Correlation of Neutrosophic Sets in Probability Spaces

Transcription

1 JMSI No orrelato of Neutrosophc Sets Probablty Spaces I.M. HNFY.. SLM O. M. KHLED ND K. M. MHFOUZ bstract I ths paper we troduce ad study the cocepts of correlato ad correlato coeffcet of eutrosophc data probablty spaces. I the case of fte spaces these results gve the results due to Salaa et al. [ ] ad study soe of ther propertes. Keywords: orrelato oeffcet Fuzzy Sets Neutrosophc Sets Itutostc Fuzzy Sets. 1. INTRODUTION I 1965 [6] Zadeh frst troduced the cocept of fuzzy sets. Fuzzy set s very uch useful ad ths oe real value [01 ] s used to represet the grade of ebershp of a fuzzy set defed o the crsp set. fter two decades taassov [19 0 1] troduced aother type of fuzzy sets that s called tutostc fuzzy set IFS whch s ore practcal real lfe stuatos. Itutostc fuzzy sets hadle coplete forato.e. the grade of ebershp fucto ad o-ebershp fucto but ot the deterate forato ad cosstet forato whch ests obvously belef syste. Sce the world s full of deteracy the eutrosophcs foud ther place to coteporary research. Saradache [ ] troduced aother cocept of precse sets called eutrosophc sets. Salaa et al. [ ] troduced ad studed the operatos o eutrosophc sets. I statstcal aalyss the correlato coeffcet plays a portat role easurg the stregth of the lear relatoshp betwee two varables. s the correlato coeffcets defed o crsp sets have bee uch dscussed t s also very coo the theory of fuzzy sets to fd the correlato betwee fuzzy sets whch accouts for the relatoshp betwee the fuzzy sets. I ths paper we dscuss ad derved a forula for correlato coeffcet defed o the doa of eutrosophc sets probablty spaces.. TERMINOLOGIES We recollect soe relevat basc prelares ad partcular the work of Saradache [ ] ad Salaa et al. [ ] /jas Uversty of SS. yrl ad Methodus Dowload Trava Date 7/19/14 :1 M

2 46 I.M. Haafy.. Salaa O. M. Khaled ad K. M. Mahfouz Saradache troduced the tutostc copoets T I ad F whch represet the - ebershp deteracy ad o-ebershp values respectvely 0 1 s o- stadard ut terval. DEFINITION.1 [6]. Let be a fed set. fuzzy set of s a object havg the for ebershp of the eleet the fucto : [01 ] defe the degree of to the set whch s a subset of. DEFINITION. [19] [0] [1]. Let be a fed set. tutostc fuzzy set of s a object havg the for: Where the fucto: : [01 ] ad : [01 ] defe respectvely the degree of ebershp ad degree of o-ebershp of the eleet to the set whch s a subset of ad for every 0 1. DEFINITION.3 [1 ]. Let be a o-epty fed set. eutrosophc set NS s a object havg the for: ad represet the degree of ebershp fucto the degree of deteracy ad the degree of o ebershp fucto respectvely of each eleet to the set. I 1991 Gerstekor ad Mako [4] defed the correlato of tutostc fuzzy sets ad a fte set } as follows: GM { ad the correlato coeffcet of fuzzy ubers was gve by: GM GM. T T T..3 1 Yu [9] defed the correlato of ad the collecto F[ab] of all fuzzy ubers whose supports are cluded a closed terval [ab] as follows: b 1 Y b a.4 a 1 ad the correlato coeffcet of fuzzy ubers was defed by Dowload Date 7/19/14 :1 M

3 JMSI No Y Y Y Y..5 I 1995 Hog ad Hwag [10] 5] defed the correlato of tutostc fuzzy sets ad a probablty space P as follows: dp HH.6 ad the correlato coeffcet of tutostc fuzzy ubers was gve by HH HH HH HH..7 Hug ad Wu [5] troduce the cocept of postvely ad egatvely correlated ad used the cocept of cetrod to defe the correlato coeffcet of tutostc fuzzy sets whch les the terval [-11] ad the correlato coeffcet of tutostc fuzzy sets ad was gve by:.8.9 Haafy ad Salaa [ ] defed the correlato of eutrosophc data a fte set } { 1 as follows: 1 HS.10 ad the correlato coeffcet of fuzzy ubers was gve by: T T HS HS.11 1 T.1 Dowload Date 7/19/14 :1 M

4 48 I.M. Haafy.. Salaa O. M. Khaled ad K. M. Mahfouz T ORRELTION OEFFIIENT OF NEUTROSOPHI SETS Let P be a probablty space ad be a eutrosophc set a probablty space : [01 ] are respectvely orel easurable fuctos satsfyg 0 1 s o-stadard ut terval [8]. 0 3 DEFINITION 3.1. For a eutrosophc sets we defe the correlato of eutrosophc sets ad as follows: dp. 3.1 Wher P s the probablty easure over. Furtherore we defe the correlato coeffcet of eutrosophc sets ad as follows: 3. T T 1 T dp ad T dp. The followg proposto s edate fro the deftos. PROPOSITION 3.1. For eutrosophc sets ad we have... If = the 1. The followg theore geeralzes both Theore 1 [3] Proposto.3 [8] ad Theore 1 [[10]] of whch the proof s rearkably sple. THEOREM 3.1. For eutrosophc sets ad we have PROOF. The equalty 0 s evdet sce 0 ad T T 0. Thus we eed oly to show that 1 or Dowload Date 7/19/14 :1 M

5 JMSI No T T 1 For a arbtrary real uber k we have 0 k dp k dp k dp dp k dp k dp. Thus by auchy Schwarz [] we ca get: dp dp dp. T T T T 1 Therefore we have 1. THEOREM 3.. 1ff = c for soe c IR. PROOF. osderg the equalty the proof of Theore 3.1 the the equalty holds f ad oly f P{ c } P{ c } P{ c } 1 for soe c IR. whch copletes the proof. THEOREM ff ad are o-fuzzy sets ad they satsfy the codto: 1 or 1 or 1. PROOF. Suppose that 0 the 0. Sce 0 the 0 ples P 0 1 whch eas that P 0 1 P 0 1ad P 0 1. If 1 the we ca get 0 ad 0. t the sae te f 1 the we ca get 0 ad 0 hece we have 1. oversely f ad are o-fuzzy sets ad 1. If 1 the we ca 0 ad 0. O the other had f 1 the we ca Dowload Date 7/19/14 :1 M

6 50 I.M. Haafy.. Salaa O. M. Khaled ad K. M. Mahfouz have 0 ad 0 whch ples 0. Slarly we ca gve the proof whe 1 or 1. THEOREM 3.4. If s a o-fuzzy set the T =1. THE PROOF s obvous. EMPLE. For a cotuous uversal set =[1] f two eutrosophc sets are wrtte respectvely. { [1]} { [1]} / Thus we have T ad T The we get It shows that eutrosophc sets ad have a good postvely correlated. ONLUSION Our a goal of ths work s propose a ethod to calculate the correlato coeffcet of eutrosophc sets whch les [0 1] gve us forato for the degree of the relatoshp betwee the eutrosophc sets. Further we dscuss soe of ther propertes ad gve eaple to llustrate our proposed ethod reasoable. Thus we geeralze soe coclusos lterature REFERENES [1].. Salaa ad S.. lblow 01 "Geeralzed eutrosophc set ad geeralzed eutrosophc topologcal spaces" oputer Scece ad Egeerg 7. [].. Salaa ad S.. lblow 01 "Neutrosophc set ad eutrosophc topologcal spaces "IOSR Joural of Matheatcs IOSR-JM [3].. Salaa Mohaed Esa ad M. M. bdeloghy 014 Neutrosophc Relatos Database Iteratoal Joural of Iforato Sceces ad tellget Systes Vol.4 No.. pp [4].. Salaa ad F. Saradache 013 Flters va Neutrosophc rsp Sets Neutrosophc Sets ad Systes Vol.1 No pp [5].. Salaa ad H. Elagay 013 Neutrosophc Flters Iteratoal Joural of oputer Scece Egeerg ad Iforato Techology Reseearch IJSEITR Vol.3 Issue1 Mar 013 pp [6].. Salaa F. Saradache ad Valer Krouov 014 Neutrosophc crsp Sets & Neutrosophc rsp Topologcal Spaces Neutrosophc Sets ad Systes Vol. pp Dowload Date 7/19/14 :1 M

7 JMSI No [7].. Salaa Sad rou ad Floret Saradache 014 Neutrosophc rsp Ope Set ad Neutrosophc rsp otuty va Neutrosophc rsp Ideals I.J. Iforato Egeerg ad Electroc usess Publshed Ole October 014 MES. [8]. Robso 1960 "No-Stadard aalyss" Prceto Uversty Press Prceto New Jersey. [9]. Yu 1993 "orrelato of fuzzy ubers". Fuzzy Sets ad Systes 55: [10] D. H. Hog S.Y. Hwag 1995 "orrelato of Itutostc fuzzy sets probablty spaces" Fuzzy Sets ad Systes 75: [11] F. Saradach 00 " ew brach of phlosophy ultple valued logc" Iteratoal Joural 83: [1] F. Saradach 003 "Neutrosophc set a geeralzato of the tutostc fuzzy sets" Proceedgs of the thrd coferece of the Europea Socety for fuzzy logc ad Techolgye EUSFLT Septaper Zttau Geaay; Uv. of ppled Sceces at Zttau Goerlt [13] F. Saradach 013 INTRODUTION TONEUTROSOPHI MESURE NEUTROSOPHI INTEGRL ND NEUTROSOPHI PROILITY EN: [14] F. Saradache1999 Ufyg Feld Logcs: Neutrosophc Logc. Neutrosophy Neutrosophc Set Neutrosophc Probablty. erca Research Press Rehoboth NM [15] F. Saradache 005 Neutrosophc set a geeralzato of the tutostcs fuzzy sets Iter. J. Pure ppl. Math [16] I. M. Haafy.. Salaa ad K. M. Mahfouz01 "orrelato of eutrosophc data" Iteratoal Refereed Joural of Egeerg ad Scece 1: [17] I. M. Haafy.. Salaa ad K. M. Mahfouz 013" Neutrosophc lasscal Evets ad Its Probablty" Iteratoal Joural of Matheatcs ad oputer pplcatos ResearchIJMRVol.3Issue 1 pp [18] I. M. Haafy.. Salaa ad K. Mahfouz 013 "orrelato oeffcet of Neutrosophc Sets by etrod Method" Iteratoals Joural of Iteratoal Joural of Probablty ad Statstcs 1 pp 9-1. [19] K. T. taassov 1983 "Itutostc fuzzy sets". etral Tech. Lbrary ulgara cadey Scece Sofa ulgara Rep. No. 1697/84. [0] K. T. taassov 1983 "Itutostc fuzzy sets". Proc. of Polsh Syp. O Iterval ad Fuzzy atheatcs Poza 3-6. [1] K. T. taassov 1986 "Itutostc fuzzy sets". Fuzzy Sets ad Systes 0: [] M. Masjed-Jae S. S. Dragor H.M. Srvastava 008 "Soe geeralzatos of the auchy-schwarz ad the auchy-uyakovsky equaltes volvg four free paraeters ad ther applcatos" RGMI Res. Rep. oll rtcle 3 pp.1 1 electroc. [3] S. S. Dragor 004 "Dscrete equaltes of the auchy-uyakovsky-schwarz type" Nova Scece Publshers Ic. Hauppauge NY. [4] T. Gerstekor J. Mako 1991 "orrelato of tutostc fuzz sets " Fuzzy Sets ad Systes 44: [5] W. L. Hug ad J. W. Wu 00 "orrelato of tutostc fuzzy sets by cetrod ethod". Iforato Sceces 144: [6] L.. Zadeh 1965 "Fuzzy sets". Iforato ad otrol 8: I.M. Haafy Departet of Matheatcs ad oputer Scece Faculty of Scece Port Sad Uversty Egypt.. Salaa Departet of Matheatcs ad oputer Scece Faculty of Scece Port Sad Uversty Egypt O. M. Khaled Departet of Matheatcs ad oputer Scece Faculty of Scece Port Sad Uversty Egypt Dowload Date 7/19/14 :1 M

8 5 I.M. Haafy.. Salaa O. M. Khaled ad K. M. Mahfouz K. M. Mahfouz Departet of Matheatcs ad oputer Scece Faculty of Scece Port Sad Uversty Egypt Dowload Date 7/19/14 :1 M