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Research Article Volume 6 Issue No. 8 Vague Contra Generalized Pre Continuous Mappings Mary Margaret. A1, AROCKIARANI. I2 Nirmala College for Women, Coimbatore, Tamil Nadu, India
Abstract: The purpose of this paper is to introduce and investigate a new class of continuous mapping in vague topological spaces namely vague contra generalized pre continuous mapping, vague generalized pre homeomorphism and vague M -generalized pre homeomorphism. We also investigated some of its properties.
Keywords: Vague topology, vague generalized pre homeomorphism, vague M-generalized pre homeomorphism, vague contra generalized pre continuous mappings.
I. INTRODUCTION tA x,1 f A xis called the “vague value” of in and
The theory of vague sets was first proposed by Gau and is denoted by VA x. Buehrer [6] as an extension of fuzzy set theory and vague sets are regarded as a special case of context-dependent fuzzy sets. Definition 2.2: [2] Let A and B be VSs of the form The basic concepts of vague set theory and its extensions defined by [3,6]. In 1970, Levine [7] initiated the study of A x,tA x,1 f A x / x X and generalized closed sets in topological spaces. The concept of B x,t x,1 f x / x X. Then fuzzy sets was introduced by Zadeh [12] in 1965. The theory B B of fuzzy topology was introduced by C.L.Chang [4] in 1967; several researches were conducted on the generalizations of a) A B if and only if tA x tB x and the notions of fuzzy sets and fuzzy topology. In 1986, the 1 f x1 f x for all x X . concept of intuitionistic fuzzy sets was introduced by A B Atanassov [1] as a generalization of fuzzy sets. b) A B if and only if and B A . c c) A x,f x,1 t x / x X . In this paper we introduce the concept of vague contra A A generalized pre continuous mapping, vague generalized pre d) A B x,tA x tB x,1 f A x 1 fB x / x X homeomorphism and vague M-generalized pre homeomorphism and we also obtain their properties and relations with counter examples. e) A B x,tA x tB x,1 f A x 1 fB x / x X
II. PRELIMINARIES For the sake of simplicity, we shall use the notion Definition 2.1:[2] A vague set A in the universe of discourse X is characterized by two membership functions given by: A x,tA x,1 f A x instead of . i) A true membership function tA : X 0,1 and ii) A false membership function f A : X 0,1. Definition 2.3:[10] A vague topology (VT in short) on X is a family of vague sets (VS in short) in X satisfying the where t A x is lower bound of the grade of membership of following axioms. x derived from the “evidence for x ”, and f A x is a lower bound of the negation of x derived from the “evidence 0, 1 G G ,for any G ,G against ” and tA x f A x 1. Thus the grade of 1 2 1 2 membership of x in the vague set is bounded by a Gi foranyfamily Gi /i J. subinterval t x ,1 f x of [0,1]. This indicates that if A A In this case the pair X , is called a vague topological the actual grade of membership x, then space (VTS in short) and any VS in is known as a vague tA x x 1 f A x. The vague set is written as open set (VOS in short) in .
A x,tA x,1 f A x / x X where the interval
International Journal of Engineering Science and Computing, August 2016 2611 http://ijesc.org/ The complement c of a VOS in a VTS is called a mapping if 1 is a VGPCS in for every VGPCS A X , f A vague closed set (VCS in short) in X . A in .
Definition 2.4:[10] A VS A x,tA,1 f A in a VTS Definition 2.7:[11] Let X , be a VTS. The vague is said to be a generalized pre closure ( vgpclA in short) for any VS A is defined as follows, i) vague semi closed set (VSCS in short) if vintvclA A, vgpclA K / K is a VGPCSin X and A K. If ii) vague pre- closed set (VPCS in short) if is VGPCS, then vgpclA A . vclvintA A, iii) vague -closed set (V CS in short) if Definition 2.8:[10] A VTS X , is said to be a vague pT1/2 vclvintvclA A, space (V pT1/2 in short) if every VGPCS in is a VCS in iv) vague regular closed set (VRCS in short) if . A vclvintA. Definition 2.9:[10] A VTS is said to be a vague Definition 2.5:[11] Let and be any two vague X , Y, gpT1/2 space (V gpT1/2 in short) if every VGPCS in is a topological spaces. A map f : X, Y, is said to be VPCS in .
1 vague continuous (V continuous in short) if f V is Definition 2.10:[5] Let and be any two
vague closed set in for every vague closed set V of topological spaces. A map is said to be
. contra continuous if is closed set in for
vague semi-continuous (VS continuous in short) if every open set in . is vague semi-closed set in for every
Definition 2.11:[8] Let f be a bijective mapping from a vague closed set of . topological space X , into a topological space Y, . vague pre-continuous (VP continuous in short) if Then is said to be generalized homeomorphism if is vague pre-closed set in for every vague closed 1 f and f are generalized continuous mapping. set of .
vague -continuous (V -continuous in short) if Definition 2.12:[9] A map f :X, Y, is said to be is vague -closed set in for every vague generalized pre closed if f V is generalized pre closed set closed set of . in for every closed set in .
vague generalized continuous (VG continuous in short) if is vague generalized closed set in for III. VAGUE HOMEOMORPHISM IN TOPOLOGICAL every vague closed set of . SPACES vague generalized semi-continuous (VGS continuous in Definition 3.1: Let be a bijective mapping from a VTS short) if is vague generalized semi-closed set in into a VTS . Then is said to be for every vague closed set of .
vague -generalized continuous (V G continuous in i) Vague homeomorphism (V homeomorphis m in short) if short)if is vague -generalized closed set in are V continuous mapping. ii) Vague pre homeomorphism (VP homeomorphism in for every vague closed set of . 1 short) if f and f are VP continuous mapping. iii) Vague generalized homeomorphism (VG Definition 2.6:[11] A map is said to be homeomorphism in short) if are VG vague generalized pre irresolute (VGP irresolute in short) continuous mapping.
International Journal of Engineering Science and Computing, August 2016 2612 http://ijesc.org/ Definition 3.2: A map f :X, Y, is said to be are VGP continuous mappings. That is the vague generalized pre closed if f V is vague generalized mapping is a VGP homeomorphism. pre closed set in for every vague closed set in Y, V X , . Example 4.6: Let and IV. VAGUE GENERALIZED PRE HOMEOMORPHISM G1 x,0.2,0.3,0.2,0.4 , . Then and Definition 4.1: A bijection mapping f :X, Y, is G2 x,0.3,0.5,0.1,0.3 called a vague generalized pre homeomorphism (VGP are VTs on and respectively. Define a 1 homeomorphism in short) if f and f are VGP continuous bijection mapping by mapping. Then is a VGP f a u and f b v. Example 4.2: Let X a,b, Y u,v and homeomorphism but not VP homeomorphism since are not a VP continuous mapping. G1 x,0.3,0.5,0.3,0.6 , G x, 0.3,0.4 , 0.2,0.6 . Then and 2 0,G1,1 Theorem 4.7: Let be a VGP are VTs on and respectively. Define a homeomorphism, then is a V homeomorphism if 0,G2 ,1 X Y bijection mapping by f : X, Y, f a u X andY are V pT1/2 space. and f b v. Then f is a VGP continuous mapping and 1 1 Proof: Let B be a VCS in Y . Then f B is a VGPCS in is also a VGP continuous mapping. Therefore is a f VGP homeomorphism. X , by hypothesis. Since X is a V space, is a VCS in . Hence is a V continuous mapping. By Theorem 4.3: Every V homeomorphism is a VGP homeomorphism but not conversely. hypothesis 1 is a VGP continuous f : Y, X, 1 1 Proof: Let be a V homeomorphism. mapping. Let A be a VCS in X . Then f A f A
Then are V continuous mappings. This implies is a VGPCS in , by hypothesis. Since is a V space,
are VGP continuous mappings. That is the f A is a VCS in . Hence is a V continuous mapping f is a VGP homeomorphism. mapping. Therefore the mapping is a V homeomorphism.
Example 4.4: Let and Theorem 4.8: Let be a VGP homeomorphism, then is a VP homeomorphis m if G1 x,0.6,0.7,0.5,0.8 , are V T space. G x, 0.7,0.8 , 0.6,0.7 . Then and gp 1/2 2 are VTs on and respectively. Define a Proof: Let be a VCS in . Then is a VGPCS in bijection mapping by , by hypothesis. Since is a VgpT1/2 space, is a f a u and f b v. Then is a VGP VPCS in . Hence is a VP continuous mapping. By homeomorphism but not V homeomorphism since hypothesis is a VGP continuous are not a V continuous mapping. mapping. Let be a VCS in . Then Theorem 4.5: Every VP homeomorphism is a VGP homeomorphism but not conversely. is a VGPCS in , by hypothesis. Since is a V gpT1/2 space, is a VPCS in . Hence is a VP continuous Proof: Let be a VP homeomorphism. mapping. Therefore the mapping is a VP homeomorphism. Then are VP continuous mappings. This implies
International Journal of Engineering Science and Computing, August 2016 2613 http://ijesc.org/ Theorem 4.9: Let f :X, Y, be a bijective V. VAGUE M-GENERALIZED PRE HOMEOMORPHISM mapping. If f is a VGP continuous mapping, then the following are equivalent. Definition 5.1: A bijection mapping f :X, Y, is called a vague M-generalized pre homeomorphism (VMGP i) is a VGP closed mapping. homeomorphism in short) if are VGP irresolute ii) is a VGP open mapping. mapping.
iii) is a VGP homeomorphism. Example 5.2: Let X a,b, Y u,v and
G1 x,0.5,0.6,0.4,0.7 , Proof: (i) (ii): Let be a bijective G x,0.6,0.7,0.6,0.8 . Then and mapping and let be a VGP closed mapping. This implies 2 are VTs on X and Y respectively. Define a f 1 : Y, X, is a VGP continuous mapping. That bijection mapping by is every VOS in X is a VGPOS in Y . Hence is a VGP open mapping. f a u and f b v. Then f is a VGP irresolute mapping and 1 is also a VGP irresolute mapping. f (ii) (iii): Let be a bijective mapping Therefore is a VMGP homeomorphism. and let be a VGP open mapping. This implies is a VGP continuous mapping. Theorem 5.3: Every VMGP homeomorphis m is a VGP homeomorphism but not conversely. Hence f and f 1 are VGP continuous mapping. That is is a VGP homeomorphis m. Proof: Let be a VMGP homeomorphism. Let B be a VCS in . This implies be (iii) (i): Let is a VGP homeomorphism. That is a VGPCS in . By hypothesis f 1B is a VGPCS in . are VGP continuous mapping. Since every VCS Hence is a VGP continuous mapping. Similarly we can in is a VGPCS in , then is a VGP closed mapping. 1 prove, f is a VGP continuous mapping. Hence Remark 4.10: The composition of two VGP homeomorphism are VGP continuous mappings. This implies is a VGP need not be a VGP homeomorphis m in general. homeomorphism.
Example 4.11:Let X a,b,Y x, yand Z p,q Example 5.4: Let and vague sets G1,G2 and G3 defined as follows: G1 x,0.7,0.8,0.6,0.7 , G1 x,0.2,0.6,0.3,0.5 , G x, 0.4,0.7 , 0.5,0.8 . Then and 2 G2 y,0.7,0.8,0.6,0.7 and are VTs on and respectively. Define a
G z, 0.4,0.8 , 0.5,0.7 . Let , bijection mapping by 3 0,G1,1 and 0,G ,1 be VTs on Then is a VGP 0,G2 ,1 3 X,Y andZ respectively. Define a bijection mapping homeomorphism. Let us consider a VS f : X, Y, by f a x and f b y , B x,0.7,0.8,0.6,0.7 in . Clearly is a defined by 1 g : Y, Z, gx p and f y q VGPCS in . But f Bnot a VGPCS in .That is is not a VGP irresolute mapping. Hence is not VMGP . Then are VGP continuous mappings Also 1 g and g are VGP continuous mappings. Hence f and g homeomorphism. are VGP homeomorphis m But the mapping is not a VGP homeomorphis m g f : X, Z, since g f is not a VGP continuous mapping.
International Journal of Engineering Science and Computing, August 2016 2614 http://ijesc.org/ Theorem 5.5: If the mapping f :X, Y, is a Proof: Since is a VMGP homeomorphis m, f 1 is also a VMGP homeomorphis m, then vgpclf 1B VMGP homeomorphis m. Therefore by Theorem 5.6, 1 1 1 1 1 for every VS in f vpclB for every VS B in Y . vpclf B f vpclB . That is for every VS in Proof: Let be a VS in . Then is a VPCS in . vpclB . This implies vpclB is a VGPCS in . Since the mapping Corollary 5.8: If the mapping is a f is a VGP irresolute mapping, f 1vpclB is a VGPCS VMGP homeomorphism, where are V space, in X . This implies vgpclf 1vpclB f 1 vpclB then vpint f B f vpintB for every VS in . . Now vgpclf 1B
1 1 c c vgpclf vpclB f vpclB.Hence Proof: For every VS in , vpintB vpclB . 1 1 vgpcl f B f vpcl B for every VS in . c c By corollary 5.7, f vpintB f vpclB
c c c c c c Theorem 5.6: If the mapping is a f vpclB vpclf B vpintf B vpint f B . VMGP homeomorphism, where X andY are V gpT1/2 space, 1 1 then vpclf B f vpclB for every VS in . Corollary 5.9: If the mapping is a VMGP homeomorphism, where are V space, Proof: Since is a VMGP homeomorphism, is a VGP
1 1 irresolute mapping. Let be a VS in . Then since then vpintf B f vpintB for every VS in is a VGPCS in , is a VGPCS in .
. Since is a V space, is a VPCS Proof: The proof is obvious. 1 1 in . Now f B f vpclB. We have Remark 5.10: The composition of two VMGP 1 1 1 vpclf B vpclf vpclB f vpclB. homeomorphism is a VMGP homeomorphis m in general. 1 1 This implies vpclf B f vpclB (*). Again, Proof: Let and g : Y, Z, 1 since is VMGP homeomorphis m, f is a VGP irresolute be any two VMGP homeomorphisms. Let A be a VGPCS in 1 1 mapping. Since vpclf B is a VGPCS in , Z . Then by hypothesis, g A is a VGPCS in . Then by 1 1 1 1 1 1 hypothesis, f g A is a VGPCS in . Hence g f f vpclf B f vpclf B is a VGPCS in 1 is a VGP irresolute mapping. Now let be a VGPCS in . . Now B f 1 f 1B Then by hypothesis, f B is a VGPCS in . Then by 1 1 1 1 f vpclf B f vpclf B. Therefore hypothesis g f B is a VGPCS in . This implies 1 1 1 vpclBvpclf vpclf B f vpclf B, g f is a VGP irresolute mapping. Hence is a since is a V space. Hence VMGP homeomorphism. That is the composition of two 1 1 1 1 VMGP homeomorphis m is a VMGP homeomorphis m in f vpclB f f vpclf B vpclf B. general. 1 1 That is f vpclB vpclf B (**). Thus form (*) VI. VAGUE CONTRA GENERALIZED PRE and (**) we get, . CONTINUOUS MAPPINGS
Corollary 5.7: If the mapping is a Definition 6.1: A map f : X, Y, is said to be a VMGP homeomorphism, where are V space, vague contra generalized pre-continuous (VCGP continuous 1 in short) mapping if f A is a VGPCS in X , for then vpcl f B f vpclB for every VS in X . every VOS A in Y, .
International Journal of Engineering Science and Computing, August 2016 2615 http://ijesc.org/ Example 6.2: Let and Proof: Let be a VCP continuous X a,b, Y u,v , mapping. Let be a VOS in . Then is a VPCS in G1 x,0.3,0.5,0.4,0.7 G x, 0.2,0.4 , 0.2,0.5 . Then and . Since every VPCS is a VGPCS, is a VGPCS in 2 0,G1,1 are VTs on and respectively. Define a . Hence, is a VCGP continuous mapping. 0,G2 ,1 X Y mapping f : X, Y, by f a u and Example 6.8: Let and f b v. Then f is a VCGP continuous mapping. G1 x,0.5,0.6,0.4,0.7 , Theorem 6.3: Every VC continuous mapping is a VCGP . Then and G2 x,0.6,0.8,0.5,0.7 continuous mapping but not conversely. are VTs on and respectively. Define a
Proof: Let be a VC continuous mapping by mapping. Let be a VOS in . Then 1 is a VCS in A f A Then is a VCGP continuous mapping but not a VCP . Since every VCS is a VGPCS, f 1A is a VGPCS in continuous mapping.
. Hence, is a VCGP continuous mapping. Theorem 6.9: Let be a mapping. Then
the following statements are equivalent. Example 6.4: Let and i) f is a VCGP continuous mapping. G1 x,0.4,0.7,0.3,0.8 , 1 G x, 0.3,0.5 , 0.2,0.6 . Then and ii) f A is a VGPOS in for every VCS in . 2 are VTs on and respectively. Define a Proof: (i) (ii): Let be a VCS in . Then Ac is a VOS mapping by f a u and 1 c in . By hypothesis, f A is a VGPCS in . Hence f b v. Then is a VCGP continuous mapping but not a is a VGPOS in . VC continuous mapping.
Theorem 6.5: Every VC continuous mapping is a VCGP (ii) (i): Let be a VOS in . Then is a VCS in . continuous mapping but not conversely. By hypothesis, is a VGPOS in . Hence is a VGPCS in . Thus is a VCGP continuous mapping. Proof: Let be a VC continuous mapping. Let be a VOS in . Then is a V CS Theorem 6.10: Let be a mapping. in . Since every V CS is a VGPCS, is a VGPCS Suppose that one of the following properties hold: in . Hence, is a VCGP continuous mapping. i) f vpclA vint f A for each VS in X . 1 1 Example 6.6: Let and ii) vpclf B f vintB for each VS B in Y . 1 1 iii) f vcl B vpint f B for each VS in . G1 x,0.2,0.6,0.3,0.7 ,
G x, 0.4,0.5 , 0.6,0.8 . Then and 2 Then is a VCGP continuous mapping. are VTs on and respectively. Define a 1 Proof: (i) (ii): Let B be a VS in . Then f B is a mapping by f a uand f b v. VS in . By hypothesis, we have Then is a VCGP continuous mapping but not a VC 1 1 continuous mapping. f vpclf B vintf f B vintB . Now 1 1 1 1 vpcl f B f f vpcl f B f vintB . Theorem 6.7: Every VCP continuous mapping is a VCGP continuous mapping but not conversely. (ii) (i): By taking the complement in (ii).
International Journal of Engineering Science and Computing, August 2016 2616 http://ijesc.org/ Suppose that (iii) holds. Let B be a VCS in Y . Then Theorem 6.12: Let be a bijective 1 vclB B . By our assumption f B mapping. Then is a VCGP continuous mapping if 1 1 1 f vclB vpintf B. But vpintf B vcl f A f vpintA for every VS in . f 1 B, hence vpintf 1B f 1B. This implies Proof: Let be a VCS in Y . Then vclA A and f 1B is a VPOS in X and hence is a VGPOS in f 1A is a VS in . By hypothesis . Thus f is a VCGP continuous mapping. vclf f 1A f vpintf 1A. Since is onto, Theorem 6.11: Let f : X, Y, be a bijective f f 1A A. Therefore A vclA mapping. Suppose that one of the following properties hold: vclf f 1 A f vpintf 1 A. Now f 1A 1 1 1 1 1 1 i) f vclB vintvpclf B for each VS B f f vpintf A vpintf A f A . in . Hence is a VPOS in and hence VGPOS in . 1 1 ii) vclvpintf B f vintB for each VS Thus is a VCGP continuous mapping. in . Theorem 6.13: If is a VCGP iii) f vclvpintA vint f A for each VS A in
X . continuous mapping, where X is a V gpT1/2 space, then the iv) f vclA vint f A for each VPOS A in . following conditions hold:
1 1 Then is a VCGP continuous mapping. i) vpclf B f vintvpclB for every VOS in . Proof:(i) (ii):Is obvious, by taking the complement in (i). ii) f 1vclvpintB vpintf 1B for every VCS (ii) (iii): Let A be a VS in X . Put B f A in . in . 1 1 This implies A f f A f B in . Now 1 Proof: i) Let be a VOS in . By hypothesis f B is a 1 1 vclvpintA vclvpintf B f vintB by 1 VGPCS in . Since is a V space, f B is a hypothesis. Therefore f vclvpintA 1 1 1 VPCS in . This implies vpclf B f B f f vintB vintB vint f A. 1 1 f vintB f vintvpclB. (iii) (iv): Let be a VPOS in . Then vpintA A. ii) Can be proved easily by taking complement in (i). By hypothesis, f vclvpintA vint f A, Therefore f vclA f vclvpintA vint f A. Theorem 6.14: i) If be a VCGP continuous mapping and g : Y, Z, is a V 1 Suppose (iv) holds: Let be a VOS in Y . Then f A is continuous mapping, then is a g f : X, Z, 1 a VOS in and vpintf A is a VPOS in . Hence, VCGP continuous mapping. 1 by hypothesis, f vclvpintf A ii) If be a VCGP continuous mapping 1 1 vintf vpintf A vintf f A and is a VC continuous mapping, then 1 . Therefore vintA A vclvpintf A is a VGP continuous mapping. 1 1 1 f f vclvpintf A f A . Now, 1 1 1 iii) If be a VGP irresolute mapping and vclvintf A vclvpintf A f A. This is a VCGP continuous mapping, then implies is a VPCS in and hence a VGPCS in . is a VCGP continuous mapping. Thus is a VCGP continuous mapping.
International Journal of Engineering Science and Computing, August 2016 2617 http://ijesc.org/ 1 1 1 1 Proof:i) Let A be VOS in Z . Then g A is a VOS in Y , f vpclB vpintf B f B.This implies by hypothesis. Since f is a VCGP continuous mapping, is a VPOS in and hence a VGPOS in . Hence 1 1 f g A is a VGPCS in X . Hence g f is a VCGP is a VCGP continuous mapping. continuous mapping. Theorem 6.17: A vague continuous mapping ii) Let be VOS in . Then is a VCS in , by is a VCGP continuous mapping if hypothesis. Since is a VCGP continuous mapping, VGPO( ) = VGPC( ).
is a VGPOS in . Hence is a VGP 1 Proof: Let A be a VOS in . By hypothesis, f A is a continuous mapping. VOS in and hence is a VGPOS in . Since VGPO( ) = VGPC( ), is a VGPCS in . Therefore iii) Let be VOS in . Then is a VGPCS in , by hypothesis. Since is a VGP irresolute mapping, is a VCGP continuous mapping. is a VGPCS in . Hence is a VCGP REFERENCES continuous mapping. [1]. Atanassov K.T, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96. Theorem 6.15: A mapping f : X, Y, is a VCGP 1 1 [2]. Biswas. R., Vague groups, Internat. J. Comput., Cognition continuous mapping if f vpclB vintf B for 4, No. 2 (2006), 20-23. every VS in . B Y [3]. Bustince. H and Burillo. P, Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems, 79 (1996), 403-405. Proof: Let be a VCS in . Then vclB B . Since [4]. Chang. C.L, Fuzzy topological spaces, J.Math.Anal.Appl., every VCS is a VPCS, this implies vpclB B . Now by 24 (1968), 182-190. hypothesis, 1 1 1 1 [5]. Dontchev. J, Contra continuous function and strongly S- f B f vpclB vint f B f B. This closed spaces, Internt. Math. Sci. 19 (1996), 15-31. 1 implies f B is a VOS in . Therefore f is a VC [6]. Gau. W.L and Buehrer. D.J, Vague sets, IEEE Trans, continuous mapping, since every VC continuous mapping is a Systems Man and Cybernet, 23 (2) (1993), 610-614. VCGP continuous mapping, is a VCGP continuous [7]. Levine. N, Generalized closed sets in topological spaces, mapping. Rend. Circ. Mat. Palermo., 19 (1970), 89–96.
Theorem 6.16: A mapping is a VCGP [8]. Maki. H, Sundram. P and Balachandram. K, On generalized homeomorphis ms in topological spaces, Bull. continuous mapping, where is a V T space if and gp 1/2 Fukuoka Univ. Ed. Part III, 40(1991), 13-21. 1 1 only if f vpcl B vpint f vcl B for every VS [9]. Maki. H, Umehara. J, Noiri. T, Every topological space is in . pre T1/2 mem Fac sci, Kochi Univ, Math, 17 (1996), 33- 42. Proof: Necessity: Let be a VS in . Then vclB is a 1 [10]. Mary Margaret. A and Arockiarani. I, Generalized pre- VCS in . By hypothesis f vcl B is a VGPOS in . closed sets in vague topological spaces, International 1 Since is a V space, f vclB is a VPOS in . Journal of Applied Research, 2(7), (2016), 893-900. 1 1 Therefore f vpclB f vclB [11].Mary Margaret. A and Arockiarani. I, Vague generalized pre continuous mappings, International Journal of 1 vpintf vclB . Multidisciplinary Research and Development, 3(8), (2016), 60-70.
Sufficiency: Let be a VCS in . Then . By [12]. Zadeh. L.A, Fuzzy Sets, Information and Control, 8 1 1 (1965), 338-353. hypothesis, f vpclB vpintf vclB 1 1 vpintf B. But vpclB B . Therefore f B
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