = 0 GMm F = F r where: Prof Hu Ng derved a equato accordg to geeral relatvty wth the help of Hu's equato ad et s formula referece [8] we derved the followg mproved Newto's formula of uversal gravtato GMm 3G M mp F = 7 4 r c r where: G s gravtatoal costat M ad m are the masses of the two obects r s the dstace betwee the two obects c s the speed of lght p s the half ormal chord for the obect m movg aroud the obect M alog wth a curve ad the value of p s gve by: p = a-e for ellpse p = a e - for hyperbola p = y / for parabola Ths formula ca gve the same results as gve by geeral relatvty for the problem of plaetary advace of perhelo ad the problem of gravtatoal defecto of a photo orbt aroud the Su It ca be wrtte as the followg form F = 0 7 GMm 3G M mp where: F = F 4 r c r It should be oted that accordg to Eq6 ad Eq7 the FTL ca be ested I some cases we should also cosder the followg gravtatoal formula cludg three terms GMm GMp wg M p F = r cr cr where: w s a costat to be determed It ca be wrtte as the followg form F 3 = 0 8 GMm 3GMp wg M p where: F = F r c r c r ut for the eample that a small ball rolls alog the cled plae the gravtatoal feld of the Earth all of the above metoed formulas caot be appled I referece [5] we preset the followg gravtatoal formula wth the varable dmeso fractal form the fractal dmeso s varable stead of costat δ F = GMm/ r 9 where: δ = 06 0 u u s the horzo dstace that the small ball rolls It ca be wrtte as the followg form F = δ where: F4 = F GMm / r I addto the gravtatoal feld equatos of Este's theory of geeral relatvty ad the gravtatoal formula ad gravtatoal equatos derved by other scholars ca also be wrtte as the form of Eq amely the form that the rght sde of the epresso s equal to zero I some cases whe dealg wth gravtatoal problem we should also cosder some prcple of coservato such as the prcple of coservato of eergy Here we wrte the prcple of coservato of eergy as the form of Eq amely the form that the rght sde of the epresso s equal to zero So do the other prcples of coservato I refereces [9] we dscussed two cases to apply the prcple of coservato of eergy drectly ad drectly To apply the prcple of coservato of eergy drectly s as follows Supposg that the tal total eergy of a closed system s equal to W 0 ad for tme t the total eergy s equal to W t the accordg to the prcple of coservato of eergy t gves W 0 = W t 0 It ca be wrtte as the followg form W t F 5 = = 0 W 0 To apply the prcple of coservato of eergy drectly s as follows Supposg that we are terested a specal physcal quatty Qot oly t ca be calculated by usg the prcple of coservato of eergy but also ca be calculated by usg other gravtatoal formula For dstgushg the values let s deote the value gve by other laws as Q whle deote the value gve Fu Yuhua Neutrosophc Eamples Physcs

32 Neutrosophc Sets ad Systems Vol 03 3 by the prcple of coservato of eergy as Q ' the the equato to apply the prcple of coservato of eergy drectly s as follows Q F 6 = = 0 Q' Now we dscuss some soltary equatos establshed oly for the soltary pots or specal cases The frst oe s the soltary equato about the gravtatoal costat S = G N m /kg =0 3 The secod oe s cosderg the deflecto agle for the problem of gravtatoal defecto of a photo orbt aroud the Su y usg geeral relatvty or mproved Newto's formula of uversal gravtato amely Eq7 the deflecto agle φ 0 reads φ 0 =75 However accordg to the epermet we should have φ =77±00 takg the average t gves φ =77 ccordg to ths epresso the correspodg soltary equato s as follows S = φ 77" = 0 4 Other soltary equatos clude: the soltary equatos establshed by the values of plaetary advace of perhelo the soltary equatos establshed by the uusual values of gravty at dfferet tmes durg total solar eclpse ad the lke Due to the lmted space they are o loger lsted pplyg least square method "partal ad temporary ufed gravtatoal theory so far" ca be epressed the followg form of "partal ad temporary ufed gravtatoal varatoal prcple so far" m Π G = W F dω W ' S = m0 5 Ω where: the subscrpt G deotes that the sutable scope s the gravty all of the equatos F = 0 deote so far dscovered derved all of the equatos related to gravty all of the equatos S = 0 deote so far dscovered derved all of the soltary equatos related to gravty ad W ad W are sutable postve weghted costats ' It should be oted that as we establsh "partal ad temporary ufed theory so far" ad the correspodg "partal ad temporary ufed varatoal prcple so far" the cludg pheomeo s allowed For eample the three terms gravtatoal formula Eq8 cludes Eq7 whle Eq7 cludes Eq6 ut we stll cosder these three equatos smultaeously Ths s because that some cases Eq7 s more coveet; as for Eq6 t s eough most cases moreover puttg Eq6 at the most promet posto epress our respect to Newto who s the greatest scetst the hstory I addto the coestg pheomeo s also allowed For eample the gravtatoal formulas of classcal mechacs the gravtatoal feld equatos of Este's theory of geeral relatvty ad the equatos of other gravtatoal theores are coestg For the soluto that s satsfyg two or more tha two theores smultaeously or solvg the problems dfferet felds smultaeously ad the lke we wll dscuss them other papers such solutos may oly be reached wth the method of varatoal prcple Now we dscuss the applcatos of varatoal prcple Eq5 Eample Settg W = ad W ' = varatoal prcple Eq5 ad other weghted costats are all equal to 0 amely applyg Eq7 ad Eq3 to derve the chagg rule for the gravtatoal coeffcet G ' stead of the gravtatoal costat G ad make the gravtatoal formula accordace wth the verse square law I refereces [0] chagg Eq7 to the followg form accordace wth the verse square law F = It gves G' Mm r G' Mm GMm 3G M mp = 4 r r c r The we have the chagg rule for the gravtatoal coeffcet G ' as follows 3GMp G ' G c r = 6 For problem of Mercury s advace of perhelo we have G G' 63 0 G For problem of gravtatoal defecto of a photo orbt aroud the Su we have G G' 5G Eample Settg W = 4 ad W 6 = varatoal prcple Eq5 ad other weghted costats are all equal to 0 amely applyg Eq9 ad Eq to determe the ukow δ Eq9 ccordg to Eq varatoal prcple Eq5 ca be smplfed to the followg form appled the law of coservato of eergy drectly Q Π = d = m Q' 0 7 Fu Yuhua Neutrosophc Eamples Physcs

33 3 Neutrosophc Sets ad Systems Vol 03 The soluto procedure ca be foud referece [9] For the fal optmum appromate soluto the value of Π calculated by the mproved uversal gravtatoal formula ad mproved Newto s secod law s equal to t s oly 0033% of the value of Π 0 calculated by the orgal uversal gravtatoal formula ad orgal Newto s secod law Eample 3 Settg W ad W ' 3 = = varatoal prcple Eq5 ad other weghted costats are all equal to 0 amely applyg Eq8 ad Eq4 to determe the ukow w Eq8 The soluto procedure ca be foud referece [0] the fal result s as follows The rage of value of w s as follows w Takg the average t gves w =0574 For the problem of gravtatoal defecto of a photo orbt aroud the Su the geeral relatvty caot gve the soluto that s eactly equal to the epermetal value whle the method preseted ths paper ca do so It should be oted that for varato prcple Eq5 f there s a eact soluto the ts rght sde ca be equal to 0 here the varatoal prcple Eq5 s eactly equvalet to F = 0 ad S = 0 see eample ad eample 3 If there s oly a appromate soluto the rght sde of varatoal prcples Eq5 ca oly be appromately equal to 0 at ths momet we ca apply the approprate optmzato method to seek the best appromate soluto ad the effect of the soluto ca be udged accordg to the etet that the value of Π s close to 0 see eample 53 Other "partal ad temporary ufed theory so far" especally "partal ad temporary ufed theory of atural scece so far" To eted the above metoed method we ca get varous "partal ad temporary ufed theory so far" For ufed dealg wth the problems of four fudametal teractos applyg least square method "partal ad temporary ufed theory of four fudametal teractos so far" ca be epressed the followg form of "partal ad temporary ufed varatoal prcple of four fudametal teractos so far" Π GESW = W F dω W S = Ω m ' m 8 where: the subscrpt GESW deotes that the sutable scope s the four fudametal teractos all of the 0 equatos F = 0 deote so far dscovered derved all of the equatos related to four fudametal teractos all of the equatos S = 0 deote so far dscovered derved all of the soltary equatos related to four fudametal teractos ad W ad W are sutable postve weghted costats For ufed dealg wth the problems of atural scece applyg least square method "partal ad temporary ufed theory of atural scece so far" ca be epressed the followg form of "partal ad temporary ufed varatoal prcple of atural scece so far" Π NTURE = ' W F dω W ' S = Ω m m 9 where: the subscrpt NTURE deotes that the sutable scope s all of the problems of atural scece all of the equatos F = 0 deote so far dscovered derved all of the equatos related to atural scece all of the equatos S = 0 deote so far dscovered derved all of the soltary equatos related to atural scece ad W ad W ' are sutable postve weghted costats I ths way the theory of everythg to epress all of atural laws descrbed by Hawkg that a sgle equato could be wrtte o a T-shrt s partally ad temporarly realzed the form of "partal ad temporary ufed varatoal prcple of atural scece so far" s already oted for "partal ad temporary ufed theory so far" ad the correspodg "partal ad temporary ufed varatoal prcple so far" the cludg pheomeo ad coestg pheomeo are allowed Here we would lke to pot out that besdes the cludg process ad coestg process the eutrosophc oe amely the smplfyg process s also allowed For eample the frst smplfyg result of "partal ad temporary ufed theory of atural scece so far" s "theory of coservato of eergy" t ca be epressed the followg form of "frst smplfyg varatoal prcple for partal ad temporary ufed theory of atural scece so far" t s shorted as "varatoal prcple of coservato of eergy" t SIMPLE Π = W t / W 0 dt m 0 NTURE = t Ths "varatoal prcple of coservato of eergy" ca be appled for ufed dealg wth may problems physcs mechacs astroomy bology egeerg ad eve may ssues socal scece For eample referece [] based o "theory of coservato of eergy" for some cases we derved Newto's secod law the law of uversal gravtato ad the lke 0 0 Fu Yuhua Neutrosophc Eamples Physcs

34 Neutrosophc Sets ad Systems Vol Further topcs are fdg more smplfyg processes smplfyg varatoal prcples ad ther combatos These wll make "partal ad temporary ufed theory of atural scece so far" smpler clearer more perfect ad more practcal For ths purpose the eutrosophy wll gve very mportat cotrbuto Refereces [] Floret Smaradache Ufyg Feld Logcs: Neutrosophc Logc Neutrosophy Neutrosophc Set Neutrosophc Probablty ad Statstcs thrd edto Xqua Phoe 003 [] D Rabousk F Smaradache Lorssova Neutrosophc Methods Geeral Relatvty Hes 005 [3] pplcatos of Smaradache s Notos to Mathematcs Physcs ad Other Sceces Edted by Yuhua Fu Lfa Mao ad Mhaly ENCZE IfoLearQuest 007 [4] Fu Yuhua Fu e Revso to Gödel's Icompleteess Theorem by Neutrosophy Iteratoal Joural of Mathematcal Combatorcs Vol ~5 [5] Fu Yuhua New soluto for problem of advace of Mercury's perhelo cta stroomca Sca No4 989 [6] Fu Yuhua Ufed water gravty wave theory ad mproved lear wave Cha Ocea Egeerg 99Vol6 No57-64 [7] Fu Yuhua ufed varatoal prcple of flud mechacs ad applcato o soltary subdoma or pot Cha Ocea Egeerg 994 Vol8 No [8] Fu Yuhua Improved Newto s formula of uversal gravtato Zrazazh Nature Joural [9] Fu Yuhua Shortcomgs ad pplcable Scopes of Specal ad Geeral Relatvty See: Usolved Problems Specal ad Geeral Relatvty Edted by: Floret Smaradache Fu Yuhua ad Zhao Fegua Educato Publshg [0] Floret Smaradache V Chrstato Fu Yuhua R Khrapko J Hutchso Ufoldg the Labyrth: Ope Problems Physcs Mathematcs strophyscs ad Other reas of Scece Hes-Phoe [] Fu Yuhua New Newto Mechacs Takg Law of Coservato of Eergy as Uque Source Law See: Cha Preprt servce system Receved: October 5th 03 ccepted: November 3rd 03 Fu Yuhua Neutrosophc Eamples Physcs

35 Neutrosophc Sets ad Systems Vol Flters va Neutrosophc Crsp Sets Salama* & Floret Smaradache** *Departmet of Mathematcs ad Computer Scece Faculty of Sceces Port Sad Uversty Egypt E-mal: **Uversty of New Meco Math & Scece Dvso 705 Gurley ve Gallup NM 8730 US E-mal: bstract I ths paper we troduce the oto of flter o the eutrosophc crsp set the we cosder a geeralzato of the flter s studes fterwards we preset the mportat eutrosophc crsp flters We also study several relatos betwee dfferet eutrosophc crsp flters ad eutrosophc topologes Possble applcatos to database systems are touched upo Keywords: Flters; Neutrosophc Sets; Neutrosophc crsp flters; Neutrosophc Topology; Neutrosophc Crsp Ultra Flters; Neutrosophc Crsp Sets Itroducto The fudametal cocept of eutrosophc set troduced by Smaradache [6 7 8] ad studed by Salama [ ] provdes a groudwork to mathematcally act towards the eutrosophc pheomea whch ests pervasvely our real world ad epad to buldg ew braches of eutrosophc mathematcs Neutrosophy has lad the foudato for a whole famly of ew mathematcal theores geeralzg both ther crsp ad fuzzy couterparts such as the eutrosophc crsp set theory Prelmares We recollect some relevat basc prelmares ad partcular the work of Smaradache [6 7 8] ad Salama et al [ ] Smaradache troduced the eutrosophc compoets T I ad F whch represet the membershp determacy ad o-membershp values respectvely where ] 0 [ - s the o- stadard ut terval 3 Neutrosophc Crsp Flters 3 Defto Frst we recall that a eutrosophc crsp set s a obect of the form = < 3> where 3 are subsets of X ad = φ = φ = φ 3 3 Let Ψ be a eutrosophc crsp set the set X We call Ψ a eutrosophc crsp flter o X f t satsfes the followg codtos: N Every eutrosophc crsp set X cotag a member of Ψ belogs toψ N Every fte tersecto of members of Ψ belogs toψ N3 φ N s ot Ψ I ths case the par X Ψ s eutrosophcally fltered byψ It follows from N ad N3 that every fte tersecto of members of Ψ s ot φ N ot empty We obta the followg results 3 Proposto The codtos N ad N are equvalet to the followg two codtos: N a The tersecto of two members of Ψ belogs toψ N a X N belogs to Ψ 33 Proposto Let Ψ be a o-empty eutrosophc subsets X satsfyg N The X N Ψ ff Ψ φn ; φn Ψ ff Ψ all eutrosophc crsp subsets of X From the above Propostos ad we ca characterze the cocept of eutrosophc crsp flter 34 Theorem Let Ψ be a eutrosophc crsp subsets a set X The Ψ s eutrosophc crsp flter o X f ad oly f t satsfes the followg codtos: Every eutrosophc crsp set X cotag a member of Ψ belogs toψ If Ψ the Ψ X Ψ Ψ φn Salama & Floret Smaradache Flters va Neutrosophc Crsp Sets

36 35 Neutrosophc Sets ad Systems Vol 03 Proof: It s clear 35 Theorem Let X φ The the set{ X N } s a eutrosophc crsp flter o X Moreover f s a o-empty eutrosophc X crsp set X the { Ψ : } s a eutrosophc crsp flter o X X N = Ψ : Sce Proof: Let { } X X N Ψ ad φn Ψ φn Ψ Ψ SupposeU V Ψ the U V Thus U V U V or U V ad 3 U 3 V for all X So 3 U V ad hece U V N 4 Comparso of Neutrosophc Crsp Flters 4 Defto Let Ψad Ψ be two eutrosophc crsp flters o a set X We say that Ψ s fer tha Ψ or Ψs coarser thaψ f Ψ Ψ If alsoψ Ψ the we say that Ψ s strctly fer tha Ψ or Ψs strctly coarser tha Ψ We say that two eutrosophc crsp flters are comparable f oe s fer tha the other The set of all eutrosophc crsp flters o X s ordered by the relato: Ψ coarser tha Ψ ths relato ducg the cluso X relato Ψ 4 Proposto Let Ψ J be ay o-empty famly of eutrosophc crsp flters o X The Ψ = JΨ s a eutrosophc crsp flter o X I fact Ψ s the greatest lower boud of the eutrosophc crsp set Ψ J the ordered set of all eutrosophc crsp flters o X 43 Remark The eutrosophc crsp flter duced by the sgle eutrosophc set X N s the smallest elemet of the ordered set of all eutrosophc crsp flters o X 44 Theorem Let Α be a eutrosophc set X The there ests a eutrosophc flter Ψ Α o X cotag Α f for ay gve fte subset { S S S } of Α the tersecto = S φn I fact Ψ Α s the coarsest eutrosophc crsp flter cotag Α Proof Suppose there ests a eutrosophc flter Α o X cotag Let be the set of all fte Ψ tersectos of members of The by aom N Ψ Α y aom N φ Ψ Α Thus for each member of we get that the ecessary codto holds Suppose the ecessary codto holds X Let Ψ = { Ψ : cotas a member of } where s the famly of all fte tersectos of members of The we ca easly check that Ψ satsfes the codtos Defto We say that the eutrosophc crsp flter Ψ defed above s geerated by ad s called a sub-base of Ψ 3 45 Corollary Let Ψ be a eutrosophc crsp flter a set X ad a eutrosophc set The there s a eutrosophc crsp / / flterψ whch s fer tha Ψ ad such that Ψ f ad s a eutrosophc set The there s a eutrosophc / crsp flterψ whch s fer tha Ψ ad such that / Ψ ff U φn for each U Ψ 46 Corollary setϕ N of a eutrosophc crsp flter o a o-empty set X has a least upper boud the set of all eutrosophc crsp flters o X f for all fte sequece Ψ J 0 of elemets of ϕ N ad all Ψ = φn 47 Corollary 3 The ordered set of all eutrosophc crsp flters o a o-empty set X s ductve If Λ s a sub-base of a eutrosophc flter Ν o X the Ψ s ot geeral the set of eutrosophc sets X cotag a elemet of Λ ; for Λ to have ths property t s ecessary ad suffcet that every fte tersecto of members of Λ should cota a elemet of Λ Hece we have the followg results 48 Theorem 3 Let β be a set of eutrosophc crsp sets o a set X The the set of eutrosophc crsp sets X cotag a elemet of β s a eutrosophc crsp flter o X f β possesses the followg two codtos: β The tersecto of two members of β cota a member of β N Salama & Floret Smaradache Flters va Neutrosophc Crsp Sets

37 Neutrosophc Sets ad Systems Vol β β adφ β φn N 49 Defto 3 Let Λ ad β be two eutrosophc sets o X satsfyg codtos β ad β We call them bases of eutrosophc crsp flters they geerate We cosder two eutrosophc bases equvalet f they geerate the same eutrosophc crsp flter 40 Remark 3 Let Λ be a sub-base of eutrosophc flter Ψ The the set β of fte tersectos of members of Λ s a base of a eutrosophc flter Ψ 4 Proposto 3 subset β of a eutrosophc crsp flter Ψ o X s a base of Ψ f every member of Ψ cotas a member of β Proof Suppose β s a base of Ν The clearly every member of Ψ cotas a elemet of β Suppose the ecessary codto holds The the set of eutrosophc sets X cotag a member of β cocdes wth Ψ by reaso of Ψ J 4 Proposto 3 / O a set X a eutrosophc crsp flterψ wth base / β s fer tha a eutrosophc crsp flterψ wth base / β f every member of β cotas a member of β Proof: Ths s a mmedate cosequece of Deftos ad 3 43 Proposto 33 / Two eutrosophc crsp flters bases β ad β o a set X are equvalet f every member of β cotas a member of β / ad every member of β / ad every / member of β cotas a member of β 5 Neutrosophc Crsp Ultraflters 5 Defto 4 eutrosophc ultraflter o a set X s a eutrosophc crsp flter Ψ such that there s o eutrosophc crsp flter o X whch s strctly fer tha Ψ other words a mamal elemet the ordered set of all eutrosophc crsp flters o X Sce the ordered set of all eutrosophc crsp flters o X s ductve Zor's lemma shows that: 5 Theorem 4 Let Ψ be ay eutrosophc ultraflter o a set X; the there s a eutrosophc ultraflter other tha Ψ 53 Proposto 4 Let Ψ be a eutrosophc ultraflter o a set X If ad are two eutrosophc subsets such that Ψ the Ψ or Ψ Proof: Suppose ot The there are eutrosophc sets ad X such that Ψ Ψ ad Ψ Let Λ = { M Ψ X : M Ψ } It s straghtforward to check that Λ s a eutrosophc crsp flter o X ad Λ s strctly fer tha Ψ sce Λ Ths cotradcto proves the hypothess that Ψ s a eutrosophc crsp ultraflter 54 Corollary 4 Let Ψ be a eutrosophc crsp ultraflter o a set X ad let Ψ be a fte sequece of eutrosophc crsp sets X If Ψ Ψ the at least oe of the Ψ = belogs to Ψ 55 Defto 5 Let Α be a eutrosophc crsp set a set X If U s ay eutrosophc crsp set X the the eutrosophc crsp set U s called trace of U o ad t s deoted by U For all eutrosophc crsp sets U ad V X we have U V = U V 56 Defto 6 Let Α be a eutrosophc crsp set a set X The the X set Λ of traces Α Ψ of members of Λ s called the trace of Λ o Α 57 Proposto 5 Let Ψ be a eutrosophc crsp flter o a set X X ad Α Ψ The the trace Ψ of Ψ o s a eutrosophc crsp flter f each member of Ψ tersects wth Proof: The result Defto 6 shows that Ψ N If M P the P = M P Thus Ψ satsfes N Hece Ψ s N e f each member of Ψ tersects wth satsfes a eutrosophc crsp flter f t satsfes 3 Salama & Floret Smaradache Flters va Neutrosophc Crsp Sets

38 37 Neutrosophc Sets ad Systems Vol Defto 7 Let Ψ be a eutrosophc crsp flter o a set X X ad Α Ψ If the trace s Ψ of Ψ o Α theψ s sad to be duced by Ψ ad Α 59 Proposto 6 Let Ψ be a eutrosophc crsp flter o a set X ducg a eutrosophc flter Ν o Α Ψ trace β o Α of a base β of Ψ s a base of Ψ Refereces X The the [] Salama ad S lblow Geeralzed Neutrosophc Set ad Geeralzed Neutrosophc Topologcal Spaces Joural computer Sc Egeerg Vol No 7 0 [] Salama ad S lbolw Neutrosophc set ad Neutrosophc topologcal space ISORJ Mathematcs Vol3 Issue4 pp [3] Salama ad S lbalw Itutostc Fuzzy Ideals Topologcal Spaces dvaces Fuzzy Mathematcs Vol7 Number pp [4] Salama ad HElagamy Neutrosophc Flters Iteratoal Joural of Computer Scece Egeerg ad Iformato Techology Reseearch IJCSEITR Vol3 IssueMar 03 pp [5] S lbow Salama & Mohmed Esa New Cocepts of Neutrosophc Sets Iteratoal Joural of Mathematcs ad Computer pplcatos Research IJMCRVol3 Issue 4 Oct [6] Floret Smaradache Neutrosophy ad Neutrosophc Logc Frst Iteratoal Coferece o Neutrosophy Neutrosophc Logc Set Probablty ad Statstcs Uversty of New Meco Gallup NM 8730 US 00 [7] Floret Smaradache troducto to the Neutrosophy probablty appled Quatum Physcs Iteratoal Coferece o Itroducto Neutrosophc Physcs Neutrosophc Logc Set Probablty ad Statstcs Uversty of New Meco Gallup NM 8730 US -4 December 0 [8] F Smaradache Ufyg Feld Logcs: Neutrosophc Logc Neutrosophy Neutrosophc Set Neutrosophc Probablty merca Research Press Rehoboth NM 999 [9] I Haafy Salama ad K Mahfouz Correlato of Neutrosophc Data Iteratoal Refereed Joural of Egeerg ad Scece IRJES Vol Issue PP [0] IM Haafy Salama ad KM Mahfouz Neutrosophc Crsp Evets ad Its Probablty Iteratoal Joural of Mathematcs ad Computer pplcatos Research IJMCR Vol 3 Issue Mar 03 pp Receved: November 03 ccepted: December 5 03 Salama & Floret Smaradache Flters va Neutrosophc Crsp Sets

39 Neutrosophc Sets ad Systems Vol Commucato vs Iformato a omatc Neutrosophc Soluto Floret Smaradache ad Ştefa Vlăduțescu Uversty of New Meco 00 College Road Gallup NM 8730 US E-mal: fsmaradache@gmalcom Uversty of Craova 3 I Cuza Street Craova Romaa E-mal: stefavladutescu@yahoocom bstract Study represets a applcato of the eutrosophc method for solvg the cotradcto betwee commucato ad formato I addto t recourse to a approprate method of approachg the cotradctos: Etescs as the method ad the scece of solvg the cotradctos The research core s the realty that the scetfc research of commucato-formato relatoshp has reached a dead ed The bvalet relatoshp commucatoformato formato-commucato has come to be cotradctory ad the two cocepts to block each other fter the crtcal eamato of coflctg postos epressed by may eperts the feld the etesc ad clusve hypothess s ssued that formato s a form of commucato The obect of commucato s the sedg of a message The message may cosst of thoughts deas opos feelgs belefs facts formato tellgece or other sgfcatoal elemets Whe the message cotet s prmarly formatoal commucato wll become formato or tellgece The argumets of supportg the hypothess are: a lgustc the most mportat beg that there s "commucato of formato" but ot "formato of commucato"; also t s clarfed ad reforced the over stuated referet that of the commucato as a process b systemc-procedural the commucato system s developg a formato system; the formg actat s a type of commucator the formato process s a commucato process c practcal the delmtato elmates the efforts of dsparate ad cosstet uderstadg of the two cocepts d epstemologcal argumets the possblty of tersubectve thkg of realty s created lgustc argumets e logcal ad realstc argumets t s oted the stuato that allows to thk coheretly a system of cocepts - dervatve seres or tegratve groups f ad argumets from hstorcal eperece the cocept of commucato has temporal prorty t appears 3 tmes Julus Caesar s wrtgs I a aomatc cocluso the ma argumets are summarzed four aoms: three are based o the pertet observatos of specalsts ad the fourth s a relevat applcato of Floret Smaradache s eutrosophc theory Keywords: eutrosophy commucato formato message etescs Clarfcato o the used methodologcal tool Wth the Etescs as a scece of solvg the coflctg ssues "etescal procedures" wll be used to solve the cotradcto I ths respect cosderg that the matter-elemets are defed ther propertes wll be eplored "The key to solve cotradctory problems We Ca argues the fouder of Etescs Ca 999 p 540 s the study of propertes about matter-elemets" ccordg to The basc method of Etescs s called eteso methodology ad "the applcato of the eteso methodology every feld s the eteso egeerg methods" Weha L & Chuya Yag 008 p 34 Wth eutrosophc lgustc systemc ad hermeeutcal methods grafted o "eteso methodology" a are "ope up the thgs" b s marked "dverget ature of matter-elemet" c "etesblty of matter-elemet" takes place ad c "eteso commucato" allows a ew cluso perspectve to ope a sequetal ragg of thgs to emphasze at a hgher level ad the cotradctory elemets to be solved "Eteso" s as postulated by We Ca Ca 999 p 538 "opeg up carred out" The subect of commucato: the message The subect of formg: the formato The formato thess as speces of message I order to fsh our basc thess that of the formato as a form of commucato ew argumets may be revealed whch corroborate wth those prevously metoed s pheomea processes the commucato ad formato occur a uque commucato system I commucato formato has acqured a specalzed profle I the formato feld the tellgece hs tur stregtheed a specfc detectable detfable ad dscrmatve profle It s therefore acceptable uder the pressure of practcal argumet that oe may speak of a geeral commucato system whch relato to the Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

40 Neutrosophc Sets ad Systems Vol message set ad cofgured the commucato process could be maged as formato system or tellgece system Uder the fluece of the systemc assumpto that a utary commucator trasmts or customze trasactoally wth aother recevg commucator a message oe may uderstad the commucatoal system as the teractoal ut of the factors that eerts ad fulfll the fucto of commucatg a message I hs books "Messages: buldg terpersoal commucato sklls" attaed 993 ts fourth edto ad 00 ts twelfth ad "Huma Commucato" 000 Joseph De Vto the reowed specalst who has proposed the ame "Commucology" for the sceces of commucato develops a cocept of a smple ad productve message The message s as cotet what s commucated s a systemc factor t s emergg as what s commucated To remember ths cotet s that the Germa Otto Kade ssted that what t s commucated to receve the ttle of "release" ccordg to Joseph De Vto through commucato meags are trasmtted "The commucated message" s oly a part of the meags De Vto 993 p 6 mog the shared meags feelgs ad perceptos are foud De Vto J 993 p 98 Lkewse formato ca be commucated De Vto 990 p 4 De Vto 000 p 347 also Fârte 004; Cupercă 009; Coocaru ragaru & Cuch 0; Cobley & Schulz 03 I a "message theory" called "geltcs" Rafael Capurro argues that the message ad formato are cocepts that desgate smlar but ot detcal pheomea I Greek "gela" meat message; from here "geltcs" or theory of the message geltcs s dfferet from geologa dealg the feld of relgo ad theology wth the study of agels R Capurro set four crtera for assessg the relatoshp betwee message ad formato The smlarty of the two eteds over three of them The message as well as the formato s characterzed as follows: s supposed to brg somethg ew ad/or relevat to the recever; ca be coded ad trasmtted through dfferet meda or messegers; s a utterace that gves rse to the recever s selecto through a release mechasm of terpretato "The dfferece betwee these two s the et: a message s seder-depedet e t s based o a heteroomc or assymetrc structure Ths s ot the case of formato: we receve a message but we ask for formato see also Capurro 0; Holgate 0 To request formato s to sed a message of requestg formato Therefore the message s smlar to the formato ths respect too I our opo the dfferece betwee them s from geus to speces: formato s a speces of message The message depeds o the trasmtter ad the formato as well Iformato s stll a specfcato of the message s a formatve message C Shao asserts that the message s the defg subect of the commucato He s the stake of the commucato because the fudametal problem of commucato s that of reproducg at oe pot ether eactly or appromately a message selected at aother pot Shao 948 p 3 The commucato process s fact the "commucato" of a comple ad multlayered message 'Thoughts terests talets epereces"duck & McMaha 0 p "formato deas belefs feelgs "Wood 009 p 9 ad p 60 ca be foud a message G Mller T M Newcomb ad ret R Rube cosder that the subect of commucato s formato: "Commucato - Mller shows meas that formato s passed from oe place to aother Mller 95 p 6 I hs tur T M Newcomb asserts: very commucato act s vewed as a trasmsso of formato Newcomb 966 p 66 ad ret R Rube argues: Huma commucato s the process through whch dvduals relatoshps groups orgazatos ad socetes create trasmt ad use formato to relate to the evromet ad oe aother Rube 99 p 8 Professor Ncolae Drăgulăescu member of the merca Socety of Iformato Scece ad Techology s the most mportat of Romaa specalsts the Scece of formato ccordg to hm "commucatg formato" s the thrd of the four processes that form the "formatoal cycle" alog wth geeratg the formato processg/storg the formato ad the use of formato The process of commucato Ncolae Drăgulaescu argues s oe of the processes whose obect s the formato p 8 also Drăgulăescu 00; Drăgulăescu 005 The same le s followed by Gabrel Zamfr too; he sees the formato as "what s commucated oe or other of the avalable laguages" Zamfr 998 p 7 as well as teacher Sultaa Craa: commucato s a "process of trasmttg a pece of formato a message" Craa 008 p 53 I geeral t s accepted that formato meas trasmttg or recevg formato However whe speakg of trasmttg formato the process s cosdered ot to be formato but commucato Therefore t s created the appearace that the formato s the product ad commucato would oly be the trasmttg process Teodoru Ştefa Io Iva ş Crsta Popa assert: "Commucato s the process of trasmttg formato so the rato of the two categores s from the basc product to ts trasmsso" Popa Teodoru & Iva I 008 p The professors Vasle Tra ad Ira Stăcugelu see commucato as a "echage of formato wth symbolc cotet" Tra & Stăcugelu 003 p 09 The commucato s a over-raged cocept ad a otologcal category more eteded tha formg or formato O the other had formato s geerated eve the global commucato process From ths pot of vew formato whose subect- Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

41 40 Neutrosophc Sets ad Systems Vol 03 message s formato s a regoal sectoral commucato Iformato s that commucato whose message cossts of ew relevat pertet ad useful sgfcaces e of formato Ths posto s shared by Doru Eache too Eache 00 p 6 The posto set by Norbert Weer cosoldated by L rllou ad edorsed by may others makes from the formato the oly cotet of the message N Weer argues that the message "cotas formato" Weer N 965 p 6 L rllou talks about "formato cotaed the message" rllou 004 p 94 ad p 8 Through commucato "formato cocepts emotos belefs are coveyed" ad commucato "meas ad subsumes formato" Rotaru 007 p0 Well-kow teachers Marus Petrescu ad Neculae Năbârou cosder that the dstcto betwee commucato ad formato must be acheved depedg o the message commucato wth a formatoal message becomes formato s a form of commucato formato s characterzed by a formatve message ad a "message s formatve as log as t cotas somethg ukow yet" Petrescu & Năbârou 006 p 5 Oe of the possble sgfcat elemets that could form the message cotet s thus the formato as well Other compoets could be thoughts deas belefs kowledge feelgs emotos epereces ews facts Commucato s "commucatg" a message regardless of ts sgfcat cotet 3 The formato thess as a form of commucato The questo of the relatoshp betwee commucato ad formato as felds of estece s the fgerprt as of commucato ad formato otology The otologcal format allows two formulas: the estece the act ad the vrtual estece The otologcal compoet of the cocepts tegrates a presece or a potecy ad a estetal fact or at a potetal of estece Zs 007; llo 007; Sta 009; urg 00; Case 03 I addto to the categoral-otologcal elemet the uclear rato of commucato-formato cocepts t shows comparatve specfctes ad regardg attrbutes ad characterstcs o three compoets epstemologcal methodologcal ad hermeeutcal I a scece whch would have frmly take a strog subect a methodology ad a specfc set of cocepts ths otologcal foudg decso would be take a aom It s kow that prcple aoms solve wth the lmts of that type of argumet called evdece clear ad dstct stuato the relatos betwee the systemc structural basc cocepts Specfcally Etescs scetsts wth a advaced vso substatated by professor We Ca aoms gover the relatoshp betwee two matter-elemets wth dverget profles For the commucato ad formato ssues that have occurred relatvely recetly about three quarters of a cetury subects of study or areas of scetfc cocer ot a scetfc authorty to settle the ssue was foud The weakesses of these sceces of soft type are vsble eve today whe after o accredted proposals of scece "comucology" - commucology Joseph De Vto "commucatcs" - "comucatque" of Metayer G formatology - Klaus Otte ad thoy Debos 970 t was resorted to the remag the ambguty of valdatg the subect "The sceces of commucato ad formato" or "The sceces of formato ad commucato" eoyg the support of some courses books studes ad dctoares Toma 999; Tudor 00; Streche 009; Ţeescu 009 Ths geerc vso of uty ad coheso wrogs both the commucato ad formato Vlăduțescu 004; Vlăduțescu 006 I practce the apparet uust overall tegratve altogether treatmet has ot a etrely ad coverg cofrmato I almost all humast uverstes of the world the facultes ad the commucato courses are prevalg cludg those of Romaa ad Cha Professor Ncolae Drăgulăescu ascertaed what Romaa s cocered that 0 colleagues commucato wth varous deomatos s taught ad oly two the formg-formato s taught The ma perspectves from whch the cotradctory relatoshp of commucato-formato was approached are the otologcal the epstemologcal ad the systemc I most cases opos were cdetal Whe t was about the dedcated studes the most commo comparatve approach was ot programmatcally made o oe or more crtera ad ether drectly ad appled I hs study "Commucato ad Iformato" 9 March 9 pp 3-3 J R Schemet starts from the observato that " the rhetorc of the Iformato ge the commucato ad formato coverge syoymous meags" O the other had he retas that there are specalsts who declare favor of statg a frmg dstcto of ther meags To clarfy eactly the relatoshp betwee the two pheomea e cocepts he eames the deftos of formato ad commucato that have marked the evoluto of the "formato studes" ad the "commucato studes" For formg formato three fudametal themes result: formato-as-thg M K ucklad formato-as-process N J elk R M Hayes Machlup & Masfeld Elster - 00 etc Iformatoas-product-of - mapulato C J Fo R M Hayes It s also oted that these three subects volve the assessg of ther ssuers a "coecto to the pheomeo of commucato" I parallel from eamg the deftos of commucato t s revealed that the specalsts "mplctly or eplctly troduce the oto of formato defg commucato" There are also three Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

42 Neutrosophc Sets ad Systems Vol 03 4 the cetral themes of defg commucato: commucato-as-trasmsso C Shao W Weaver E Emery C Cherry erelso G Steer commucato-as-sharg-process R S Gover W Schramm commucato-as-teracto G Gerber L Thayer Comparg the s thematc odes Schemet emphaszes that the lk betwee formato ad commucato s "hghly comple" ad dyamc "formato ad commucato s ever preset ad coected" Schemet 993 p 7 I addto order that formato est the potetal for commucato must be preset The result at the otologcal level of these fdgs s that the estece of formato s strctly codtoed by the presece of commucato That s for the formato to occur commucato must be preset Commucato wll precede ad always codto the estece of formato d more detaled: commucato s part of the formato otology Otologcally formato occurs commucato also as potecy of commucato Vlăduțescu 00 J R Schemet s focused o fdg a way to cesus a coheret mage leadg to a theory of commucato ad formato "Toward a Theory of Commucato ad Iformato" - Schemet 993 p 6 He avods to coclusvely assertg the temporal ad lgustc prorty the otologcal precedece ad the ampltude of commucato relato to formato The study cocludes that "Iformato ad commucato are socal structures" "two words are used as terchageable eve as syoyms" t s argued Schemet 993 p 7 "The study of formato ad commucato share cocepts commo" both of them commucato formato "symbol cogto cotet structure process teracto techology ad system are to be foud" - Schemet 993 p 8 3 "Iformato ad commucato form dual aspects of a broader pheomeo" Schemet JR 993 p 8 I other words we uderstad that: a lgustcally "words" "terms" "otos" "cocepts" "dea of" commucato ad formato are syoyms; b as area of study the two resort the same coceptual arseal Stuato produced by these two elemets of the cocluso allows our opo a herarchy betwee commucato ad formato If t s true that otologcally ad temporally the commucato precedes formato f ths latter pheomeo s a eteso smaller tha the frst f evetual sceces havg commucato as obect respectvely formato beeft from the oe ad the same coceptual vocabulary the the formato ca be a form of commucato Despte ths le followed coheretly by the lgustc categorcalotologcal coceptual ad deftoal epstemologcal argumets brought the reasog the thrd part of the cocluso postulates the estece of a uque pheomeo whch would clude commucato ad formato 3 "Iformato ad commucato form two aspects of the same pheomeo "- Schemet JR 993 p 8 Ths pheomeo s ot amed The coclusve le followed by the argumets ad the prevous coclusve elemets eabled us to artculate formato as oe of the forms of commucato Cofrmatvely the fact that J R Schemet does ot ame a pheomeo stuated over commucato ad formato gves us the possblty of attractg the argumet order to stregthe our thess that formato s a form of commucato That s because a category of pheomea ecompassg commucato ad formato caot be foud J R Schemet teds towards a levelg perspectve ad of covergece the commucato ad formato otology Istead M Norto supports a emphaszed dfferetato betwee commucato ad formato He belogs to those who see commucato as oe of the processes ad oe of the methods "for makg formato avalable" The two pheomea "are trcately coected ad have some aspects that seem smlar but they are ot the same" Norto 000 p 48 ad p 39 Harmut Mokros ad ret R Rube 99 lay the foudato of a systemc vso ad levelg uderstadg of the commucato-formato relatoshp Takg to accout the cotet of reportg as a core elemet of the teral structure of commucato ad formato systems they mark the formato as a crtero for the radography of relatoshp The systemc-theoretcal olear method of research fouded 983 by R Rube s appled to the subect represeted by the pheomea of commucato ad formato Research lays the "Iformato ge" ad creates a formatoal reportg mage The ma mert of the vestgato comes from the relevace gve to the o-subordato betwee commucato ad formato terms of a upolar commucato that relates to a levelg formato Iterestg s the approach of formato three costtuet aspects: "formatoe" potetal formato - that whch ests a partcular cotet but ever receved a sgfcace the system "formato" actve formato the system ad "formato" formato created socally ad culturally the system The levelg formato s related to a ufed commucato Hofkrcher 00; Flord 0; Fuchs 03; Hofkrcher 03 O each level of formato there s commucato Iformato ad commucato s co-preset: commucato s heret to formato Iformato has heret propertes of commucato Research brgs a systemc-cotetual elucdato to the relatoshp betwee commucato ad formato ad oly subsdarly a frm otologcal postog I ay case: formato commucato ever msses I the most mportat studes of the professor Sta Petrescu: "Iformato the fourth weapo" 999 ad "bout tellgece Espoage-Couterespoage" 007 formato s uderstood as "a type of commucato" Petrescu 999 p 43 ad stuated the broader cotet Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

43 4 Neutrosophc Sets ad Systems Vol 03 of "kowledge o the teral ad teratoal formato evromet " Petrescu 007 p 3 4 omatc cocluso: four aoms of commucato-formato otology 4 The message aom We call the otologcal segregato aom o the subect or the Tom D Wlso - Solomo Marcus aom the thess that ot ay commucato s formato but ay formato s commucato Wheever the message cotas formato the commucatoal process wll acqure a formatoal profle Moreover the commucatoal system becomes formatoal system Dervatvely the commucator becomes the "former" ad the commucatoal relatoshp turs to formatoal relatoshp The teractoal bass of socety eve the Iformato ge s the commucatoal teracto Most socal teractos are o-formatoal I ths respect T D Wlso has oted: We frequetly receve commucatos of facts data ews or whatever whch leave us more cofused tha ever Uder formal defto these commucatos cota o formato Wlso 987 p 40 cademca Solomo Marcus takes to accout the udeable estece of a commucato "wthout a trasfer of formato" Marcus 0a p 0; Marcus 0b For commucatos that do ot cota formato we do ot have a separate ad specfc term Commucatos cotag formato or ust formato are called formg Commucato volves a kd of formato but as Jea audrllard stated pud Dâcu 999 p 39 "t s ot ecessarly based o formato" More specfcally ay commucato cotas cogto that ca be kowledge data or formato Therefore commucato formato may be mssg may be adacet cdetal or collateral Commucato ca be formatoal ature or ts destato That commucato whch by ts ature ad orgazato s commucato of formato s called formg The ma process ra Iformato System s formg The fucto of such a system s to form The actats ca be formats producers-cosumers of formato trasmtters of formato etc The formato acto takes detty by the cover eabled oto-categoral by the verb "to form" I hs tur Petros Gelepths cosders the two cocepts commucato ad formato to be crucal for "the study of formato system" Gelepths 999 p 69 Cofrmg the formato aom as post reductost message as reduced obect of commucato Sore rer substatates: commucato system actually does ot echage formato rer 999 p 96 Sometmes wth the commucato system formato s o loger echaged However commucato remas; commucato system preserves ts valdty whch dcates ad subsequetly proves that there ca be commucato that does ot volve formato ates 006; Deca 006; Chapma & Ramage 03 O the other had the a whe the Iformato System fuctoal prcples such as "eed to kow"/"eed to share" are troduced b whe rug processes for collectg aalyzg ad dssematg formato c whe the beefcares are decders "decso maker" "mstry" "govermet" "polcymakers" ad d whe the cagess tem occurs ths Iformato System wll become Itellgece System see Gll Marr & Phyta 009 p 6 p 7 p p 7 Sms & Gerber 005 p 46 p 34; Gll P& Phyta 006 p 9 p 36 p 88; Johso 00 p 5 p 6 p 6 p 39 p 79; Maor 009; Maor 00 Peter Gll shows that "Secrecy s the Key to Uderstadg the essece of tellgece" Gll 009 p 8 ad Professor George Crsta Maor emphaszes: " tellgece collectg ad processg formato from secret sources rema essetal" Maor 00 p Sherma Ket W Laqueur M M Lowethal G-C Maor etc start from a comple ad multlayered cocept of tellgece uderstood as meag kowledge actvty orgazato product process ad formato Subsequetly the questo of otology epstemology hermeeutcs ad methodology of tellgece occurs Lke Peter Gll G-C Maor does poeerg work to separate the otologcal approach of tellgece from the epstemologcal oe ad to aalyze the "epstemologcal foudato of tellgece" Maor 00 p 33 ad p 43 The tellgece must be also cosdered terms of otologcal aom of the obect I ths regard otceable s that oe of ts meags perhaps the crtcal oe places t some way the formato area I our opo the formato that has crtcal sgfcace for accredted operators of the state ecoomc facal ad poltcal power ad holds or acqures cofdetal secret feature s or becomes tellgece Iformato from tellgece systems ca be by tself tellgece or ed up beg tellgece after some specalzed processg "Itellgece s ot ust formato that merely ests" Marcă & Iva 00 p 08 Maraa Marcă ad Io Iva assert t s acqured after a "coscous act of creato collecto aalyss terpretato ad modelg formato" Marcă & Iva 00 p 05 4 Lgustc aom secod aom of commucato-formato otologcal segregato ca be draw relato to the lgustc argumet of the acceptable grammatcal cotet Rchard Varey cosders that uderstadg "the dfferece betwee commucato ad formato s the Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

44 Neutrosophc Sets ad Systems Vol cetral factor" ad fds the lgustc cotet the crtero to valdate the dfferece: we speak of gvg formato to whle commucate wth other Varey 997 p 0 The trasmsso of formato takes place "to" or to someoe ad commucato takes place "wth" log wth ths varat of grammatcal cotet t mght also emerge the stuato of acceptablty of some statemets relato to the obect of the commucato process respectvely the obect of the formato process The statemet "to commucate a message formato" s acceptable Istead the statemet "to form commucato" s ot The phrase "commucato of messages-formato" s vald but the phrase "formg of commucato" s ot Therefore laguage bears kowledge ad "lead us" Mart Hedegger states to ote that lgustcally commucato s more otologcal etesve ad that formato otology s subsumed to t Heo 03; Gîfu & Crstea 03; Goru & Goru 0 The otcal ad otologcal ature of laguage allows t to epress the estece ad to acheve a fuctoalgrammatcal specfcato Laguage allows oly grammatcal esteces s message the formato ca be "commucated" or "commucable" There s also the case whch a pece of formato caot be "commucated" or "commucable" Related commucato caot be "formed" The sematc feld of commucato s therefore larger rcher ad more versatle Ştefa uzărescu 006 Commucato allows the "commucable" 43 Teleologcal aom I addto to the aom of segregatg commucato of formg relato to the obect message t may be stated as a aom a Magoroh Maruyama's cotrbuto to the demythologzato of formato I the artcle "Iformato ad Commucato Poly Epstemologcal System" "The Myths of Iformato" he states: The trasmsso of formato s ot the purpose of commucato I Dash culture for eample the purpose of commucato s frequetly to perpetuate the famlar rather tha to troduce ew formato Maruyama 980 p 9 The otologcal aom of segregato relato to the purpose determes formato as that type of commucato wth low emergece whch the purpose of the teracto s trasmttg formato 44 The eutrosophc commucato aom Uderstadg the frame set by the three aoms we fd that some commucatoal elemets are heterogeeous ad eutral relato to the crtero of formatvty I a speech some elemets ca be suppressed wthout the message sufferg formatoal alteratos Ths meas that some message-dscursve meags are redudat; others are ot essetal relato to the ores-the practcal course or of practcal touch the order of reasog Redudaces ad o-uclear sgfcatoal compoets ca be elded ad formatoal ad the message remas formatoally uchaged Ths proves the estece of cores wth eutral eutrosophc meags I the epstemologcal foudatos of the cocept of eutrosophy we refer to Floret Smaradache s work Ufyg Feld Logcs Neutrosophc Logc Neutrosophy Neutrosophc Set Neutrosophc Probablty ad Statstcs 998 Smaradache 998; Smaradache 999; Smaradache 00; Smaradache 005; Smaradache 00a; Smaradache 00b; Smaradache & Părou 0 O the operato of ths pheomeo are based the procedures of tetual cotracto of groupg of seral regstrato of assocatg summarzg sytheszg tegratg We propose to uderstad by eutrosophc commucato that type of commucato whch the message cossts of ad t s based o eutrosophc sgfcatoal elemets: o-formatoal redudat eldable cotradctory complete vague mprecse cotemplatve o-practcal of relatoal cultvato Iformatoal commucato s that type of commucato whose purpose s sharg a formatoal message The ssuer's fudametal approach s formatoal commucato to form To form s to trasmt formato or specfcally the professor s Ile Rad words: "to form that s ust sed formato" Moldova 0 p 70 also Rad 005; Rad 008 I geeral ay commucato cotas some or certa eutrosophc elemets suppressble redudat eldable o-uclear elemets ut whe eutrosophc elemets are prevalg commucato s o loger formatoal but eutrosophc Therefore the eutrosophc aom allows us to dstgush two types of commucato: eutrosophc commucato ad formatoal commucato I most of the tme our commucato s eutrosophc The eutrosophc commucato s the rule The formatoal commucato s the ecepto I the ocea of the eutrosophc commucato damate slads of formatoal commucato are dstgushed Refereces [] llo P 007 Iformatoal cotet ad formato structures: a pluralst approach I Proceedgs of the Workshop o Logc ad Phlosophy of Kowledge Commucato ad cto The Uversty of the asque Coutry Press pp 0- [] ates M 006 Fudametal forms of formato Joural of merca Socety for Iformato Scece ad Techology [3] elk N J 978 Iformato cocepts for Iformato Scece Joural of Documetato [4] rer S 999 What s a Possble Otologcal ad Epstemologcal Framework for a true Uversal Ifor- Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

45 44 Neutrosophc Sets ad Systems Vol 03 mato Scece I W Hofkrsher Ed The Quest for a ufed Theory of Iformato msterdam: Gordo ad reach Publshers [5] rllou L 004 Scece ad Iformato Theory d ed New York: Dober Publcatos [6] urg M 00 Theory of Iformato World Scetfc Publshg [7] uzărescu Ştefa 006 Regmul teraţoal al formaţe Revsta de Iformatcă Socală 35-3 [8] Ca We 999 Eteso Theory ad ts pplcato Chese Scece ullet [9] Capurro R 0 geletcs - Message Theory I R Capurro & J Holgate Eds Messages ad Messegers: ge-letcs as a pproach to the Pheomeology of Commucato pp 5-5 Vol 5ICIE Seres: Much Germay [0] Case D O 0 Lookg for Iformato 3rd ed gley: Emerald Group Publshg [] Chapma D & Ramage M 03 Itroducto: The Dfferece That Makes a Dfferece Trple C -5 [] Cupercă E M 009 Pshosocologa veţ cotdee ucureşt: Edtura NIMV [3] Cobley P & Schulz P J 03 Itroducto I P Cobley & P J Schulz Eds Theores ad Models of Commucato pp -6 erl/osto: Walter de Gruyter [4] Coocaru S ragaru C & Cuch O M 0 The Role of Laguage Costructg Socal Realtes The pprecatve Iqury ad the Recostructo of Orgasatoal Ideology Revsta de Cercetare ş Iterveţe Socală 36 3 [5] Craa Sultaa 008 Dcţoar de comucare massmeda ş ştţa comucăr ucureşt: Edtura Meroa [6] Dâcu V S 999 Comucarea smbolcă Clu-Napoca: Edtura Daca [7] Deca D 006 Pragmatc versus Stactc Idetfcato of Thematc Iformato Dscourse Scetfc ullet of the Poltehcs of Tmsoara Trasactos o Moder Laguages 5 - [8] DeVto J 993 Messages Harper Colls College Publshers [9] DeVto J 000 Huma Commucato ddso Wesley Logma [0] DeVto J 98 Commucology New-York: Harper ad Row [] Dobreau Crstel 00 Prevetg surprse at the strategc level uletul Uverstăţ Naţoale de părare Ca-rol I [] Drăgulăescu N 00 Emergg Iformato Socety ad Hstory of Iformato Scece Romaa Joural of the merca Socety for Iformato Scece ad Techology 53 [3] Drăgulăescu N 005 Epstemologcal pproach of Cocept of Iformato Electcal Egeerg ad Iformato Scece Hypero Scetfc Joural 4 [4] Duke S W & McMaha D T 0 The ascs of Commucato: relatoal perspectve Sage [5] Elster D 00 Iformato als Prozess Trple C [6] Eache D 00 Iformaţa de la prmul cal troa la cel de-al dolea cal troa Paraşutşt [7] Fârte G I 004 Comucarea O abordare praeologcă Iaş: Edtura Demurg [8] Flord L 0 The Phlosophy of Iformato Oford Uversty Press [9] Fruză S 0 Does commucato costruct realty? Revsta de Cercetare ș Iterveţe Socală [30] Fuchs C 03 Iteret ad socety: Socal theory formato age Lodo: Routledge [3] Gelepths P 999 rudmetary theory of formato I W Hofkrsher Ed The Quest for a ufed Theory of Iformato msterdam: Gordo ad reach [3] Gîfu D & Crstea D 03 Towards a utomated Semotc alyss of the Romaa Poltcal Dscourse Computer Scece 6 [33] Gll P & Phyta S 006 Itellgece a secure world Cambrdge: Polty Press [34] Gll P Marr S & Phyta S 009 Itellgece Theory: Key questos ad debates New York: Routledge [35] Goru & Goru H T 0 Publc-Prvate: Publc Sphere ad Ctzeshp Joural of US-Cha Publc dmstrato [36] Heo J 03 Emergece of Iformato Commucato ad Laguage I P Votas et al Eds Iformato Modellg ad Kowledge ases XXIV pp mstedam: IOS Press [37] Hofkrcher W 00 ufed theory of formato: outle truagora 64 [38] Hofkrcher W 03 Emerget Iformato Whe a Dfferece Makes a Dfferece Trple C [39] Holgate J 0 The Hermesa Paradgm: mythologcal perspectve o ICT based o Rafael Capurro s geltcs ad Vlem Flusser s Commucology I R Capurro & J Holgate Eds Messages ad Messegers: geletcs as a pproach to the Pheomeology of Commucato pp Vol 5 ICIE Seres: Much: Fk [40] Johso L K Ed 00 The Oford of Natoal Securty Itellgece Oford Uversty Press [4] L Wehua & Yag Chuya 008 Eteso Iformato-Kowledge-Strategy System for Sematc Iteroperablty Joural of Computers [4] Maor George Crsta 009 Icerttude Gâdre strategcă ş relaţ teraţoale î secolul XXI ucureşt: Edtura Rao [43] Maor George Crsta 00 U războ al mţ Itellgece servc de formaţ ş cuoaştere strategcă î secolul XXI ucureşt: Edtura Rao [44] Marcus S 0b Elargg the Perspectve: Eergy Securty Va Equlbrum Iformato ad Computato Eergy Securty 7-78 [45] Marcus S 0 Îtâlr cu/meetgs wth Solomo Marcus ucureşt: Edtura Spadugo [46] Marescu Valeta 0 Itroducere î teora comucăr ucureşt: Edtura C H eck [47] Marcă M & Iva I 00 Itellgece de la teore către ştţă Revsta Româă de Stud de Itellgece [48] Maruyama M 980 Iformato ad Commucato Poly-Epstemologcal Systems I K Woodward Ed Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

46 Neutrosophc Sets ad Systems Vol 03 The Myths of Iformato Routledge ad Kega Paul [49] Métayer G 97 La Commucatque Pars: Les édtos d orgasato [50] Mller G 95 Laguage ad commucato New York: Mc-Graw-Hll [5] Mokros H & Rube D 99 Uderstadg the Commucato-Iformato Relatoshp: Levels of Iformato ad Cotets of valabltes Scece Commucato [5] Moldova L 0 Idc uralstce Itervu cu prof uv dr Ile Rad Vatra veche Sere ouă [53] Newcomb T M 966 pproach to the study of commucatve acts I G Smth Ed Commucato ad culture New York: Holt Rehart ad Wsto [54] Norto M 000 Itroductory cocepts of Iformato Scece Iformato Today [55] Otte K W & Debos 970 Toward a Metascece of Iformato: Iformatology Joural SIS [56] Păvălou Cathere 00 Elemete de deotologe a evaluăr î cotetul creşter caltăţ actulu educaţoal Forţele terestre [57] Petrescu Marus & Năbârou Neculae 006 Maagemetul formaţlor vol I Târgovşte: Edtura blotheca [58] Petrescu Sta 999 Iformaţle a patra armă ucureşt: Edtura Mltară [59] Petrescu Sta 007 Despre tellgece Spoa Cotraspoa ucureşt: Edtura Mltară [60] Popa C Ştefa Teodoru & Iva Io 008 Măsur orgazatorce ş structur fucţoale prvd accesul la formaţ ucureşt: Edtura NI [6] Rad Ile 005 Juralsmul cultural î actualtate Clu- Napoca: Edtura Trbua [6] Rad Ile 008 Cum se scre u tet ştţfc Dscplele umastce Iaş: Polrom [63] Rotaru Ncolae 007 PSI-Comucare ucureşt: Edtura NI [64] Rube D 99a The Commucato-formato relatoshp System-theoretc perspectve Joural of the merca Socety for Iformato Scece [65] Rube D 99b Commucato ad huma behavor New York: Pretce Hall [66] Schemet J R 993 Commucato ad formato I J R Schemet & D Rube Eds Iformato ad ehavor Vol 4 etwee Commucato ad Iformato New ruswck NJ: Trasacto Publshers [67] Shao C E 948 Matematcal Theory of Commucato The ell System Techcal Joural [68] Sms J E & Gerber 005 Trasformg US Itellgece Washgto DC: Georgetow Uversty Press [68] Smaradache F 998 Ufyg Feld Logcs Neutrosophc Logc Neutrosophy Neutrosophc Set Neutrosophc Probablty ad Statstcs Reboboth: merca Research Press [69] Smaradache F 999 Ufyg Feld Logcs: Neutrosophc Logc Phlosophy -4 [70] Smaradache F 00 Neutrosophy a ew rach of Phlosophy Multple Valued Logc [7] Smaradache F 005 Ufyg Feld Logcs: Neutrosophc Logc Neutrosophy Neutrosophc Set Neutrosophc Probablty Ifte Study [7] Smaradache F 00a Strategy o T I F Operators Kerel Ifrastructure Neutrosophc Logc I F Smaradache Ed Multspace ad Multstructure Neutrosophc Trasdscplarry 00 Collected Papers of Sceces pp Vol 4 Hako: NESP [73] Smaradache F 00b Neutrosophc Logc as a Theory of Everythg Logcs I F Smaradache Ed Multspace ad Multstructure Neutrosophc Trasdscplarry 00 Collected Papers of Sceces pp Vol 4 Hako: NESP [74] Smaradache F & Părou T 0 Neutrosofa ca reflectare a realtăţ ecoveţoale Craova: Edtura Stech [75] Smaradache F 005 Toward Dalectc Matter Elemet of Etescs Model Iteret Source [76] Sta L V 009 Iformato ad Iformatoal Systems wth the Moder attle feld uletul Uverstăț Națoale de părare Carol I [77] Streche M 009 Terms of Lat Org the feld of Commucato Sceces Stud ş cercetăr de Oomastcă ş Lecologe SCOL II - 03 [78] Toma Gheorghe 999 Tehc de comucare ucureşt: Edtura rtprt [79] Tra V & Stăcugelu I 003 Teora comucăr ucureşt: comucarero [80] Tudor Doa 00 Mapularea ope publce î coflctele armate Clu-Napoca: Daca [8] Ţeescu la 009 Comucare ses dscurs Craova: Edtura Uverstara [8] Vlăduţescu Ştefa 00 Iformaţa de la teore către ştţă Propedeutcă la o ştţă a formaţe ucureşt: Edtura Ddactcă ş Pedagogcă [83] Vlăduţescu Ştefa 004 Comucologe ş Mesagologe Craova: Edtura Stech [84] Vlăduţescu Ştefa 006 Comucarea uralstcă egatvă ucureşt: Edtura cademe [85] Weer N 965 Cyberetcs 3 th ed MIT Press [86] Wlso T D 987 Treds ad ssues formato scece I O oyd-arrett & P raham Eds Meda Kowledge ad Power Lodo: Croom Helm [87] Wood J T 009 Commucato Our Lves Wadsworth/Cegage Learg [88] Zamfr G 998 Comucarea ş formaţa î sstemele de strure asstată de calculator d domeul ecoomc Iformatca Ecoomcă 7 7 [89] Zs C 007 Coceptos of formato scece Joural of the merca Socety of Iformato Scece ad Techology Receved: November ccepted: November 9 03 Floret Smaradache&Ştefa Vlăduţescu Commucato vs Iformato a omatc Neutrosophc Soluto

47 Neutrosophc Sets ad Systems Vol Novel Image Segmetato lgorthm ased o Neutrosophc Flterg ad Level Set Yahu Guo ad bdulkadr Şegür School of Scece Techology & Egeerg Maagemet St Thomas Uversty 640 NW 37th veue Mam Gardes Florda US E-mal: yguo@stuedu Departmet of Electrc ad Electrocs Egeerg Frat Uversty Elazg Turkey E-mal: ksegur@fratedutr bstract Image segmetato s a mportat step mage processg ad aalyss patter recogto ad mache vso few of algorthms based o level set have bee proposed for mage segmetato the last twety years However these methods are tme cosumg ad sometme fal to etract the correct regos especally for osy mages Recetly eutrosophc set NS theory has bee appled to mage processg for osy mages wth determat formato I ths paper a ovel mage segmetato approach s proposed based o the flter NS ad level set theory t frst the mage s trasformed to NS doma whch s descrbed by three membershp sets T I ad F The a flter s ewly defed ad employed to reduce the determacy of the mage Fally a level set algorthm s used the mage after flterg operato for mage segmetato Epermets have bee coducted usg dfferet mages The results demostrate that the proposed method ca segmet the mages effectvely ad accurately It s especally able to remove the ose effect ad etract the correct regos o both the ose-free mages ad the mages wth dfferet levels of ose Keywords: Image segmetato Neutrosophc set Drectoal alpha-mea flter Level set Itroducto Image segmetato s a essetal process ad s also oe of the most dffcult tasks mage processg feld It s defed as a process dvdg a mage to dfferet regos such that each rego s homogeeous but the uo of ay two adacet regos s ot homogeeous Image segmetato approaches are based o ether dscotuty ad/or homogeety The approaches based o dscotuty ted to partto a mage by detectg solated pots les ad edges accordg to the abrupt chages of the testes The approaches based o homogeety clude thresholdg clusterg rego growg ad rego splttg ad mergg [] Neutrosophy set NS provdes a powerful tool to deal wth the determacy ad the determacy s quattatvely descrbed usg a membershp [] I eutrosophc set a set s descrbed by three subsets: <> <Neut-> ad <t-> whch s terpreted as truth determacy ad falsty set It provdes a ew tool to descrbe the mage wth ucerta formato whch had bee appled to mage processg techques [3] such as mage segmetato thresholdg ad deose I ths paper we proposed a ovel mage segmetato method based o NS theory The mage s mapped to NS doma ad a ew flter drectoal alpha-mea flter s defed NS doma ad used to remove the determace o the mage Fally the mage o NS doma s segmeted usg the method based o level set actve cotour model The remader of ths paper s orgazed as follows The et secto descrbes the eutrsosophc mage the drectoal alpha-mea flter ad the segmetato algorthm tegrated wth level set model Secto three reports the epermets ad the relevat dscusso Cocludg remarks are draw Secto four Proposed method Neutrosophc mage mage mght have a few determate regos such as ose shadow ad boudary It s hard for the classc sets to terpret the determate regos o mages clearly I a eutrosophc set a subset I s amed as determate set ad employed to terpret the determacy the mage eutrosophc mage s descrbed by three membershp sets T I ad F The pel P the mage doma s trasformed to the eutrosophc set doma deoted as P NS ad P NS={T I F } T I ad F are the membershp values belogg to brght pel set determate set ad o-brght pel set respectvely whch are defed as follows [4]: Yahu Guo&bdulkadr Şegür Novel Image Segmetato lgorthm ased o Neutrosophc Flterg ad Level Set

48 Neutrosophc Sets ad Systems Vol gw gwm T = g g wma wm δ δm I = δ δ ma m F = T 3 where gw s the mea value of testy the local eghborhood whose sze s w w δ s the absolute value of the dfferece betwee testy g ad ts local mea value gw The value of I measures the determacy degree of P NS Drectoal α-mea operato I [3] a α-mea operato was defed o a eutrosophc mage ad t removed ose effcetly However t mght blur the mage ad reduce the cotrast whch could reduce the performace of the segmetato To overcome ths drawback a drectoal α-mea operato deoted as DM s ewly proposed to remove the ose effect ad preserve the edges at the same tme The fucto of the drectoal mea flter DMF s defed as [5]: R GTh GTv σ DMF = R GTh GTv > σ 4 R3 GTv GTh > σ where GTh ad GTv are the orm of the gradet at of the subset T at the horzotal ad vertcal drecto respectvely σ s a threshold value ad selected as 00here The drectoal α-mea flter DMF s defed usg the subset T ad I as: T I < α DMF = DMF I α 5 3 Level set Level set method was proposed [6] ad appled for mage segmetato [7] The level set method tracks the evoluto of the boudares betwee dfferet obects whch are embedded as the zero level set The level set actve cotour models ca be dvded to two classes: edge based ad rego based The edge based model tres to fd a curve wth the mamum edge dcator value whch ca mmze the eergy J C [8]: fucto ' J C = C s g Im C s ds 6 where g s a edge dcator fucto C s the boudary ad t ca be represeted mplctly as the zero level set of a true postve fucto φ : Ω R Ω s the doma of mage The evoluto equato of boudary C ca be derved as: φ φ = φ dv g Im v Im t φ 7 φ 0 y = φ0 y Ω where v s a term for creasg the evoluto speed to reach the boudary The rego based model uses the sde/outsde mea values to compose the eergy fucto [9]: F C c c = μ L C μ S C λ Im c S ds λ Im c ds S where c ad c are the mea testes of the regos sde ad outsde the boudary C respectvely L ad S are the legth of C ad the area sde C S ad S are the rego sde ad outsde of C respectvely The assocated level set flow ca be represeted as: φ 9 φ μdv v = δ φ φ t λ Im c λ Im c φ 0 y = φ0 y δ φ φ = 0 φ where δ s the Drac fucto ad deotes the eteror ormal to the boudary Ω dv s the dvergece fucto o the mage Usually edge based approaches ofte suffer from ose especally whe the mage has a low sgal/ose rato; whle the rego based approaches are more adaptve to ose or vashg boudares due to cosderg the etre formato of the regos to buld a eergy fucto 4 Segmetato algorthm based o eutrosophc set ad level set NSLS segmetato algorthm s proposed based o the drectoal α-mea flter ad level set o eutrosophc mage Frstly the mage s trasferred to the NS doma The the DMF s processed the NS mage Fally the boudary of rego s segmeted usg the level set actve cotour algorthm based o the rego model The eergy fucto s defed usg the T subset after DMF processg 8 Yahu Guo&bdulkadr Şegür Novel Image Segmetato lgorthm ased o Neutrosophc Flterg ad Level Set

49 48 Neutrosophc Sets ad Systems Vol 03 = μ μ F C c c L C S C T c S ds λ λ T c ds S 0 3 Epermetal results ad dscusso To test the performace of the proposed method a few of mages ad mages wth dfferet ose levels are employed The NFLS method s compared wth that of the segmetato algorthm based o the tradtoal level set [9] whch s oted as TLS d Segmetato result of TLS at 500 teratos e Segmetato result of TLS at 000 teratos Fgure 3: Comparso result o the osy lver tumour mage wth dfferet teratos a Image wth Gaussa ose varace = 5 b Segmetato result of NSLS at 50 teratos a Orgal mage b Segmetato result of NSLS at 800 teratos c Segmetato result of TLS at d Segmetato result of LS at 50 teratos 00 teratos Fgure 4: Comparso result o the osy three cells mage wth dfferet teratos c Segmetato result of TLS at 800 teratos d Segmetato result of TLS at 500 teratos Fgure : Comparso result o the lver tumour mage wth dfferet teratos a Orgal mage b The segmetato result of NSLS at 50 teratos c The segmetato result of TLS at 50 teratos Fgure : Comparso result o the cells mage wth same teratos epermet s performed to compare the tme cosumpto of NSLS ad LS methods The NSLS takes less tha 33 secods per mage o average for a MD Pheomtm 9500 Quad-core Processor GHz Table compares the computatoal tme o dfferet mages for dfferet algorthms The NSLS takes less terato ad fewer CPU tmes tha the TLS method Image TLS NSLS Iterato CPU tme s Iterato CPU tme s Lver tumour Cells Nosy lver tumour Nosy cells Table : Comparso of the CPU tmes o mages wth ad wthout ose a Image wth Gaussa ose varace = 5 c Segmetato result of NSLS at 000 teratos From the comparsos t ca be see clearly that the NFLS method has better performace o the mage segmetato tha the tradtoal level set method wth hgh segmetato accuracy ad low terato tme O osy mages the NFLS segmets the most obects wth etre shape whle the performaces of the tradtoal method are affected by the ose ad some obects are dvded to several regos The results by the NSLS are smoother ad more coected Furthermore the boudary Yahu Guo&bdulkadr Şegür Novel Image Segmetato lgorthm ased o Neutrosophc Flterg ad Level Set