A Note on Quasi-coincidence for Fuzzy Points of Fuzzy Topology on the Basis of Reference Function

Transcription

1 I.J. Mathematcal Scences and Computng, 2016, 3, Publshed Onlne July 2016 n MECS ( DOI: /jmsc Avalable onlne at A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon Kangujam Pryokumar Sngh a, Bhmraj Basumatary b,* a Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, BTC, Assam,783370, Inda b Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, BTC, Assam,783370, Inda Abstract In ths artcle our man am s to revst the defnton of fuzzy pont and fuzzy quas-concdent of fuzzy topology whch s accepted n the lterature of fuzzy set theory. We analyse some results and also prove some proposton wth extended defnton of complementaton of fuzzy sets on the bass of reference functon and some new defntons have also been ntroduced whenever possble. In ths work the man efforts have been made to show that the exstng defnton of complement of fuzzy pont and defnton of fuzzy quas-concdent are not acceptable. Index Terms: Fuzzy Pont, Fuzzy Quas-Concdent, Fuzzy Topology Publshed by MECS Publsher. Selecton and/or peer revew under responsblty of the Research Assocaton of Modern Educaton and Computer Scence 1. Introducton Fuzzy set theory was dscovered by Zadeh [1] n The theory of fuzzy sets actually has been a generalzaton of the classcal theory of sets n the sense that the theory of sets should have been a specal case of the theory of fuzzy sets. But unfortunately t has been accepted that for fuzzy set A and ts complement A C, nether A A C s empty set nor A A C s the unversal set. Whereas the operatons of unon and ntersecton of crsp sets are ndeed specal cases of the correspondng operaton of two fuzzy sets, they end up gvng pecular results whle defnng A A C and A A C. In ths regard Baruah [2, 3] has forwarded an extended defnton of complement of fuzzy sets whch enable us to defne complement of fuzzy sets n a way that gve us A A C s empty and A A C s unversal set. * Correspondng author. Basumatary B E-mal address: brbasumatary14@gmal.com

2 50 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon Chang [4] ntroduced fuzzy topology. After the ntroducton of fuzzy sets and fuzzy topology, several researches were conducted on the generalzatons of the notons of fuzzy sets and fuzzy topology. In fuzzy topology also many results are not same as general topology. It s seen that when we used the exstence defnton of fuzzy set n fuzzy boundary then closure of any fuzzy set s not equal to unon of a fuzzy set and ts boundary. The man reason behnd t s expresson of complement of exstng defnton of fuzzy set. Present author [5] has expressed ncely n ths regard wth extended defnton of fuzzy set and showed that closure of a fuzzy set s equal to unon of a fuzzy set and ts boundary. In ths artcle we apply the extended defnton of complementaton of fuzzy sets on the bass of reference functon to gve defnton of fuzzy pont and quas-concdence of fuzzy topology and try to prove some results on quas-concdence for fuzzy pont so that we can get result more accuracy than before. 2. Related Work To avod dffculty of complement of fuzzy set many new deas were developed. The concept of ntutonstc fuzzy set was ntroduced by Atnassov [6] as a generalzaton of fuzzy set. By observng ths dea n 1997 Coker [7] ntroduced the concept of ntutonstc fuzzy topology. Pu and Lu [8] were dscussed on Fuzzy topology I neghbourhood structure of a fuzzy pont and Moore-Smth convergence. Coker and Demrc [9] were explaned very ncely on ntutonstc fuzzy ponts. In ths artcle we would dscuss fuzzy pont and fuzzy quas concdent on the bass of extended defnton of fuzzy set. 3. Baruah s Defnton of Complementaton of Fuzzy Sets Baruah [2, 3] gave an extended defnton of complementaton of fuzzy set. Accordng to Baruah [2, 3] to defne a fuzzy set, two functons namely fuzzy membershp functon and fuzzy reference functon are necessary. Fuzzy membershp value s the dfference between fuzzy membershp functon and fuzzy reference functon. Let µ 1 (x) and µ 2 (x) be two functons such that 0 µ 2 (x) µ 1 (x) 1. For fuzzy number denoted by {x, µ 1 (x), µ 2 (x); xϵu}, we call µ 1 (x) as fuzzy membershp functon and µ 2 (x) a reference functon such that (µ 1 (x) - µ 2 (x)) s the fuzzy membershp value. 4. Basc Operatons Let A={x, µ 1 (x), µ 2 (x); xϵu} and B={x, µ 3 (x), µ 4 (x); xϵu} be two fuzzy sets defned over the same unverse U. 1. A B ff µ 1 (x) µ 3 (x) and µ 4 (x) µ 2 (x) for all xϵu. 2. A B ={x, max(µ 1 (x), µ 3 (x)), mn(µ 2 (x), µ 4 (x))} for all xϵu. 3. A B ={x, mn(µ 1 (x), µ 3 (x)), max(µ 2 (x), µ 4 (x))} for all xϵu. If for some xϵu, mn(µ 1 (x), µ 3 (x)) max(µ 2 (x), µ 4 (x))}, then our concluson wll be A B= 4. A C ={x, µ 1 (x), µ 2 (x); xϵu} C ={x, µ 2 (x), 0; xϵu} {x, 1, µ 1 (x); xϵu} 5. If D = {x, µ(x), 0; xϵu} then D C ={x, 1, µ(x); xϵu} for all xϵu. we shall dscuss some propostons regardng fuzzy topology consderng the concepts of reference functon, whch are as follows: 4.1. Proposton

3 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon 51 For fuzzy sets A, B, C over the same unverse X, we have the followng proposton 1.1 A B, B C A C 1.2 A B A, A B B 1.3 A A B, B A B 1.4 A B A B=A 1.5 A B A B=B 4.2. Proposton Let τ={a : ϵi} be a collecton of fuzzy sets over the same unverse U. Then A ={x, max(µ 1 ), mn(µ 2 ); xϵu} A ={x, mn(µ 1 ), max(µ 2 ); xϵu} 4.3. Proposton Let τ={a : ϵi} be a collecton of fuzzy sets over the same unverse U. Then 3.1. { 3.2. { A } C = A } C = A C {A } C 4.4. Proposton For a fuzzy set A={x, µ(x), γ (x); xϵu}. (A C ) C =A Proposton For a fuzzy set A 1. A A C = 2. A A C =U Defnton: Let X and Y be two non empty sets and f: X Y be a functon

4 52 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon Let B={y, µ B (y), γ B (y): yϵy} be fuzzy set on Y, then premage of B under f denoted by f -1 (B), s fuzzy set n X defned by f -1 (B) ={x, f -1 ( µ B )(x), f -1 ( γ B )(x): xϵx}, where f -1 ( µ B )(x)=µ B (f(x)) and f -1 ( γ B )(x)=γ B (f(x)). If A={x, µ A (x), γ A (x): xϵx} be fuzzy set n X, then mage of A under f s denoted by f(a) and defned as f( A)(y)= { xϵx, f(x)=y, µ A (x), γ A (x)} Theorem Let f be a functon from X to Y. Then 1. B 1 B 2 f -1 [B 1 ] f -1 [B 2 ], B 1 and B 2 are fuzzy sets n Y. 2. A 1 A 2 f[a 1 ] f[a 2 ], A 1 and A 2 are fuzzy sets n X. 3. B f[f -1 [B]] for any fuzzy subset B n Y. 4. A f -1 [f[a]] for any fuzzy subset A n X. 5. f -1 [ B]= f -1 [B] 6. f -1 [ B]= f -1 [B] 7. f[ A]= f[a] 8. f[ B]= f[b] 4.8. Theorems Let f be a functon from X to Y. Then 1. f -1 [1 U ]=1 U. 2. f -1 [0 U ]=0 U. 3. f -1 [B C ]={f -1 [B]} C for any fuzzy set B n Y. 4. {f[a]} C f[a C ] for any fuzzy set A n X. It s clearly seen that the above theorems and propostons are also true when we use our extended defnton of fuzzy set. usng our extended defnton of fuzzy set we would lke to dscussed on fuzzy pont and fuzzy quasconcdent of fuzzy topology Defnton A fuzzy topology on a nonempty set X s a famly τ of fuzzy set n X satsfyng the followng axoms (T1) 0 X, 1 X τ (T2) G 1 G 2 τ, for any G 1, G 2 τ (T3) G τ, for any arbtrary famly {G : G τ, I}. In ths case the par (X,τ) s called a fuzzy topologcal space and any fuzzy set n τ s known as fuzzy open set n X and clearly every element of τ C s sad to be fuzzy closed set. Example: Let X={a, b}.

5 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon 53 Let M={(a, 0.4, 0), (b, 0.5, 0)}, N={(a, 0.6, 0), (b, 0.8, 0)} Then the famly δ={0 X, 1 X, M, N} s a fuzzy topology Defnton Let X be a non empty set and p be a fxed element of X. Let rϵ(0, 1) and sϵ[0, 1) such that r-s 1, then the fuzzy set p r s(y)={ x, p r (x), p s (x); xϵx} s called fuzzy pont n X, where p r (x)=r, when x=y, otherwse zero, denotes the membershp functon and p s =r, when x=y, otherwse zero, denotes the reference functon. Note: Let A={x, µ A (x), γ A (x); xϵx}. The fuzzy pont p r s={ x, p r (x), p s (x); xϵx} s contaned n A f and only f µ A (x) p r (x) and ) γ A (x) p s (x) Defnton 1. Let A and B are two fuzzy sets n X then A and B are sad to be ntersectng to each other f and only f there exsts a pont xϵx such that A B ф. 2. Also, two fuzzy sets A and B are sad to be equal f and only f pϵa pϵb, for fuzzy pont p n X Propertes Let us consder the famly of fuzzy sets{a : ϵi} n X and P be fuzzy pont on X. Then 1. If Pϵ {A : ϵi}, then for If ϵi we have If PϵA. 2. f(p C )=(f(p)) C It s seen that these propertes are easly verfed f the complementaton s defned on the bass of reference functon Defnton A fuzzy pont p s sad to be quas-concdent wth the fuzzy set A f p A C, denoted by pqa Proposton Let (X, δ) be fuzzy topology. Let A and B be two fuzzy sets then AqB at x BqA at x. Proof Case 1 when reference functon s zero. Let A={x, µ 1 (x), 0; xϵx} and B={x, µ 2 (x), 0; xϵx} Let AqB at x A B C at x (B C ) C A C at x B A C at x Hence, BqA at x Conversely let BqA at x BqA at x B A C at x (A C ) C B C at x A B C at x

6 54 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon Thus AqB at x Case 2 when reference functon s not zero. Let A={x, µ 1 (x), γ 1 (x); xϵx} and B={x, µ 2 (x), γ 2 (x); xϵx} And B C ={x,1, µ 2 (x); xϵx} {x, γ 2 (x), 0; xϵx}. Let AqB at x A B C at x Membershp value of A Membershp value of B C, at x ( µ 1 (x)- γ 1 (x) ) (1- µ 2 (x)) + γ 2 (x) 1-( µ 1 (x)- γ 1 (x) ) 1- (1- µ 2 (x)) + γ 2 (x) 1-( µ 1 (x)- γ 1 (x) ) (µ 2 (x)) - γ 2 (x) Membershp value of A C Membershp value of B BqA at x Conversely let BqA then A C ={x,1, µ 1 (x); xϵx} {x, γ 1 (x), 0; xϵx}. BqA at x B A C at x Membershp value of B Membershp value of A C at x (µ 2 (x) - γ 2 (x)) (1- µ 1 (x)) + γ 1 (x) 1-(µ 2 (x) - γ 2 (x)) 1- (1- µ 1 (x)) + γ 1 (x) 1-(µ 2 (x) - γ 2 (x)) (µ 1 (x)) γ 1 (x) Membershp value of B C Membershp value of A, at x AqB, at x Hence AqB at x BqA at x Proposton Let (X, δ) be fuzzy topology. Let A and B be two fuzzy sets then AqB BqA. Proof we can prove ths proposton by followng prove of the proposton Proposton Let (X, δ) be fuzzy topology. Let A, B and C be fuzzy sets, f A B then CqA CqB. Proof Case 1 when reference functon s zero. Let A={x, µ 1 (x), 0; xϵx}, B={x, µ 2 (x), 0; xϵx} and C={x, µ 3 (x), 0; xϵx}. We have A B so clearly µ 1 (x) µ 2 (x). CqA C A C Snce A B B C A C. So CqA C A C B C C B C CqB Case 2 When reference functon s not zero.

7 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon 55 Let A={x, µ 1 (x), γ 1 (x); xϵx}, B={x, µ 2 (x), γ 2 (x); xϵx} and C={x, µ 3 (x), γ 3 (x); xϵx}. Also, A C ={x, 1, µ 1 (x); xϵx} {x, γ 1 (x), 0; xϵx}, B C ={x,1, µ 2 (x); xϵx} {x, γ 2 (x), 0; xϵx}. CqA C A C Membershp value of C Membershp value of A C (µ 3 (x) - γ 3 (x)) (1- µ 1 (x)) + γ 1 (x) Agan as A B B C A C Membershp value of B C Membershp value of A C (1- µ 2 (x)) + γ 2 (x) (1- µ 1 (x)) + γ 1 (x). Hence CqA (µ 3 (x) - γ 3 (x)) (1- µ 1 (x)) + γ 1 (x) (1- µ 2 (x)) + γ 2 (x) (µ 3 (x) - γ 3 (x)) (1- µ 2 (x)) + γ 2 (x) C B C CqB Therefore when A B then CqA CqB Proposton Let (X, δ) be fuzzy topology. Let A, B fuzzy sets, f A B then pqa pqb. Proof Prove s straghtforward Proposton Let (X, δ) and (Y, Г) be two fuzzy topologcal spaces and let A and B be fuzzy sets. Let f be a functon from X to Y then. Aqf -1 (B) f(a)qb. AqB f(a)q f(b). f -1 (A) q f -1 (B) AqB () Proof We have Aqf -1 (B) A (f -1 (B)) C A (f -1 (B C )) f(a) f(f -1 (B C )) f(a) B C f(a)qb Conversely let f(a)qb. f(a)qb f(a) B C f -1 (f (A)) f -1 ( B C ) A (f -`1 ( B)) C Aqf -1 ( B)

8 56 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon Hence Aqf -1 (B) f(a)qb. () Proof Let AqB A B C f(a)(y)= {A(x); xϵx: f(x)=y} {B C (x); xϵx: f(x)=y}, as A B C =( {B(x); xϵx: f(x)=y}) C =(f(b)) C f(a)q f(b) Hence AqB f(a)q f(b) () Proof The prove s straghtforward followng prove of and. 5. Concluson In ths artcle we attempted extended defnton of complementaton of fuzzy set on the bass of reference functon to gve defnton of fuzzy pont and quas-concdence for fuzzy pont because there are some drawbacks n the exstng defnton of complement of fuzzy sets. In ths artcle the defnton of complement of fuzzy set proposed by Baruah [2, 3] can be seen as a partcular case of what we are gvng. We gve value our defnton of complement of an extended defnton of fuzzy set wth example whch was dscussed very ncely by Baruah [2, 3] and show that ndeed our defnton satsfes all those propertes that complement of a set really does n classcal sense. By observng ths dea we used extended defnton of fuzzy set to defne fuzzy pont and quas concdent for fuzzy pont because complement of fuzzy set plays mportant role to defne fuzzy pont and quas concdent for fuzzy pont. The man purpose of ths artcle s to revst and comment on some results assocated wth the exstng defnton of complementaton of fuzzy sets. These results whch are assocated wth exstng defnton of fuzzy sets are dscussed from the standponts of the new defnton of complementaton of fuzzy sets on the bass of reference functon also some proposton are proved. It s accepted that these new defntons would be able to remove the drawbacks that exst. We have seen that f we used the extended defnton of fuzzy set and complement of an extended defnton of fuzzy set then we arrve at the concluson that the fuzzy sets too follow the set theoretc axoms. We hope that ths work wll help for further work of fuzzy topology. References [1] Zadeh L. A., Fuzzy sets, Informaton and Control, 8, (1965). [2] Baruah H. K., Towards Formng a Feld of Fuzzy Sets, Int. Jr. of Energy, Informaton and Communcatons, Vol. 2, ssue 1, Feb. (2011). [3] Baruah H. K., The Theory of Fuzzy Sets: Belef and Realtes, Int. Jr. of Energy, Informaton and Communcatons, 1-22Vol. 2, ssue 2, May (2011). [4] Chang C. L., Fuzzy Topologcal Space, Journal of Mathematcal Analyss and Applcaton 24, (1968). [5] Basuatary B., Towards Formng the Feld of Fuzzy Closure wth Reference to Fuzzy Boundary, JPMNT,

9 A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon 57 Vol. 4, No.1, pp [6] Atanassov K. T., Intutonstc fuzzy sets, Fuzzy sets and systems, 20, 87-96, (1986). [7] Coker D., An ntroducton to Intutonstc fuzzy topologcal spaces, Fuzzy sets and systems, 88, 81-89, (1997). [8] Pu P. M. and Lu Y. M., Fuzzy Topology I Neghbourhood Structure of a Fuzzy Pont and OORE-Smth Convergence, J.Math.Anal Appl. 76(1980), No.2, [9] Coker D.and Demrc M., On Intutonstc Fuzzy Ponts, Notes IFS 1(1995), No. 2, [10] Sh W. and Lu K., A Fuzzy Topology for Computng the Interor, Boundary and Exteror of Spatal Objects Quanttatvely n GIS, Computers and Geoscences 33 (2007) [11] Baruah H. K., The Randomness-Fuzzness Consstency Prncple, Int. Jr. of Energy, Informaton and Communcatons, Vol. 1, Issue 1, November, (2010). [12] Baruah H. K., In Search of the Root of Fuzzyness: The Measure Theoretc Meanng of Partal Presence, Annals of Fuzzy Mathematcs and Informatcs, Vol. 2 No. 1, (July 2011), pp [13] Lupanez F. G., Quasconcdent for Intutonstc Fuzzy Ponts, Int. J. of Mathematcs and Mathematcal Scences 2005:10(2005) [14] Patl D. B. and Dongre Y. V., A Fuzzy Approach for Text Mnng, I. J. Mathematcal Scences and Computng, 2015, 4, [15] Palanappan N., Fuzzy Topology, CRC Press, Florda, Authors Profles Kangujam Pryokumar Sngh was born at Kakwa Lamdabong, Imphal, Manpur (Inda). At present he s the faculty member of Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, Assam, Inda. He was the former Head, Department of Mathematcal Scences and Dean, Faculty of Scence and Technology, Bodoland Unversty. He publshed one book and more than 35(thrty fve) research papers n nternatonal Journals n the feld of Pure and Appled Mathematcs. He attaned and presented many papers n nternatonal and natonal semnars and conferences n the feld of Mathematcs. Bhmraj Basumatary s a research scholar of Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, BTC, Assam, Inda. He has publshed more than 6(sx) research artcles. How to cte ths paper: Kangujam Pryokumar Sngh, Bhmraj Basumatary,"A Note on Quas-concdence for Fuzzy Ponts of Fuzzy Topology on the Bass of Reference Functon", Internatonal Journal of Mathematcal Scences and Computng(IJMSC), Vol.2, No.3, pp.49-57, 2016.DOI: /jmsc