International Journal of Information Research and Review, November, 2016
International Journal of Information Research and Review Vol. 03, Issue, 11, pp. 3058-3065, November, 2016
Research Article
A NEW CLASS OF GENERALIZED BIPOLAR VAGUE SETS
Cicily Flora, S. and Arockiarani, I.
Department of Mathematics, Nirmala College for Women, Coimbatore-18
ARTICLE INFO ABSTRACT
The focus of this paper is to formulate a new class of generalized bipolar vague closed sets in bipolar Article History:
vague topological spaces and confered its properties.
Received 28th August, 2016 Received in revised form 22nd September, 2016 th Accepted 24 October, 2016 Published online November, 30th 2016
Keywords:
Bipolar Vague Topology, Bipolar Vague Continuity, Bipolar Vague Generalized Closed Set.
Copyright © 2016, Cicily Flora and Arockiarani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
Generalized closed sets in general topology were studied by Levine (Levine, 1970). After the introduction of the concept of fuzzy sets by Zadeh (Zadeh, 1965) several researches were conducted on the generalizations of the notion of the fuzzy set. In the traditional fuzzy sets, the membership degree of element range over the interval [0, 1]. There are several kinds of fuzzy set extensions in the fuzzy set theory, for example, intuitionistic fuzzy sets, interval-valued fuzzy sets, vague sets etc. Bipolar-valued fuzzy sets, which was introduced by Lee(9, 10) is an extension of fuzzy sets whose membership degree range is enlarged from the interval [0, 1] to [-1, 1]. The notion of vague set theory introduced by Gau W.L and Buehrer D.J (Gau, 1993), as a generalization of Zadeh’s fuzzy set.
The objective of this paper is to introduce the concept of bipolar vague generalized closed set.
2. PRELIMINARIES
Definition 2.1(Lee, 2000): Let X be the universe. Then a bipolar valued fuzzy sets, A on X is defined by positive membership function A , that is A : X [0,1], and a negative membership function A , that is A : X [1,0] . For the sake of simplicity, we shall use the symbol A {x, A (x), A (x) :x X}.
Definition 2.2(Lee, 2000): Let A and B be two bipolar valued fuzzy sets then their union, intersection and complement are defined as follows:
(i) (x) max{ (x), (x)} A B A B . (ii) (x) min{ (x), (x)} A B A B (iii) A B (x) min{ A (x), B (x)}
*Corresponding author: Cicily Flora, S. Department of Mathematics, Nirmala College for Women, Coimbatore-18. 3059 Cicily Flora and Arockiarani. A new class of generalized bipolar vague sets
(iv) A B (x) max{ A (x),B (x)} (v) A (x) 1 A (x) and A (x) 1 A (x) for all x X .
Definition 2.3(7): A vague set A in the universe of discourse U is a pair (t A , f A ) where t A :U [0,1], f A :U [0,1]are the mapping such that t A f A 1for all u U .The function t A and f A are called true membership function and false membership function respectively. The interval [t A ,1 f A ] is called the vague value of u in A , and denoted by vA (u) , i.e vA (u) [t A (u),1 f (u)].
Definition 2.4(Gau, 1993): Let A be a non-empty set and the vague set A and B in the form
A {x,t A (x),1 f A (x) : x X}, B {x,tB (x),1 f B (x) : x X}.
Then
(i) A B if and only if t A (x) tB (x) and 1 f A (x) 1 f B (x)
(ii) AB { max(t A (x),tB (x)),max(1 f A (x),1 f B (x)) / x X} .
(ii) A B { min(t A (x),tB (x)),min(1 f A (x),1 f B (x)) / x X}.
(iv) A {x, f A (x),1 t A (x) :x X} .
Definition 2.5(Arockiarani, 2016): Let X be the universe of discourse. A bipolar-valued vague set A in X is an object having the form A {x,[t A (x),1 f A (x)],[ 1 f A (x),t A (x)] : x X} where [t A ,1 f A ] : X [0,1] and [ 1 f A ,t A ] : X [1,0] are the mapping such that t A f A 1and 1 t A f A . The positive membership degree [t A (x),1 f A (x)]denotes the satisfaction region of an element x to the property corresponding to a bipolar-valued set A and the negative membership degree [ 1 f A (x),t A (x)] denotes the satisfaction region of x to some implicit counter property of . For a sake of simplicity, we shall use the notion of bipolar vague set and . A v A [t A ,1 f A ] v A [1 f A ,t A ]
Definition 2.6(4): A bipolar vague set A [v A ,v A ] of a set U with v A 0 implies that t A 0,1 f A 0 and v A 0 implies that t A 0, 1 f A 0 for all xU is called zero bipolar vague set and it is denoted by 0 .
Definition 2.7(4): A bipolar vague set A [v A ,v A ] of a set U with v A 1 implies that t A 1,1 f A 1 and v A 1 implies that t A 1, 1 f A 1 for all xU is called unit bipolar vague set and it is denoted by 1.
3. BIPOLAR VAGUE TOPOLOGICAL SPACE
Definition 3.1: Let A x, [t A ,1 f A ][1 f A , t A ] and B x, [tB ,1 fB ][1 fB , tB ] be two bipolar vague sets then their union, intersection and complement are defined as follows:
1. where A B {(x,[t AB (x),1 f AB (x)], [1 f AB (x),t AB (x)]) / x X} and t AB (x) max{t A (x),t B (x)}, t AB (x) min{t A (x),tB (x)} , 1 f AB (x) max{1 f A (x),1 f B (x)} . 1 f AB (x) min{1 f A (x),1 f B (x)}
2. A B {(x,[t AB (x),1 f AB (x)], [1 f AB (x),t AB (x)]) / x X}where and t AB (x) min{t A (x),tB (x)}, t AB (x) max{t A (x),tB (x)} , 1 f AB (x) min{1 f A (x),1 f B (x)} . 1 f AB (x) max{1 f A (x),1 f B (x)} 3. A {(x,[ f A (x),1 t A (x)],[1 t A (x), f A (x)]) / x X} for all x X . 3060 International Journal of Information Research and Review Vol. 03, Issue, 11, pp. 3058-3065, November, 2016
Definition 3.2: Let A and B be two bipolar vague sets defined over a universe of discourse X . We say that A B if and only if t (x) t (x), 1 f (x) 1 f (x) and t (x) t (x) , 1 f (x) 1 f (x) for all x X . A B A B A B A B
Definition 3.3: A bipolar vague topology BVT on a nonempty set X is a family BV of bipolar vague set in X satisfying the following axioms
1. 0, 1BV
2. G1 G2 BV , for any G1 ,G2 BV
3. Gi BV , for any arbitrary family{Gi :Gi BV ,i I}.
In this case the pair (X , BV ) is called a bipolar vague topological space and any BVS in BV is known as bipolar vague open set in X . The complement A of a bipolar vague open set (BVOS) A in a bipolar vague topological space (X , BV ) is called a bipolar vague closed (BVCS) in X .
Example 3.4: Let a b , X {a,b}, A x, , [0.4,0.6][0.6, 0.5] [0.6,0.8][0.6, 0.3]
a b B x, , , a b . C x, , [0.2,0.4][0.6, 0.4] [0.4,0.7][0.6, 0.2] [0.5,0.6][0.7, 0.5] [0.7,0.9][0.7, 0.5]
Then the family BV = {0, 1, A, B, C}of bipolar vague sets in X is a BVT on X .
Definition 3.5: Let (X , BV ) be a bipolar vague topological space and A x,[t A ,1 f A ][1 f A ,t A ] be a BVS in X . Then the bipolar vague interior and bipolar vague closure of A are defined by,
Bvcl(A) {K : K is a BVCS in X and A K}, Bv int(A) {G :G is a BVOS in X and G A}
Note that bvcl(A) is a BVCS and bvint(A) is a BVOS in X . Further,
1. A is a BVCS in X iff Bvcl(A) A, 2. A is a BVOS in X iff Bvint(A) A
Example 3.6: Let X {a, b, c},
a b c A x, , , [0.4,0.6][0.5, 0.4] [0.3,0.7][0.4,0.3] [0.5,0.6][0.7,0.5] a b c B x, , , [0.5,0.7][0.6, 0.5] [0.6,0.8][0.5,0.4] [0.6,0.9][0.8,0.6]
Then the family BV {0, 1, A, B} of a bipolar vague sets in X is a BVT on X .
a b c If F x, , , then [0.2,0.4][0.6, 0.4] [0.4,0.5][0.4,0.2] [0.5,0.7][0.7,0.5]
Bvint(F) {G :G is a BVOS in X and G F} 0 , and Bvcl(F) {K :K is a BVCS in X and F K} 1.
Proposition 3.7: For any BVS A in (X , BV ) we have 3061 Cicily Flora and Arockiarani. A new class of generalized bipolar vague sets
1. Bvcl(A) Bvint(A) . 2. Bvint(A) Bvcl(A) . Proof:
Let A x,[t A ,1 f A ],[1 f A ,t A ] and suppose that BVOS's contained in A are indexed by the family { x,[t ,1 f ],[1 f ,t ]:i J} .Then, Bv int( A) x,[t ,1 f ],[ 1 f , t ] and hence Gi Gi Gi Gi Gi Gi Gi Gi Bv int( A) x,[ f ,1 t ],[ 1 t , f ] Gi Gi Gi Gi ……… (1)
Since A x,[ f ,1 t ],[1 t , f ] where t t , 1 f 1 f and 1 f 1 f , t t , for every A A A A Gi A Gi A Gi A Gi A i J we obtain that{ x,[ f ,1 t ],[1 t , f ]:i J} is the family of BVCS's containing A , that is, Gi Gi Gi Gi Bvcl(A) x,[ f ,1 t ],[ 1 t , f ] Gi Gi Gi Gi ………………. (2)
Hence from (1) and (2) we get Bvcl(A) Bvint(A) .
(2) Follows from (1).
Proposition 3.8: Let (X , BV ) be a BVTS and A , B be are BVS’s in X . Then the following properties hold:
1) Bvint(A) A 2) A Bvcl(A) 3) A B Bvint(A) Bvint(B) 4) A B Bvcl(A) Bvcl(B) 5) Bvint(Bvint(A)) Bvint(A) 6) Bvcl(Bvcl(A)) Bvcl(A) 7) Bvint(A B) Bvint(A) Bvint(B) 8) Bvcl(A B) Bvcl(A) Bvcl(B) 9) Bvint(1) 1 10) Bvcl(0) 0
Definition 3.9: Let (X , BV ) and (Y , BV ) be two bipolar vague topological spaces and : X Y be a function. Then is said to be bipolar vague continuous iff the preimage of each bipolar vague open set in Y is a bipolar vague open set in X .
Proposition 3.10: Let A , {Ai :i J} be a bipolar vague set in X , and B , {B j : j K} be a bipolar vague set in Y , and : X Y a function. Then
(a) A1 A2 (A1) (A2 ), 1 1 (b) B1 B2 (B1 ) (B2 ) 1 1 1 1 (c) (Bi ) (Bi ) and (Bi ) (Bi ) .
Proof: Obvious
Proposition 3.11: The following are equivalent to each other.
1. : X Y is bipolar vague continuous. 2. 1 (Bvint(B)) Bvint( 1 (B)) for each BVOS B in Y . 3. Bvcl( 1 (B)) 1 (Bvcl(B)) for each BVOS B in Y .
3062 International Journal of Information Research and Review Vol. 03, Issue, 11, pp. 3058-3065, November, 2016
Proof: (1) (2) Given :X Y is bipolar vague continuous.
1 1 Then we have to show that (Bvint(B)) Bvint( (B)) for each BVOS B in Y . Let B y, [tB ,1 f B ],[1 f B ,tB ] be a BVOS in Y and Let Bvint(B) { y,[t ,1 f ][ 1 f , t ] :i I}, where t t , 1 f 1 f and Hi Hi Hi Hi Hi B Hi B 1 f 1 f , t t , for every i I . By the definition of continuity 1 (Bvint(B)) is a bipolar vague open set in BV . Hi B Hi B Now,
1 (Bvint(B)) { 1 ( y,[t ,1 f ][ 1 f ,t ] )} Hi Hi Hi Hi ={( x,[ 1 (t ), 1 (1 f )][ 1 ( 1 f ), 1 (t )] )} Hi Hi Hi Hi = {( x,[( 1 (t )),( 1 (1 f ))][( 1 (1 f )),( 1 (t ))] )} H i Hi Hi Hi 1 . Bv int( (B))
Given 1 1 , for each BVOS in . Let (2) (1) (Bvint(B)) Bvint( (B)) B Y B y,[t B ,1 f B ][1 f B ,t B ] be a BVOS in Y . We know that B is bipolar vague open in Y if and only if Bvint(B) B and hence 1 (Bv int(B)) 1 (B) . But according to our assumption 1 (Bv int(B)) Bv int( 1 (B)), therefore we get 1 (B) Bv int( 1 (B)) . Hence 1 (B) Bv int( 1 (B) , i.e., 1 (B) is a BVS in X and this proves that is a bipolar vague continuous .
(1) (3) Given : X Y is bipolar vague continuous.
Let B y,[t B ,1 f B ][1 f B ,t B ] be a BVOS in Y . Let Bvcl(B) { y,[t ,1 f ],[ 1 f , t ]:i I}, Ki Ki Ki Ki where t t , 1 f 1 f Ki B Ki B and 1 f 1 f , t t , for each i I . Since is a bipolar vague continuous iff the inverse image of each BVCS Ki B Ki B in Y is a BVCS in X , therefore 1 (Bvcl(B)) is a BVCS in X Now,
1 (Bvcl(B)) { 1 ( y,[t ,1 f ][ 1 f ,t ] )} Ki Ki Ki Ki ={( x,[ 1 (t ), 1 (1 f )][ 1 ( 1 f ), 1 (t )] )} Ki Ki Ki Ki = {( x,[( 1 (t )),( 1 (1 f ))][( 1 (1 f )),( 1 (t ))] )} Ki Ki Ki Ki 1 . Bvcl( (B))
1 1 (3) (1) Given Bvcl( (B)) (Bvcl(B)) , for each BVOS B in Y .Let B y,[t B ,1 f B ][1 f B ,t B ] be a BVCS in Y . Since Bvcl(B) B . But it is given that Bvcl( 1 (B)) 1 (Bvcl(B)), hence Bvcl( 1 (B)) 1 (B) .Hence 1 (B) Bvcl( 1 (B) , i.e., 1 (B) is a BVCS in X and this proves that is a bipolar vague continuous .
4. GENERALIZED BIPOLAR VAGUE CLOSED SETS
Definition 4.1: Let (X , BV ) be a bipolar vague topological space. A bipolar vague set A in (X , BV ) is said to be a generalized bipolar vague closed set if Bvcl(A) G whenever A G and G is bipolar vague open. The complement of a generalized bipolar vague closed set is generalized bipolar vague open set.
Definition 4.2: Let (X , BV ) be a bipolar vague topological space and A be a bipolar vague set in X . Then the generalized bipolar vague closure (GBvcl for short) and generalized bipolar vague interior (GBvint for short) of A are defined by,
1) GBvcl(A) {G:G is a generalized bipolar vague closed set in X and A G}, 3063 Cicily Flora and Arockiarani. A new class of generalized bipolar vague sets
2) GB int(A) {G:G is a generaliz ed bipolar vague open set in X and A G}
Remark 4.3: Every BVCS is generalized bipolar vague closed but not conversely.
Example 4.4: Let X {a, b} and BV {0, 1, G}is a BVT on X where a b G x, , .Then bipolar vague set [0.4,0.6][0.4, 0.2] [0.2,0.4][0.5,0.3]
a b A x, , is a generalized bipolar vague closed but not BVC in X . [0.5,0.6][0.3, 0.2] [0.5,0.8][0.5,0.2]
Proposition 4.5: Let (X , BV ) be a bipolar vague topological space. If A is a generalized bipolar vague closed set and A B Bvcl(A), then B is a generalized bipolar vague closed set.
Proof: Let G be a bipolar vague open set in (X , BV ) , such that B G . Since A B, A G. Now A is a generalized bipolar vague closed set and Bvcl(A) G. But Bvcl(B) Bvcl(A). Since Bvcl(B) Bvcl(A) G, Bvcl(B) G. Hence, B is a generalized bipolar closed set.
Proposition 4.6: If A is a bipolar vague open set and generalized bipolar vague closed set in (X , BV ) , then A is a bipolar vague closed set in X .
Proof: Let A is a bipolar vague open set in X . Since A A , by hypothesis Bvcl(A) A.But from definition A Bvcl(A). Therefore Bvcl(A) A . Hence A is bipolar vague closed set in X .
Proposition 4.7: Let (X , BV ) be a bipolar vague topological space. A bipolar vague set A is a generalized vague open set if and only if B Bvint(A) , whenever B is a bipolar vague closed set and B A .
Proof: Let A be a generalized bipolar vague open set and B be a bipolar vague closed set, such that B A. Now, B A A B and A is a generalized bipolar vague closed set implies that Bvcl(A) B. That is, B (B) Bvcl(A) .
But . Thus, Bvcl(A) Bvint(A) B Bvint(A)
Conversely, suppose that A is a bipolar vague set, such that B Bvint(A) , whenever B is a bipolar vague closed set and B A . Let A B whenever B is a bipolar vague open set. Now, A B B A . Hence by assumption, B Bvint(A) . That is, Bvint(A) B . But Bvint(A) Bvcl(A).Hence, Bvcl(A) B . That is, A is a generalized bipolar vague closed set. Therefore, A is a generalized bipolar vague open set.
Proposition 4.8: If Bvint(A) B Aand if A is a generalized bipolar vague open set then B is also a generalized bipolar vague open set.
Proof: Now, A B Bv int(A) Bvcl(A). As A is a generalized bipolar vague open, A is generalized bipolar vague closed set. By proposition 4.5, B is a generalized bipolar vague closed set. That is, B is a generalized bipolar vague open set.
Definition 4.9: Let (X , BV ) and (Y , BV ) be any two bipolar vague topological spaces
1. A map : (X , BV ) (Y , BV ) is said to be generalized bipolar vague continuous if the inverse image of every bipolar vague open set in (Y , BV ) is a generalized vague open set in (X , BV ) . 3064 International Journal of Information Research and Review Vol. 03, Issue, 11, pp. 3058-3065, November, 2016
2. A map : (X , BV ) (Y , BV ) is said to be generalized bipolar vague irresolute if the inverse image of every generalized bipolar vague open in (Y , BV ) is a generalized bipolar vague open set in (X , BV ) .
Proposition 4.10: Let (X , BV ) and (Y, BV ) be any two bipolar vague topological spaces. Let : (X , BV ) (Y , BV ) be a generalized bipolar vague continuous function mapping. Then for every bipolar vague set A in X , (GBvcl ( A)) Bvcl ( ( A)) .
Proof: Let A be a bipolar vague set in (X , BV ) . Since Bvcl( (A)) is a bipolar vague closed set and is a generalized bipolar vague continuous mapping, 1 (Bvcl( (A))) is a generalized bipolar vague closed set and 1 (Bvcl( (A))) A . 1 Now, GBvcl(A) (Bvcl( (A))) . Therefore, (GBvcl ( A)) Bvcl ( ( A)) .
Proposition 4.11: Let (X , BV ) and (Y, BV ) be any two bipolar vague topological spaces. Let : (X , BV )
(Y , BV ) be a generalized bipolar vague continuous mapping. Then for every bipolar vague set A in Y , GBvcl( 1 (A)) 1 (Bvcl(A)) .
1 1 Proof: Let A be a bipolar vague set in (Y, BV ) . Let B (A) . Then, (B) ( (A)) A , By proposition 4.10, 1 1 1 1 (GBvcl( (A))) Bvcl( ( (A))) . Thus, GBvcl( (A)) (Bvcl(A)) .
Proposition 4.12: Let (X , BV ) and (Y, BV ) be any two bipolar vague topological spaces. If : (X , BV ) (Y , BV ) is a bipolar vague continuous mapping, then it is a generalized bipolar vague continuous mapping.
1 Proof: Let A be a bipolar vague open set in (Y, BV ) . Since is a bipolar vague continuous mapping, (A) is a bipolar 1 vague open set in (X , BV ) . Every bipolar vague open set is a generalized bipolar vague open set. Now, (A) is a generalized bipolar vague open set in (X , BV ) . Hence, is a generalized bipolar vague continuous mapping.
The converse of the above need not be true as shown in example
a b Example 4.13: Let X {a, b}, Y {u, v} and A x, , , [0.4, 0.7][0.6, 0.3] [0.2, 0.5][0.7,0.4] a b B y, , . [0.5,0.7][0.6, 0.2] [0.4,0.7][0.5,0.3]
Then BV {0, 1, A} and BV {0, 1, B} are BVT on X and Y respectively. Define a mapping : (X , BV ) by an . Then is generalized bipolar vague continuous mapping but not bipolar vague (Y , BV ) (a) u (b) v continuous mapping.
Proposition 4.14: Let (X , BV ) and (Y, BV ) be any two bipolar vague topological spaces. If : (X , BV ) (Y , BV ) is a generalized bipolar vague irresolute mapping, then it is a generalized bipolar vague continuous mapping.
Proof: Let A be a bipolar vague open set in (Y, BV ) . Since every bipolar vague open is generalized bipolar vague open set in 1 (Y, BV ) , but is a generalized bipolar vague irresolute mapping, (A) is a generalized bipolar vague open set in (X , BV ) . Hence, is a generalized bipolar vague continuous mapping.
Proposition 4.15: Let (X , BV ) , (Y, BV ) and (Z, BV ) be any three bipolar vague topological spaces. Let : (X , BV )
(Y , BV ) be a generalized bipolar vague irresolute mapping and 1 : (Y, BV ) (Z, BV ) be a generalized bipolar vague continuous mapping. Then 1 is a generalized bipolar vague continuous mapping.
3065 Cicily Flora and Arockiarani. A new class of generalized bipolar vague sets
1 Proof: Let A be a bipolar vague open set in (Z, BV ) . Since 1 is a generalized bipolar vague continuous mapping, 1 (A) 1 1 is a generalized bipolar vague open set in (Y, BV ) . Since is a generalized bipolar vague irresolute mapping, ( 1 (A)) is a generalized bipolar open set in (X , BV ) . Thus, 1 is a generalized bipolar vague continuous mapping. REFERENCES
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