Neutrosophic Crisp Probability Theory & Decision Making Process

Transcription

1 4 Neutrosophi Crisp Probability Theory & Deision Making Proess Salama Florentin Smarandahe Department of Math and Computer Siene Faulty of Sienes Port Said University Egypt Math & Siene Department University of New Mexio Gallup New Mexio US bstrat Sine the world is full of indeterminay the neutrosophis found their plae into ontemporary researh In neutrosophi set indeterminay is quantified expliitly and truth-membership indeterminay-membership and falsity-membership are independent For that purpose it is natural to adopt the value from the seleted set with highest degree of truth-membership indeterminay membership and least degree of falsity-membership on the deision set These fators indiate that a deision making proess takes plae in neutrosophi environment In this paper we introdue and study the probability of neutrosophi risp sets fter giving the fundamental definitions and operations we obtain several properties and disuss the relationship between them These notions an help researhers and make great use in the future in making algorithms to solve problems and manage between these notions to produe a new appliation or new algorithm of solving deision support problems Possible appliations to mathematial omputer sienes are touhed upon Keyword Neutrosophi set Neutrosophi probability Neutrosophi risp set Intuitionisti neutrosophi set Introdution Neutrosophy has laid the foundation for a whole family of new mathematial theories generalizing both their lassial and fuzzy ounterparts [ ] suh as the neutrosophi set theory The fundamental onepts of neutrosophi set introdued by Smarandahe in [ ] and Salama et al in [ ] provides a natural foundation for treating mathematially the neutrosophi phenomena whih pervasively exist in our real world and for building new branhes of neutrosophi mathematis Critial Review Volume XII 0

2 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 5 In this paper we introdue and study the probability of neutrosophi risp sets fter giving the fundamental definitions and operations we obtain several properties and disuss the relationship between neutrosophi risp sets and others Terminology We reollet some relevant basi preliminaries and in partiular the work of Smarandahe in [ ] and Salama et al [ ] Smarandahe introdued the neutrosophi omponents T I F whih represent the membership indeterminay and non-membership values respetively whih are inluded into the nonstandard unit interval Example [7 9] Let us onsider a neutrosophi set a olletion of possible loations (positions) of partile x and let and B two neutrosophi sets One an say by language abuse that any partile x neutrosophially belongs to any set due to the perentages of truth/indeterminay/falsity involved whih varies between 0 and For example: x (05 0 0) belongs to (whih means a probability of 50% that the partile x is in a probability of 0% that x is not in and the rest is undeidable); or y (0 0 ) belongs to (whih normally means y is not for sure in ); or z (0 0) belongs to (whih means one does know absolutely nothing about z affiliation with ) More general x((0-0) (04 045) [050-05{00408}) belongs to the set whih means: with a probability in between 0-0% the partile x is in a position of (one annot find an exat approximation beause of various soures used); with a probability of 0% or 4% or 8% x is not in ; the indeterminay related to the appurtenane of x to is in between 40-45% or between 50-5% (limits inluded) The subsets representing the appurtenane indeterminay and falsity may overlap and in this ase n-sup = 0% + 5% + 8% > 00 Definition [4 5 ] neutrosophi risp set (NCS for short) an be identified to an ordered triple whih are subsets on X and every risp set in X is obviously a NCS having the form Critial Review Volume XII 0

3 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess Definition [] The objet having the form is alled ) Neutrosophi Crisp Set with Type I if it satisfies and (NCS-Type I for short) ) Neutrosophi Crisp Set with Type II if it satisfies and and X (NCS-Type II for short) ) Neutrosophi Crisp Set with Type III if it satisfies and X (NCS-Type III for short) Definition Neutrosophi Set [7]: Let X be a non-empty fixed set neutrosophi set (NS for short) is an objet having the form ( x) ( x) ( x) where ) and the degree of non- x x and x x) the degree of indeterminay (namely x membership (namely x where and represent the degree of membership funtion (namely ) respetively of eah element x X to the set 0 ( x) ( x) ( x) 0 ( x) ( x) ( x) Neutrosophi Intuitionisti Set of Type [8]: Let X be a non-empty fixed set neutrosophi intuitionisti set of type (NIS for short) set is an objet having the form ( x) ( x) ( x) x x x whih represent the degree of membership funtion (namely indeterminay (namely x where and x x ) the degree of ) and the degree of non-membership (namely ) respetively of eah element x X to the set where 0 ( x) ( x) ( x) and the funtions satisfy the ondition and x x x ( x) ( x) ( x) Neutrosophi Intuitionisti Set of Type [4]: Let X be a non-empty fixed set neutrosophi intuitionisti set of type (NIS for short) is an objet having the form ( x) ( x) ( x) x x x whih represent the degree of membership funtion (namely where and x ) the degree of Critial Review Volume XII 0

4 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 7 indeterminay (namely x x ) and the degree of non-membership (namely ) respetively of eah element x X to the set where 05 ( x) ( x) ( x) and the funtions satisfy the ondition and x x 05 ( x) x 05 ( x) x 05 0 x ( x) ( x) ( ) neutrosophi risp with three types the objet an be identified to an ordered triple whih are subsets on X and every risp set in X is obviously a NCS having the form Every neutrosophi set ( x) ( x) ( x) on X is obviously a NS having the form ( x) ( x) ( x) Salama et al in [4 5 ] onstruted the tools for developed neutrosophi risp set and introdued the NCS X in X Remark N The neutrosophi intuitionisti set is a neutrosophi set but the neutrosophi set is not a neutrosophi intuitionisti set in general Neutrosophi risp sets with three types are neutrosophi risp set N The Probability of Neutrosophi Crisp Sets If an experiment produes indeterminay that is alled a neutrosophi experiment Colleting all results inluding the indeterminay we get the neutrosophi sample spae (or the neutrosophi probability spae) of the experiment The neutrosophi power set of the neutrosophi sample spae is formed by all different olletions (that may or may not inlude the indeterminay) of possible results These olletions are alled neutrosophi events In lassial experimental the probability is number of times event ours total number of trials Similarly Smarandahe in [ 7 8] introdued the Neutrosophi Experimental Probability whih is: Critial Review Volume XII 0

5 8 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess number of times total number of trials event ours number Critial Review Volume XII 0 of times indeterminay total number of trials ours number of times event does total number of trials not our Probability of NCS is a generalization of the lassial probability in whih the hane that an event to our is: ) true ) indetermin ate ) false on a sample spae X or NP ) ) ) ) ( subspae of the universal set endowed with a neutrosophi probability defined for eah of its subsets forms a probability neutrosophi risp spae Definition Let X be a non- empty set and be any type of neutrosophi risp set on a spae X then the neutrosophi probability is a mapping NP : X 0 NP ) ) ) ) that is the probability of a neutrosophi risp set ( that has the property that Remark (ppp ) 0 In ase if ) where if is NCS then 0 ( ) ) p p p P ) In ase if is NCS-Type I then 0 P ( ) ) ) p 0 o The Probability of NCS-Type II is a neutrosophi risp set where 0 ( ) ) P ) 4 The Probability of NCS-Type III is a neutrosophi risp set where 0 ( ) ) P ) Probability xioms of NCS xioms The Probability of neutrosophi risp set and NCS-Type III on X NP ) ) ) ) where P ) 0 ) 0 ) 0 or ( (ppp ) 0 ) where ( if p p p 0 p o The probability of neutrosophi risp set and NCS-Type IIIs on X NP ) ) ) ) where ( 0 ( ) p( ) p( p )

6 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 9 Bounding the probability of neutrosophi risp set and NCS-Type III NP ) ) ) ) where P ( ) 0 ) 0 ) 0 ( 4 ddition law for any two neutrosophi risp sets or NCS-Type III if B) ( ) B ) B ) ) B ) ) P ( ) B ) ) ( B B then NP B) ) N ( N B) ) B ) ) B ( B ) ) B ) N N ) N Sine our main purpose is to onstrut the tools for developing probability of neutrosophi risp sets we must introdue the following Probability of neutrosophi risp empty set with three types ( N ) short) may be defined as four types: Type : Type : Type : Type 4: ) N ) ) ) X ) 00 ; N ) ) X ) X ) 0 ; N ) ) ) ) 00 0 ; N ) ) X ) ) 0 0 Probability of neutrosophi risp universal and NCS-Type III universal sets ( NP X ) for short) may be defined as four types ( N Type : Type : X N ) X ) ) ) 0 0 ; X N ) X ) X ) ) 0 ; Type : X N ) X ) X ) X ) ; Type 4: X N ) X ) ) X ) 0 for ) Remark X N ) N N ) ON where N O N are in Definition [] or equals any type for N Definition (Monotoniity) Let X be a non-empty set and NCSS and B in the form B B B with B NP ) ) ) ) NP B) B ) B ) ) ( ( B Critial Review Volume XII 0

7 40 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess then we may onsider two possible definitions for subsets ( B ) Type: or Type: NP ) B) ) B ) ) B ) and ) ) ( B NP ) B) ) B ) ) B ) and ) ) ( B Definition Let X be a non-empty set and NCSs and B in the form B B B be NCSs B Then B) may be defined two types as Type: B) B ) B ) B ) or Type: NP B) B ) B ) ) ( B B) may be defined two types as: Type: or Type : NP B) B ) B ) ) ( B NP B) B ) B ) ) ( B ) may be defined by three types: Type: ( NP ) ) ) ) = )( )( ) or Type: ) ) ) ) or Type: NP ) ) ) ) ( ( Critial Review Volume XII 0

8 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 4 Proposition be NCSs on a non- Let and B in the form B B B B empty set X Then Proposition ) ) ( or ) or = any type of N X N B) ( ) B )( ) B ) P ( ) ) ( B ) ) ) NP ( B) B ) B ) B ) N Let and B in the form B B B B are NCSs on a nonempty set X and p Then p N are NCSs p) ; n( X ) n( X ) n( X ) p N ) 0 n( X ) n( X ) Example Let X a d by a b d B a p a d and B are two neutrosophi risp events on X defined then see that NP ( ) NP ( B) p) one an ompute all probabilities from definitions If b and d Then B are neutrosophi risp sets on X B and B) N B d and B) N Example Let X { d e f } { d}{ e}{ f } D { b}{ e }{ f d} be a NCS-Type Critial Review Volume XII 0

9 4 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess B { }{ d}{ e} be a NCT-Type I but not NCS-Type II III C { b}{ d}{ e f a} be a NCS-Type III but not NCS-Type I II E { d e}{ d}{ e f a} F { d e} { e f d b} We an ompute the probabilities for NCSs by the following: NP ( ) NP ( D) NP ( B) NP ( C) NP ( E) 4 4 Remark 5 F) 0 The probabilities of a neutrosophi risp set are neutrosophi sets Example Let X { d} { b}{ }{ d} B { a}{ }{ d b} are NCS-Type I on X and U { b}{ d}{ } U { }{ }{ } are NCS-Type III on X; then d d we an find the following operations Union intersetion omplement differene and its probabilities a) Type: B { a}{ }{ d b} NP ( B) } and Type : B { a}{ }{ d b} NP ( B) } B) may be equals Type: NP ( B) Type : NP ( B) Type : NP ( B) b) Type : B { b}{ }{ d} B) } and Type : B { a b}{ }{ d} NP ( B) } Critial Review Volume XII 0

10 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 4 ) Type: { d} { d}{ } NCS-Type III set on X NP ( ) Type: { d} { d}{ b} NCS-Type III on X NP ( ) Type: { d} { }{ b} NCS-Type III on X NP ( ) d) Type : B { d}{ d}{ } be NCS-Type III on X B ) Type : Type : B { d}{ }{ a} NCS-Type I on X and B ) B { d}{ d}{ a} NCS-Type III on X and B ) e) Type : U U { }{ d}{ } NCS-Type III NP ( U U ) d { Type : U U { }{ }{ } d NP U U f) Type: U U { b}{ d}{ } NCS-Type III NP ( U U ) d Type: U U { b}{ }{ } NCS-Type III and NP ( U U ) d b ( ) { g) Type : U { d}{ b}{ } NCS-Type III and NP ( U ) Type : U { d}{ d}{ } NP ( U ) b Type: U { d}{ b}{ } b NP ( U ) NCS-Type III and NCS-Type III and h) Type: U { d}{ d}{ } NCS-Type III and NP ( U ) Type: U { d}{ }{ } NCS-Type III and NP ( U ) Type: U { d}{ d}{ } NCS-Type III NP ( U ) Critial Review Volume XII 0

11 44 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess Probabilities for events NP ( ) NP ( B) NP ( U) NP ( U ) NP ( U ) NP ( U ) e) ( B) { b d}{ d}{ } be a NCS-Type III NP ( B) be a neutrosophi set f) NP ( ) B) NP ( ) B) g) B) ) B) B) } h) NP ( ) NP ( ) NP ( B) NP ( B ) Probabilities for Produts The produt of two events given by B {( a)( a)}{( )}{( d d)( d b)} Remark 4 and NP ( B) B {( a)( b)}{( )}{( d d)( d)} and NP ( B ) U {( a)( a)( b)( b)}{( )( d)}{( d d)( d )} a and NP ( U 4 ) U U {( a)( a)( b)( b)( )( )}{( )( d )}{( d d)( d)} and NP ( U U ) The following diagram represents the relation between neutrosophi risp onepts and neutrosphi sets: Probability of Neutrosophi Crisp Sets Generalized Neutrosophi Set Intuitionisti Neutrosophi Set Neutrosophi Set Critial Review Volume XII 0

12 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 45 Referenes: [] K tanassov Intuitionisti fuzzy sets in V Sgurev ed ITKRS Session Sofi June 98 Central Si and Tehn Library Bulg ademy of Sienes 984 [] K tanassov Intuitionisti fuzzy sets Fuzzy Sets and Systems [] K tanassov Review and new result on intuitionisti fuzzy sets preprint IM-MFIS--88 Sofi 988 [4] Salama Basi Struture of Some Classes of Neutrosophi Crisp Nearly Open Sets and Possible ppliation to GIS Topology Neutrosophi Sets and Systems 05 Vol 7 pp8- [5] Salam Mohamed Eis S Elhafeez and M M Lotfy Review of Reommender Systems lgorithms Utilized in Soial Networks based e- Learning Systems & Neutrosophi System Neutrosophi Sets and Systems 05 Vol 8 pp 5-44 [] Salama and Said Broumi Roughness of Neutrosophi Sets Elixir ppl Math pp -7 [7] Salam Mohamed bdelfattah and Mohamed Eisa Distanes Hesitany Degree and Flexible Querying via Neutrosophi Sets International Journal of Computer ppliations Volume 0 No 0 04 pp [8] M M Lofty Salam H El-Ghareeb and M Eldosuky Subjet Reommendation Using Ontology for Computer Siene CM Curriul International Journal of Information Siene and Intelligent System Vol 04 pp [9] Salam Haithem El-Gharee yman M Maine and Florentin Smarandahe Introdution to Develop Some Software Programs for dealing with Neutrosophi Sets Neutrosophi Sets and Systems 04Vol 4 pp 5-5 [0] Salam F Smarandahe and M Eisa Introdution to Image Proessing via Neutrosophi Tehnique Neutrosophi Sets and Systems 04 Vol 5 pp 59- [] S lblowi Salama and Mohmed Eisa New Conepts of Neutrosophi Sets International Journal of Mathematis and Computer ppliations Researh (IJMCR) Vol 4 Issue 04 pp 59- [] Salam Mohamed Eisa and M M bdelmoghny Neutrosophi Relations Database International Journal of Information Siene and Intelligent System () 04 pp -4 [] Salam Haitham El-Gharee yman M Manie and M M Lotfy Utilizing Neutrosophi Set in Soial Network nalysis e-learning Systems Critial Review Volume XII 0

13 4 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess International Journal of Information Siene and Intelligent System () 04 pp -7 [4] I M Hanafy Salama and K Mahfouz Correlation of Neutrosophi Dat International Refereed Journal of Engineering and Siene (IRJES) Vol Issue 0 pp 9- [5] I M Hanafy Salama and K M Mahfouz Neutrosophi Classial Events and Its Probability International Journal of Mathematis and Computer ppliations Researh (IJMCR) Vol Issue Marh 0 pp 7-78 [] Salama and S lblowi Generalized Neutrosophi Set and Generalized Neutrosophi Topologial Spaes Journal Computer Si Engineering Vol No 7 0 pp 9- [7] Salama and S lblowi Neutrosophi Set and Neutrosophi Topologial Spaes ISOR J Mathematis Vol Issue 0 pp -5 [8] Salama Neutrosophi Crisp Point & Neutrosophi Crisp Ideals Neutrosophi Sets and Systems Vol No 0 pp [9] Salama and F Smarandahe Filters via Neutrosophi Crisp Sets Neutrosophi Sets and Systems Vol No 0 pp 4-8 [0] Salam and H Elagamy Neutrosophi Filters International Journal of Computer Siene Engineering and Information Tehnology Reseearh (IJCSEITR) Vol Issue 0 pp 07- [] Salam F Smarandahe and Valeri Kroumov Neutrosophi risp Sets & Neutrosophi Crisp Topologial Spaes Neutrosophi Sets and Systems Vol pp [] Salam Mohamed Eisa and M M bdelmoghny Neutrosophi Relations Database International Journal of Information Siene and Intelligent System () 04 [] Salam Florentin Smarandahe and S lblowi New Neutrosophi Crisp Topologial Conepts Neutrosophi Sets and Systems (epted 04) [4] Salam Said Broumi and Florentin Smarandahe Neutrosophi Crisp Open Set and Neutrosophi Crisp Continuity via Neutrosophi Crisp Ideals IJ Information Engineering and Eletroni Business 04 Vol pp-8 [5] Salam Said Broumi and Florentin Smarandahe Some Types of Neutrosophi Crisp Sets and Neutrosophi Crisp Relations IJ Information Engineering and Eletroni Business 04 [] Salam Haithem El-Gharee yman M Maine and Florentin Smarandahe Introdution to Develop Some Software Programes for dealing with Neutrosophi Sets Neutrosophi Sets and Systems 04Vol pp 5-5 Critial Review Volume XII 0

14 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess 47 [7] Salam Florentin Smarandahe and S lblowi The Charateristi Funtion of a Neutrosophi Set Neutrosophi Sets and Systems 04 Vol pp 4-8 [8] Salam Mohamed bdelfattah and Mohamed Eisa Distanes Hesitany Degree and Flexible Querying via Neutrosophi Sets International Journal of Computer ppliations Issue 4 Vol May 04 [9] Salam F Smarandahe and Valeri Kroumov Neutrosophi Closed Set and Continuous Funtions Neutrosophi Sets and Systems 04 [0] Salam Said Broumi and Florentin Smarandahe Some Types of Neutrosophi Crisp Sets and Neutrosophi Crisp Relations I J Information Engineering and Eletroni Business 04 [] Salam Florentin Smarandahe Neutrosophi Ideal Theory Neutrosophi Loal Funtion and Generated Neutrosophi Topology In Neutrosophi Theory and Its ppliations Colleted Papers Volume EuropaNov Bruxelles 04 pp -8 [] M E bd El-Monsef Nasef Salama Extensions of fuzzy ideals Bull Calutta Math So 9 No pp [] ME bd El-Monsef Nasef Salama Some fuzzy topologial operators via fuzzy ideals Chaos Solitons Fratals Vol No pp [4] M E bd El-Monsef Nasef Salama Fuzzy L-open sets and fuzzy L-ontinuous funtions nalele Universitatii de Vest din Timisoar Seria Matematia-Informati Vol 4 No pp - 00 [5] I M Hanafy and Salama unified framework inluding types of fuzzy ompatness Conferene Topology and nalysis in ppliations Durban - July 004 Shool of Mathematial Sienes UKZN [] Salama Fuzzy Hausdorff spaes and fuzzy irresolute funtions via fuzzy ideals V Italian-Spanish Conferene on General Topology and its ppliations June lmeri Spain [7] ME bdel Monsef Kozae Salama and H Elagamy Fuzzy Ideals and Bigranule Computing 0th Conferene of Topology and its ppliations 007 Port Said Univ Egypt [8] Salama Intuitionisti Fuzzy Ideals Theory and Intuitionisti Fuzzy Loal Funtions CTC 08 the 4th Biennial Computational Tehniques and ppliations Conferene th July 008 ustralian National University Canberr CT ustralia [9] Salama Fuzzy Bitopologial Spaes Via Fuzzy Ideals Blast 008 ugust University of Denver Denver CO US [40] Salama New Form of Fuzzy Compat spaes and Related Topis via Fuzzy Idealization Journal of fuzzy System and Mathematis Vol 4 No 00 pp -9 Critial Review Volume XII 0

15 48 Salam Florentin Smarandahe Neutrosophi Crisp Probability Theory & Deision Making Proess [4] Salama and Hassan On Fuzzy Regression Model The Egyptian Journal for ommerial Studies Volume 4 No 4 pp [4] Salama and S lblowi Neutrosophi Set Theory and Neutrosophi Topologial Ideal Spaes The First International Conferene on Mathematis and Statistis ICMS 0 [4] Salama New Form of Fuzzy Hausdorff Spae and Related Topis via Fuzzy Idealization IOSR Journal of Mathematis (IOSR-JM ) Volume Issue 5 (Sep-Ot 0) pp -4 [44] Salama and F Smarandahe Neutrosophi Crisp Set Theory Eduation Publishing 05 [45] M E bd El-Monsef M Kozae Salam H Elagamy Fuzzy Biotopolgial Ideals Theory IOSR Journal of Computer Engineering (IOSRJCE) Vol Issue 4 0 pp -5 [4] I M Hanafy Salama M bdelfattah and Y Wazery Seurity in Mant Based on Pki using Fuzzy Funtion IOSR Journal of Computer Engineering Vol Issue 0 pp 54-0 [47] M E bd El-Monsef Kozae Salam and H M Elagamy Fuzzy Pairwise L-Open Sets and Fuzzy Pairwise L-Continuous Funtions International Journal of Theoretial and Mathematial Physis Vol No Marh 0 pp 9-7 [48] Florentin Smarandahe Neutrosophy and Neutrosophi Logi First International Conferene on Neutrosophy Neutrosophi Logi Set Probability and Statistis University of New Mexio Gallup NM 870 US 00 [49] Florentin Smarandahe Unifying Field in Logis: Neutrosophi Logi Neutrosophy Neutrosophi risp Set Neutrosophi Probability merian Researh Press Rehoboth NM 999 [50] Florentin Smarandahe Neutrosophi set a generalization of the intuituionistis fuzzy sets Inter J Pure ppl Math pp [5] Florentin Smarandahe Introdution To Neutrosophi Measure Neutrosophi Integral and Neutrosophi Probability 05 unmedu/ ebooks-otherformatshtm [5] M Bhowmik and M Pal Intuitionisti Neutrosophi Set Relations and Some of its Properties Journal of Information and Computing Siene 5() pp [5] L Zadeh Fuzzy Sets Inform and Control 8 pp [54] F Smarandahe Operators on Single-Valued Neutrosophi Oversets Neutrosophi Undersets and Neutrosophi Offsets Journal of Mathematis and Informatis Vol 5-7 0; Critial Review Volume XII 0