Generalized Near Rough Connected Topologized Approximation Spaces

Transcription

1 Global Journal of Pure and Applied Mathematics ISSN Volume 13 Number 1 (017) pp Research India Publications Generalized Near Rough Connected Topologized Approximation Spaces Maha Al-Soudi and Radwan Abu-Gdairi Department of Mathematics Faculty of Science Zarqa University Zarqa Jordan Abstract Rough set theory has been introduced by Pawlak [18] It is considered as a base for all researches in the rough set area introduced after this date Most of these researches concentrated on developing results and techniques based on Pawlak's results [18] In this paper we apply topological concepts to introduce definitions for generalized b-approximations generalized b-boundary regions and generalized near rough connected topologized approximation spaces from topological view The basic concepts of some generalized near open generalized near rough and generalized near exact sets are sufficiently illustrated Moreover proved results examples and counter examples are provided eywords: Generalized b- approximations Generalized b-rough set Generalized b-connected spaces 1 INTRODUCTION One of the most powerful notions in system analysis is the concept of topological structures [10] and their generalizations Rough set theory introduced by Pawlak in 198 [18] is a mathematical tool that supports also the uncertainty reasoning but qualitatively In this paper we shall integrate some ideas in terms of concepts in topology Topology is a branch of mathematics whose concepts exist not only in almost all branches of mathematics but also in many real life applications We

2 8410 Maha Al-Soudi and Radwan Abu-Gdairi believe that topological structure will be an important base for modification of knowledge extraction and processing PRELIMINARIES A topological space [10] is a pair ( X consisting of a set X and family τ of subsets of X satisfying the following conditions: (T1) φ τ and X τ (T) τ is closed under arbitrary union (T3) τ is closed under finite intersection Throughout this paper ( X denotes a topological space the elements of X are called points of the space the subsets of X belonging to τ are called open sets in the space the complement of the subsets of X belonging to τ are called closed sets in the space and the family of all open sets of ( X is denoted by τ and the family of all closed sets of ( X is denoted by C( X ) For a subset A of a space ( X Cl ( A ) denote the closure of A and is given by Cl ( A ) = { F X : A F and F C ( X )} Evidently Cl ( A ) is the smallest closed subset of X which contains A Note that A is closed iff A = Cl( A) Int ( A ) denote the interior of A and is given by Int ( A ) = { G X : G A and G τ} Evidently Int ( A ) is the largest open subset of X which contained in A Note that A is open iff A = Int( A) The boundary of a subset A X is denoted by BN ( A ) and is given by BN ( A) = Cl( A) Int( A) We shall recall some concepts about some near open sets which are essential for our present study Definition 1 A subset A of a space ( X is called: i) Semi-open [1] ( briefly s open ) if A Cl( Int( A)) ii) Pre-open [14] ( briefly iii) α -open [15] if A Int( Cl( Int( A))) p open ) if A Int( Cl( A)) iv) β -open [1] ( = semi-pre-open [ ] ) if A Cl( Int( Cl( A))) v) b - open [ ] A Int( Cl( A)) Cl ( Int ( A ))

3 Generalized Near Rough Connected Topologized Approximation Spaces 8411 The complement of a s open (resp p open α -open β -open and b -open) set is called a s closed ( resp p closed α -closed β -closed and b -closed ) set ( ) τ The family of all s open ( resp p open α -open β -open and b -open) sets of X is denoted by SO (X ) ( resp PO (X ) α O(X ) β O(X ) and BO( X ) ) The family of all closed) sets of ( BC( X )) s closed (resp p closed α - closed β -closed and b - X is denoted by SC (X ) (resp PC (X ) α C(X ) β C(X ) and The semi-closure (resp α -closure pre-closure β -closure and b -closure) of a subset A of ( X denoted by scl ( A ) (resp Cl ( A α ) Cl p ( A ) βcl ( A ) and bcl ( A ) ) and defined to be the intersection of all semi-closed (resp α - closed p -closed β -closed and b -closed) sets containing A The semiinterior (resp α -interior pre-interior β -interior and b -interior) of a subset A of ( X denoted by s Int ( A ) (resp Int ( A α ) Int p ( A ) Int ( A ) β and b Int ( A ) ) and defined to be the union of all semi-open (resp α -open p -open β -open and b - open) sets contained in A Definition A subset A of a space ( X is said to be: (i) generalized closed[11] (briefly g -closed) if Cl ( A ) U whenever A U and U is open in ( X (ii) generalized semi-closed[6] (briefly gs -closed) if Cl s ( ) A U and U is open in ( X (iii) generalized β -closed[7] (briefly g β -closed) if βcl ( A ) U whenever A U and U is open in ( X (iv) α -generalized closed[13] (briefly A U and U is open in ( X α g -closed) if αcl ( A ) U whenever (v) generalized pre-closed[16] (briefly gp -closed) if Cl p ( A ) U whenever A U and U is open in ( X (vi) generalized B -closed[] (briefly gb -closed) if Cl B ( A ) U whenever A U and U is open in ( X The complement of a g -closed (resp gs -closed g β -closed gp -closed α g -closed and -closed) set is called g -open (resp gs -open g β -open gp -open α g -open and -open) The family of all g -open (resp gs -open gp -

4 841 Maha Al-Soudi and Radwan Abu-Gdairi open α g -open g β -open and -open) sets of ( X is denoted by go( X ) (resp gso( X ) gpo ( X ) α go( X ) g β O( X ) and gbo( X )) The family of all g - closed (resp gs -closed gp -closed α g -closed g β -closed and -closed) sets of ( X is denoted by gc( X ) (resp gsc ( X ) gpc ( X ) α gc( X ) g β C( X ) and gbc ( X )) The generalized interior (briefly g -interior) of A is denoted by g Int ( A ) and is defined by Int ( A g ) = { G X : G A G is a g -open set} the generalized near interior (briefly -interior) of A is denoted by Int ( A ) for all j { s p β b } and is defined by Int ( A ) = { G X : G A G is a -open set}and the generalized α interior (briefly gα -interior) of A is denoted by gα Int ( A ) and is defined by gα Int ( A ) = { G X : G A G is a α g -open set} The generalized closure (briefly g -closure) of A is denoted by gcl ( A ) and is defined by Cl ( A g ) = { F X : A F F is a g -closed set} the generalized near closure (briefly -closure) of A is denoted by Cl ( A ) for all j { s p β b } and is defined by Cl ( A ) = { F X : A F F is a closed set} and the generalized α closure (briefly gα -closure) of A is denoted by Cl g ( A α ) and is defined by Cl ( A g ) { F X : A F F α = is a α g -closed set} The generalized boundary (briefly g -boundary) region of A is denoted by g BN ( A ) and is defined by gbn ( A) = gcl( A) gint( A) and the generalized near boundary (briefly -boundary) region of A is denoted by BN ( A ) for all j { s p α β B} and is defined by BN ( A) = Cl( A) Int( A) The generalized exterior (briefly g -exterior) of A is denoted by Ext g ( A ) and is defined by gext ( A) = X gcl ( A) and the generalized near exterior (briefly -exterior) of A is denoted by Ext( A ) for all j { s p α β b } and is defined by Ext( A) = X Cl( A) 3 GENERALIZED NEAR APPROXIMATIONS The present section is devoted to introduce the concept of generalized near approximations

5 Generalized Near Rough Connected Topologized Approximation Spaces Generalized near lower and generalized near upper approximations Definition 311 [9] Let = ( X R) be an approximation space with general relation R and τ k is the topology associated to Then the triple ( X R τ k ) is called a topologized approximation space Definition 31 [9] Let = ( X R τ k ) be a topologized approximation space If A X then the lower approximation (resp upper approximation ) of A is defined by RA = Int( A) (resp RA = Cl( A) ) Definition 313 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized lower approximation (briefly g -lower approximation) of A is denoted by R A and is defined by R A = Int( A) g g g Definition 314 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized near lower approximation (briefly -lower approximation) of A is denoted by R A and is defined by R A = Int( A) where j { s p α β} Definition 315 Let = ( X R τ k ) be a topologized approximation space If A X then the generalized b lower approximation (briefly -lower approximation) of A is denoted by R A and is defined by R A = Int( A) Definition 316 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized upper approximation (briefly g -upper approximation) of A is denoted by R g A and is defined by R g A = Cl( A) g Definition 317 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized near upper approximation (briefly -upper approximation) of A is denoted by R A and is defined by R A = Cl( A) where j { s p α β}

6 8414 Maha Al-Soudi and Radwan Abu-Gdairi Definition 318 Let = ( X R τ k ) be a topologized approximation space If A X then the generalized b upper approximation (briefly -upper approximation) of A is denoted by R A = Cl A R A and is defined by ( ) Theorem 311 Let = ( X R τ k ) be a topologized approximation space If A X then the implications between lower approximation and -lower approximations of A are given by the following diagram for all j { s p α β B} R A R gp A R A R A g α g β R gs A Figure 311 Relation between the -lower approximations Proof R A = Int ( A ) = { G gpo ( X ): G A} { G g βo ( X ): G A} gp gp R A = Int ( A ) Int ( A ) = R A gp gp g β g β Theorem 31 Let = ( X R τ k ) be a topologized approximation space If A X then the implications between -upper approximations of A are given by the following diagram for all j { s p α β B} R gβ A R A R R g α A R gp gs A A Figure 31 Relation between the -upper approximations

7 Generalized Near Rough Connected Topologized Approximation Spaces 8415 Proof (i) R gs A = Cl( A) = { F gsc( X ): A F} { F C( X ): A F} gs R gs A Cl( A) = R A (ii) R gpa = Cl ( A ) = { F gpc ( X ): A F} { F C ( X ): A F} gp R gpa Cl( A) = R A (iii) R A = Cl( A) = { F C( X ): A F} { F gβc( X ): A F} R A Cl( A) = R g β A g β 3 Generalized b-boundary regions In this section we obtain some rules to find generalized b-boundary regions Definition 31 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized near boundary (briefly -boundary) region of A is denoted by BN ( A ) and is defined by R BN ( A) = R ( A) R ( A) where j { s p α β} R Definition 3 Let = ( X R τ k ) be a topologized approximation space If A X then the generalized b boundary (briefly -boundary) region of A is denoted by BN ( A ) and is defined by R BN ( A) = R ( A) R ( A) R Definition 33 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized near positive (briefly -positive) region of A is denoted by POS R ( A ) and is defined by POS R ( A ) = R A where j { s p α β} Definition 34 Let = ( X R τ k ) be a topologized approximation space If A X then the generalized b positive (briefly -positive) region of A is denoted by POS R ( A ) and is defined by POS ( A ) = R A R

8 8416 Maha Al-Soudi and Radwan Abu-Gdairi Definition 35 [] Let = ( X R τ k ) be a topologized approximation space If A X then the generalized near negative (briefly -negative) region of A is denoted by NEG ( A ) and is defined by R NEG ( A ) = X R A where j { s p α β} R Definition 36 Let = ( X R τ k ) be a topologized approximation space If A X then the generalized b negative (briefly -negative) region of A is denoted by NEG ( A ) and is defined by R NEG ( A ) = X R A R Theorem 31 k = ( X R τ k ) be a topologized approximation space If A X then BN ( A) BN ( A) for all j { s p α β B} R Proof obvious R Theorem 3 Let k = ( X R τ k ) be a topologized approximation space If A X then the implications between boundary and -boundary of A are given by the following diagram for all j { s p α β B} BN ( A ) BN ( A) BN ( A) BN ( A) Rg β R R gs BN ( A ) R gp R g α Proof Obvious Figure 31 Relations between the -boundaries

9 Generalized Near Rough Connected Topologized Approximation Spaces Generalized near rough and generalized near exact sets In this section we used topological concepts to introduce definitions to generalized b-rough and generalized b-exact sets Definition 331 [] Let ( X be a topological space and A X Then i) A is totally -definable ( -exact) set if Int ( A ) = A = Cl ( A ) ii) A is internally -definable set if A = Int( A) iii) A is externally -definable set if A = Cl( A) iv) A is - indefinable set if A Int( A) A Cl( A) where j { s p α β} Definition 33 Let ( X be a topological space and A X Then v) A is totally -definable ( -exact) set if Int ( A ) = A = Cl ( A ) vi) A is internally -definable set if A = Int( A) vii) A is externally -definable set if A = Cl( A) viii) A is - indefinable set if A Int( A) A Cl( A) Theorem 331 Let ( X be a topological space and A then it is - exact X If A is an exact set Proof Let A be exact set then Cl ( A ) = A = Int ( A ) Now Cl ( A ) = { F X : A F F C ( X )} { F X : A F F C( X )} Also Int ( A ) = { G X G τ} { G X : G A G O( X )} = Int ( A ) Therefore Cl ( A ) Cl ( A ) A Int ( A ) Int ( A ) Since A is exact we get Cl ( A ) = A = Int ( A ); Hence A is -exact

10 8418 Maha Al-Soudi and Radwan Abu-Gdairi 4 GENERALIZED NEAR ROUGH CONNECTED TOPOLOGIZED APPROXIMATION SPACES The present section is devoted to introduce various level of connectedness in approximation spaces with general binary relations using some classes of generalized near closed sets Proposition 41 Let κ ( X R = be a topologized approximation space If A and B are two subsets of X then: i) R A A R A ii) R ϕ R ϕ ϕ = = and R X = R X = X iii) If A B then R A R B iv) If A B then R A R B R X A = X R A v) ( ) R X A = X R A vi) ( ) Definition 41 Let κ ( X R Then = be a topologized approximation space and A X i) A is called totally -definable ( -exact) set if R A = A = R A ii) A is called internally -definable set if A = R A iii) A is called externally -definable set if A = R A iv) A is called -indefinable ( -rough) set if A R A and A R A Definition 4 Let κ ( X R = be a topologized approximation space Then κ is said to be generalized j-rough (briefly -rough) disconnected if there are two nonempty subsets A and B of X such that A B = X and A R B = R A B = ϕ Where j { s p α β b} The space κ ( X R = is said to be -rough connected if it is not -rough disconnected

11 Generalized Near Rough Connected Topologized Approximation Spaces 8419 = is -rough disconnected If and only if there exists a nonempty -exact proper subset X where j { s p α β b} Proposition 4 A topologized approximation space κ ( X R Proof We shall prove this theorem in the case of j Suppose κ ( X R = b = is -rough disconnected topologized approximation space then there exist two nonempty subsets A and B of X such that A B = X and A R B = R A B = ϕ But A R A hence A B = ϕ Thus A = X B Also A = X R B since A R B = φ and A R B A B = X Hence A = R A and B = R B Similarly B = R B and A = R A Therefore there exists a nonempty -exact proper subset A of X Conversely suppose that A is a nonempty -exact proper subset of X then we get B = X A is also a nonempty -exact proper subset of X Hence A B = X and A R B = A B = R A B = ϕ Thus κ ( X R = is -rough disconnected Example 41 Let κ ( X R = be a topologized approximation space such that X = {1 3 4} and R be a binary relation defined on X such that R = {(11)(13)(3)(33)(34)(4 4) } Thus / {{1 3}{3}{3 4}{4} } τ = { X φ{3}{4}{3 4}{13}{13 4}} So X R = and { X φ } BO( X ) = {3}{4}{13}{3}{4}{3 4}{13}{134}{34} { φ X } BC ( X ) = {1}{}{4}{1 }{1 3}{1 4}{4}{1 3}{1 4} = { X φ = { X φ } BO ( X g ) {1}{3}{4}{13}{1 4}{3}{4}{3 4}{1 3}{134}{3 4}} BC ( X g ) {1}{}{4}{1}{13}{14}{3}{ 4}{13}{1 4}{34} Since A = {1} is a nonempty -exact proper subset of X then the space ( X R κ = is -rough disconnected

12 840 Maha Al-Soudi and Radwan Abu-Gdairi Proposition 43 The implications between generalized near rough disconnected topologized approximation spaces are given by the following figure gα rough disconnected gs rough disconnected gp rough disconnected rough disconnected g β rough disconnected Definition 43 Let κ ( X R 1 Figure 41 = = ( Y R Q be two topologized approximation spaces Then for every j { s p α β b} a mapping f : κ Q is called -rough continuous if 1 f ( RV) R f 1 ( V) 1 means the inverse image of V Q for every subset V of Y in Q Where f Theorem 41 Let f : κ Q be a mapping from a topologized approximation space κ = ( X R1 τ ) to a topologized approximation space Q = ( Y R τ Q ) Then the following statements are equivalent Where j { s p α β b} i) f is -rough continuous ii) The inverse image of each internally -definable set in κ iii) The inverse image of each externally -definable set in κ iv) f ( R1 A ) Rf ( A) v) R1 f ( B) f ( RB) R R for every subset A of X in κ for every subset B of Y in Q Proof We shall prove this theorem in the case of j definable set in Q is internally definable set in Q is externally = b

13 Generalized Near Rough Connected Topologized Approximation Spaces 841 (i) (ii) Let f be -rough continuous and let V be an internally R definable set in Q Then R V = V By (i) we get ( ) ( ) ( ) 1 and f ( V ) f V = f R V R f V Then ( ) ( ) But 1 R f ( V ) f 1 ( V ) f V R f V ( ) = ( ) Therefore f ( V ) f V R f V κ 1 1 is a subset of X in κ Hence is internally -definable set in (ii) (i) Let A be a subset of Y in Q Since R A A then f R A f A But R A is internally R definable set in Q then by ( ) ( ) (ii) we get f ( R A) f ( A) Hence 1 ( ) 1 f R A R f ( A) f 1 ( A) is internally -definable set in κ contained in 1 since R 1 1 f ( A) the largest internally -definable set contained in f ( A) Thus ( ) ( ) f R A R f A for every subset A of Y in Q Therefore f is 1 1 -rough continuous is (ii) (iii) Let F be an externally internally definable set in κ R definable set in Q then Y F is R definable Thus by (ii) we have f ( Y F ) Since f ( Y F ) = X f ( F ) then X f ( F ) definable set in κ Hence f ( F ) is internally - is internally - is externally -definable set in κ Similarly we can prove (iii) (ii) (ii) (iv) Let A be a subset of X in κ then f ( A) R definable set in Q Hence Y R f ( A) R Q Thus by (ii) we get f Y R f ( A) = X f R f -definable set in κ and so R f ( A) R is an externally is internally definable set in ( ) ( ( A) ) ( ) is internally f is externally -definable set

14 84 Maha Al-Soudi and Radwan Abu-Gdairi ( ) containing A in κ Thus A R A f R f ( A) since R1 A is the 1 smallest externally -definable set containing A in κ Hence Therefore f ( R A) R f ( A) ( 1 ) ( ( )) ( ) f R A f f R f A R f A 1 for every subset A in κ (iv) (v) Let B be a subset of Y in Q Let A f ( B) of X in κ By (iv) we get ( ) ( ) ( ( )) 1 = then A is a subset f R A R f A = R f f B R B Hence R1 A f ( RB) Thus R1 A R 1 f ( B) f ( RB) Therefore R1 f ( B) f ( RB) = for every subset B of Y in Q (v) (ii) Let G be an internally R definable set in Q then B = Y G is externally definable set in Q Thus by (v) we get R ( ) ( ) R f B f R B 1 1 Since B is externally R definable set then f ( R B) = f ( B) Thus R f B f B f 1 B R f 1 B then 1 ( ) ( ) But ( ) 1 ( ) R1 f ( B) = f ( B) Hence f ( B) Since f ( B) = f ( Y G) = X f ( G) then X f ( G) -definable set in κ Therefore f ( G) Example 4 Let κ ( X R = = ( Y R is externally -definable set in κ is externally is internally -definable set in κ 1 Q Q be two topologized approximation spaces such that X = {1 3 4} Y = { a b c d} R 1 = {(11)()(44)(1)(1)} and R {( aa )( d d)( ab )} = { X ϕ{4}{1 }{1 4} } and τ { ϕ{ }{ }{ } τ Define a mapping f : Q such that = Then Q = Y d ab abd Hence f (1) = a f () = d f (3) = b and f (4) = c

15 Generalized Near Rough Connected Topologized Approximation Spaces 843 Then f is not a -rough continuous mapping since V = { a b} is an internally R definable set in Q but f ( V ) = {1 4} is not an internally -definable set in κ Proposition 4 Let κ ( X R 1 = and = ( Y R Q be two topologized approximation spaces If f : κ Q is a -rough continuous mapping then the inverse image of each exact set in Q is -exact set in κ Q Proof Let A be an exact set in Q then A is both internally and externally R definable set in Q Hence f ( A) is both internally and externally - definable set in κ Therefore f ( A) is a -exact set in κ 3 CONCLUSIONS In this paper we used -open sets to introduce the definition of -rough connected topologized approximation space REFERENCES [1] M E Abd El-Monsef S N El-Deeb R A Mahmoud β-open sets β-continuous mappings Bull Fac Sc Assuit Univ 1(1983) [] M E Abd El-Monsef A M ozae R A Abu-Gdairi Generalized near approximations in topological spaces International Journal of Mathematical Archive (3) Mar 011 pages: [3] M E Abd El-Monsef A M ozae R A Abu-Gdairi Generalized near rough probability in topological spaces Int J Contemp Math Sciences Vol6 011 no [4] D Andrijević Semi-preopen sets ibid 38(1986) 4-3 [5] D Andrijević On b-open sets Mat Vesnik 48(1996) [6] S P Arya T M Nour Characterizations of s-normal spaces Indian Jpure appl math 1 (8) (1990) [7] J Dontchev On generalizing semi-preopen sets Mem Fac Sci ochi Univ (Math) 16 (1995) [8] M Ganster and M Steiner On τ b -closed sets Appl Gen Topol 8(007) no 43-47

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