Generalized Near Rough Probability. in Topological Spaces

Transcription

1 It J Cotemp Math Scieces, Vol 6, 20, o 23, Geealized Nea Rough Pobability i Topological Spaces M E Abd El-Mosef a, A M ozae a ad R A Abu-Gdaii b a Depatmet of Mathematics, Faculty of Sciece Tata Uivesity, Egypt b Depatmet of Mathematics, Faculty of Sciece Jeash Uivesity, Joda Rabu_gdaii@yahoocom Abstact Rough set theoy has bee itoduced by Pawla [3] It is cosideed as a base fo all eseaches i the ough set aea itoduced afte this date Most of these eseaches cocetated o developig esults ad techiques based o Pawla's esults [3] I this pape we shall itoduce geealized ough pobability fom topological view The basic cocepts of some geealized ea ope, geealized ea ough, ad geealized ea exact sets ae itoduced ad sufficietly illustated Moeove, poved esults, examples ad coute examples ae povided The topological stuctue which suggested i this pape opes up the way fo applyig ich amout of topological facts ad methods i the pocess of gaula computig eywods: Topological space; Geealized ea ough set, Geealized ea ough pobability Itoductio Oe of the most poweful otios i system aalysis is the cocept of topological stuctues [6] ad thei geealizatios Rough set theoy, itoduced by Pawla i 982 [3], is a mathematical tool that suppots also the ucetaity easoig but qualitatively I this pape, we shall itegate some ideas i tems of cocepts i topology Topology is a bach of mathematics, whose cocepts exist ot oly i almost all baches of mathematics, but also i may eal life applicatios We believe that topological stuctue will be a impotat base fo modificatio of owledge extactio ad pocessig

2 00 M E Abd El-Mosef, A M ozae ad R A Abu-Gdaii 2 Pelimiaies A topological space [6] is a pai (, ) X cosistig of a set X ad family of subsets of X satisfyig the followig coditios: (T) φ ad X (T2) is closed ude abitay uio (T3) is closed ude fiite itesectio Thoughout this pape ( X, ) deotes a topological space, the elemets of X ae called poits of the space, the subsets of X belogig to ae called ope sets i the space, the complemet of the subsets of X belogig to ae called closed sets i the space, ad the family of all ope sets of ( X, ) is deoted by ad the family of all closed sets of ( X, ) is deoted by C( X ) Fo a subset A of a space ( X, ), Cl ( A ) deote the closue of A ad is give by Cl ( A ) = { F X : A F ad F C ( X )} Evidetly, Cl ( A ) is the smallest closed subset of X which cotais A Note that A is closed iff A = Cl( A) It ( A ) deote the iteio of A ad is give by It( A) = { G X : G A ad G } Evidetly, It ( A ) is the lagest ope subset of X which cotaied i A Note that A is ope iff A = It( A) We shall ecall some cocepts about some ea ope sets which ae essetial fo ou peset study Defiitio 2 A subset A of a space ( X, ) is called: i) Semi-ope [8] ( biefly s ope ) if A Cl( It( A)) ii) Pe-ope [0] ( biefly p ope ) if A It( Cl( A)) iii) α -ope [] if A It( Cl( It( A))) iv) β -ope [] ( = semi-pe-ope [ 2] ) if A Cl( It( Cl( A))) The complemet of a s ope ( esp p ope, α -ope ad β -ope ) set is called s closed ( esp p closed, α -closed ad β -closed ) set The family of all s ope ( esp p ope, α -ope ad β -ope) sets of ( X, ) is deoted by SO (X ) ( esp PO (X ), α O(X ) ad β O(X ) ) The family of all s closed ( esp p closed, α - closed ad β -closed ) sets of ( X, ) is deoted by SC (X ) (esp PC (X ), α C(X ) ad β C(X ) ) The semi-closue (esp α -closue, pe-closue, semi-pe-closue) of a subset A of ( X, ), deoted by scl ( A ) (esp Cl ( A α ), Cl p ( A ), ( ) spcl A ) ad defied to be the itesectio of all semi-closed (esp α -closed, p -closed, sp -closed) sets cotaiig A The semi-iteio (esp α -iteio, pe-iteio, semi-pe-iteio) of a subset A of ( X, ), deoted by It ( A ) (esp It ( A ) s α, It p ( A ), ( ) sp It A ) ad defied to be the uio of all semi-ope (esp α -ope,

3 Geealized ea ough pobability 0 p -ope, sp -ope) sets cotaied i A Defiitio 22 A subset A of a space ( X, ) is said to be: (i) geealized closed[7] (biefly, g -closed) if Cl ( A ) U wheeve A U ad U is ope i ( X, ), (ii) geealized semi-closed[3] (biefly, -closed) if Cl s ( A ) U wheeve A U ad U is ope i ( X, ), (iii) geealized semi-peclosed[] (biefly, p -closed) if Cl sp ( A ) U wheeve A U ad U is ope i ( X, ), (iv) α -geealized closed[9] (biefly, α g -closed) if αcl ( A ) U wheeve A U ad U is ope i ( X, ), (v) geealized peclosed[2] (biefly, gp -closed) if Cl p ( A ) U wheeve A U ad U is ope i ( X, ) The complemet of a g -closed (esp -closed, p -closed, gp -closed ad α g -closed) set is called g -ope (esp -ope, p -ope, gp -ope ad α g -ope) The family of all g -ope (esp -ope, gp -ope, α g -ope ad p -ope) sets of ( X, ) is deoted by go ( X ) (esp gso ( X ), gpo ( X ), α go ( X ) ad po ( X )) The family of all g -closed (esp -closed, gp -closed, α g -closed ad p -closed) sets of ( X, ) is deoted by gc ( X ) (esp gsc ( X ), gpc ( X ), α gc ( X ), ad pc ( X ) ) The geealized iteio (biefly g -iteio) of A is deoted by g It ( A ) ad is defied by It ( A g ) = { G X : G A, G is a g -ope}, ad the geealized ea iteio (biefly -iteio) of A is deoted by It ( A ) fo all ad is defied by It ( A ) = { G X : G A, G is a -ope} The geealized closue (biefly g -closue) of A is deoted by gcl ( A ) ad is defied by Cl ( A g ) = { F X : A F, F is a g -closed set}, ad the geealized ea closue (biefly -closue) of A is deoted by Cl ( A ) fo all ad is defied by Cl ( A ) = { F X : A F, F is a closed set} The geealized bouday (biefly g -bouday) egio of A is deoted by g BN ( A ) ad is defied by gbn ( A) = gcl( A) git( A) ad the geealized ea bouday (biefly -bouday) egio of A is deoted by BN ( A ) fo all ad is defied by BN ( A) = Cl( A) It( A)

4 02 M E Abd El-Mosef, A M ozae ad R A Abu-Gdaii 3 Pawla's appoach Coside the appoximatio space ( U, R) the uivese ad R is a equivalece elatio The ode tiple S ( U, p) =, whee U is a set called = is called the stochastic appoximatio space [], whee p is a pobability measue; ay subset of U will be called a evet The pobability measue p has the followig popeties: p( φ ) = 0, p( U) = ad if A = X i i = p( A) = p( ) i= X i is a obsevable set i, the It is clea that A is a uio of disjoit sets, sice R is a equivalece elatio Pawla itoduced the defiitios of the lowe ad uppe pobabilities of a evet A i the stochastic appoximatio space S = ( U, p) These defiitios ae: - The lowe pobability of A is deoted by p (A) ad is give by p ( A) = p( RA) - The uppe pobability of A is deoted by p (A) ad is give by p ( A) = p( RA) Clealy, 0 p ( A), p( A) Geealized ea ough pobability i topological spaces Defiitio [5] Let = ( X, R) be a appoximatio space with geeal elatio R ad is the topology associated to The the tiple ( X, ) is called a topologized appoximatio space Defiitio 2 [5] Let ( U, R) = be a appoximatio space with geeal elatio R ad is the topology associated to The the ode -tuples S = U, is called the topologized stochastic appoximatio space ( ) Geealized ea ough pobability We obtai some ules to fid lowe ad uppe pobabilities i a topologized stochastic appoximatio spaces fo all Defiitio Let A be a evet i the topologized stochastic appoximatio space S = ( U, ) The the lowe (esp uppe) pobability of A fo all is give by

5 Geealized ea ough pobability 03 p ( A ) = p ( It ( A )) (esp p ( A ) = p ( Cl ( A )) ) Defiitio 2 Let A be a evet i the topologized stochastic appoximatio S = U, The the ough pobability of A fo all space ( ) is deoted by p * ( A ) ad is give by p * ( A) = p( A), p( A) Popositio Let A be a evet i the topologized stochastic appoximatio space S = ( U, ), the the implicatios betwee the lowe pobability of A ae give by the followig diagam fo all j { s, α, β} pa ( ) α p( A ) g β pa ( ) g ( ) gp pa Figue The elatio betwee the lowe pobability Poof The poof is obvious Popositio 2 Let A be a evet i the topologized stochastic appoximatio space S = ( U, ), the the implicatios betwee the uppe pobability of A ae give by the followig diagam fo all j { s, α, β} g β p( A ) p ( A ) p gp ( A ) gα p( A ) Figue 2 The elatio betwee the uppe pobability Poof The poof is obvious 2 Geealized ea ough distibutio fuctio We shall itoduce the cocept of geealized ea ough (biefly ough) distibutio fuctio of a adom vaiable X fo all

6 0 M E Abd El-Mosef, A M ozae ad R A Abu-Gdaii I the followig defiitio we defie the lowe ad the uppe distibutio fuctios of a adom vaiable X fo all Defiitio 2 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the lowe distibutio (esp uppe distibutio) fuctio of X fo all is give by F ( x) = p( X x) (esp F( x) = p( X x) ) Defiitio 22 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the ough distibutio fuctio of X fo all is deoted by * ( x ) ad is give by F F * ( x) = F( x), F( x) Example 2 Coside the expeimet of choosig oe fom fou cads have umbeed fom oe to fou The collectio of the five elemets foms the outcome space Hece, U = {,2,3,}, Let R be a biay elatio defied o U such that R = (,),(, 2),(,3),(2,3),(3,3),(3, ),(, 2) { } Thus U / R = {{, 2, 3},{3},{3, },{2} } Let ( U, R) = be a appoximatio space ad is the topology associated to Thus, = { U, φ,{2},{3},{2,3},{3,},{,2,3},{2,3,}} The SO( U ) = U, φ,{2},{3},{,2},{,3},{2,3},{3,},{,2,3},{,3,},{2,3,}, { } = { φ U }, { U φ = { U φ } SC ( U ),,{},{2},{},{,2},{,},{2,},{3,},{,2,},{,3,} SO ( U g ) =,,{2},{3},{,2},{,3},{2,3},{3,},{,2,3},{,3,},{2,3,}}, SC ( U g ),,{},{2},{},{,2},{,},{2,},{3,},{,2,},{,3,} Defie the adom vaiable X to be the umbe o the chose cad We ca costuct Table 53 which cotais the lowe ad the uppe pobabilities of a adom vaiable X = x fo j = s Table 2 px X 2 3 ( = x ) 0 p ( X = x ) 2 The the lowe distibutio fuctio of X is 0

7 Geealized ea ough pobability 05 0 < x < 2, F( x) = 2 x < 3, 2 3 x < Ad the uppe distibutio fuctio of X is 0 < x <, x < 2, 2 F( x) = 2 x < 3, 3 x <, 5 x < 3 Geealized ea ough expectatio We shall itoduce the geealized ea ough (biefly ough) expectatio of a adom vaiable X fo all We shall defie the lowe ad the uppe expectatios of a adom vaiable X fo all i the followig defiitio Defiitio 3 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the lowe (esp uppe) expectatio of X fo all is give by (esp = E ( X ) = x p( X = x ) = = E ( X ) = x p( X = x )) = Defiitio 32 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the ough expectatio of X fo all is deoted by E * ( X ) ad is give by E * ( X ) = E( X ), E( X ) ; The ough expectatio of X also deoted by =, fo all *

8 06 M E Abd El-Mosef, A M ozae ad R A Abu-Gdaii Geealized ea ough vaiace ad geealized ea ough stadad deviatio We shall itoduce the ea ough (biefly ough) vaiace ad the ea ough (biefly ough) stadad deviatio of a adom vaiable X fo all Defiitio Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the lowe (esp uppe) vaiace of X fo all is give by 2 ( ) = ( ) (esp V X E X 2 V ( X ) = E( X ) ) Defiitio 2 Let X be a adom vaiable i the topologized stochastic S = U, The the ough vaiace of X fo all appoximatio space ( ) is deoted by V * ( X ) ad is give by V * ( X ) = V ( X ), V ( X ) Defiitio 3 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the lowe (esp uppe) stadad deviatio of X fo all is give by σ ( X ) = V ( X ) (esp σ ( X ) = V ( X )) Defiitio Let X be a adom vaiable i the topologized stochastic S = U, The the ough stadad deviatio of appoximatio space ( ) X fo all is deoted by σ * ( X ) ad is give by σ * ( X ) = σ( X ), σ( X ) Example Coside the same expeimet as i Example 2 Fom Table 2 it is easy to see the followig: - Neithe of the lowe ad the uppe pobabilities summed to oe fo j = s - The value 2 of X has exact pobability sice ( ) = px ( ) = at px X = 2 - The lowe ad the uppe expectatios of X ae: = E( X ) = 25, = E( X ) = The ough mea (o ough expectatio) of X equals: * = 25, The lowe ad the uppe vaiaces of X ae:

9 Geealized ea ough pobability 07 V 29 ( X ) = =, V ( X ) = The ough vaiace of X equals: V * ( X ) = 0906, Fially, the ough stadad deviatio of X is give by: σ * ( X ) = 0952, Geealized ea ough momets We shall defie the ea ough (biefly ough) momet of a adom vaiable X fo all Defiitio 5 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the lowe (esp uppe) th momet of X about the lowe mea (esp the uppe mea ) fo all, also called the geealized lowe (esp geealized uppe) th cetal momet, is defied as ( ) = (esp ( ) = E ( X ) = x p( X = x ) ( ) ( )), Whee = 0,,2, = E X = x p X = x = th The lowe (esp uppe) momet of X about oigi is defied as ( = E X ) (esp ( = E X )), whee = 0,,2, Defiitio 52 Let X be a adom vaiable i the topologized stochastic th appoximatio space S = ( U, ) The the ough momet of X * fo all is deoted by ad is defied by * ( X ) =, I the followig defiitio we shall defie the lowe ad the uppe momet geeatig fuctio of a adom vaiable X fo all Defiitio 53 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the lowe (esp uppe) momet geeatig fuctio of X fo all is defied by

10 08 M E Abd El-Mosef, A M ozae ad R A Abu-Gdaii tx ( ) tx M X t = E e = e p X = x = () ( ) tx M X t = E e = e p X = x = () ( )) tx (esp ( ) Defiitio 5 Let X be a adom vaiable i the topologized stochastic appoximatio space S = ( U, ) The the ough momet geeatig fuctio of X fo all is defied by M * () t = M (), t M () t X X X Example 5 Coside the same expeimet as i Example 2 Fom Table 2 it is easy to see the followig: th - The lowe momet of X about the lowe mea is ( ) = E X = x p X = x = ( ) + ( ), whee = 0,,2, + ( ) ( ) = 3 7 = - The uppe th momet of X about the uppe mea is ( ) = E X = x p X = x = ( 9) + ( 5) + 2( ) + (3) = + ( ) ( ) () th - The lowe momet of X about oigi is, whee = 0,,2, = E ( X ) = (2 + 3 ), whee = 0,,2, th - The uppe momet of X about oigi is = E ( X ) = ( 2 2(3) ), whee, = 0,,2 - The lowe momet geeatig fuctio of X is tx tx - M X () t = E ( e ) = e p( X = x ) = ( 2t 3t e + e ) = - The uppe momet geeatig fuctio of X is tx ( ) tx M X t = E e = e p X = x = () ( )

11 Geealized ea ough pobability = ( t t 2 t t e + e + e + e ) 5 Coclusios I this pape, we used topological cocepts to itoduce defiitios to geealized ough pobability, geealized ough distibutio fuctio, geealized ough expectatio, etc The topological applicatios which itoduced help fo measuig geealized ough pobability, geealized ough expectatio, etc Refeeces [] M E Abd El-Mosef, S N El-Deeb, R A Mahmoud, β-ope sets, β- cotiuous mappi, Bull Fac Sc Assuit Uiv, 2(983), [2] D Adijević, Semi-peope sets, ibid 38(986), 2-32 [3] S P Aya, T M Nou, Chaacteizatios of s-omal spaces, Idia Jpue appl math, 2 (8) (990), [] J Dotchev, O geealizig semi-peope sets, Mem Fac Sci ochi Uiv (Math), 6 (995), 35-8 [5] M J Iqela, O Topological Stuctues ad Ucetaity, PhD Thesis, Tata Uivesity, (200) [6] J elley, Geeal topology, Va Nostad Compay, 955 [7] N Levie, Geealized closed sets i topology, Red Cic math palemo, 9 (2) (970), [8] N Levie, Semi-ope sets ad semi-cotiuity i topological spaces, Ame Math Mothly, 70 (963), 36- [9] H Mai, R Devi ad Balachada, Associated topologies of geealized α-closed sets ad α-geealized closed sets, Mem Fac, Sci ochi Uiv(Math), 5 (99), 5-63

12 0 M E Abd El-Mosef, A M ozae ad R A Abu-Gdaii [0] A S Mashhou, M E Abd El-Mosef, S N El-Deeb, O pe cotiuous ad wea pe cotiuous mappi, Poc Math Phys Soc Egypt, 53(982), 7-53 [] O Njastad, O some classes of ealy ope sets, Pacific J Math 5(965), [2] T Noii, H Mai ad J Umehaa, Geealized peclosed fuctio, Mem Fac Sci ochi Uiv (Math), 9 (998), 3-20 [3] Z Pawla, Rough sets, It J of Ifomatio ad Compute Scieces, (5) (982) [] Z Pawla, Rough pobability, Bulleti of the Polish Academy of Scieces, Mathematics, 32 (98) Received: Septembe, 200