THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS. Hee Chan Choi


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1 KagweoKyugki Math. Jour. 4 (1996), No. 2, pp THE COMPLETENESS OF CONVERGENT SEQUENCES SPACE OF FUZZY NUMBERS Hee Cha Choi Abstract. I this paper we defie a ew fuzzy metric θ of fuzzy umber sequeces, ad prove that the space of coverget sequeces of fuzzy umbers is a fuzzy complete metric space i the fuzzy metric θ. 1. Itroductio D. Dubois ad H. Prade itroduced the otios of fuzzy umbers ad defied its basic operatios [2]. R. Goetschel, A. Kaufma, M. Gupta ad G. Zhag [37] have doe much work about fuzzy umbers. Let R be the set of all real umbers ad F (R) all fuzzy subsets defied o R. G. Zhag [57] defied the fuzzy umber ã F (R) as follows : (1) ã is ormal, i.e., there exists x R such that ã(x) = 1, (2) for every λ (0, 1], a λ = {x ã(x) λ} is a closed iterval, deoted by [a λ, a+ λ ]. Now, let us deote the set of all fuzzy umbers defied by G. Zhag as F (R). I this paper, we will use Zhag s fuzzy distace ρ of fuzzy umbers [57] as follows : ρ(ã, b) = λ [ a 1 b 1, a η b η a + η b + η ] for ay ã, b F (R), where meas max. The purpose of this paper is to prove that the space of coverget sequeces of fuzzy umbers is a fuzzy complete metric space i some fuzzy metric θ. Received Jue 18, Mathematics Subject Classificatio: 40A05. Key words ad phrases: fuzzy metric(distace), fuzzy complete metric space.
2 118 Hee Cha Choi I sectio 2, we quote basic defiitios ad theorems from [1,5,6] which will be eeded i the proof of mai theorem. I sectio 3, we prove mai theorem: The space of coverget sequeces of fuzzy umbers is a fuzzy complete metric space i a fuzzy metric θ defied by θ(a i, A j ) = λ [ (a i ) 1 (a j) 1, (a i ) η (a j ) η (a i ) + η (a j ) + η ] where A i = {ã i } ad A j = {ã j } are coverget sequeces of fuzzy umbers with respect to the fuzzy metric ρ. 2. Defiitios ad prelimiaries I this sectio, we quote basic defiitios [1,5,6,7] ad theorems, proved i [6,7], which will be eeded i the proof of mai theorem. Let F (R) be the set of all fuzzy subsets defied o R. Defiitio 2.1. Let ã F (R). ã is called a fuzzy umber if ã has the followig properties: (1) ã is ormal, i.e., there exists x R such that ã(x) = 1. (2) For every λ (0, 1], a λ = {x ã(x) λ} is a closed iterval, deoted by [a λ, a+ λ ]. Let F (R) be the set of all fuzzy umbers o the real lie R. By the decompositio theorem of fuzzy sets ã = λ [ a λ, ] a+ λ for every ã F (R). If we defie ã(x) by { 1 for x = k ã(x) = 0 for x k (k R), the k F (R) ad k = λ[k, k].
3 The completeess of coverget sequeces space 119 Defiitio 2.2. Let ã, b, c F (R). We defie as follows: (1) c = ã+ b if c λ = a λ +b λ ad c+ λ = a+ λ +b+ λ for every λ (0, 1]. (2) c = ã b if c λ = a λ b+ λ ad c+ λ = a+ λ b λ for every λ (0, 1]. (3) For every k R ad ã F (R), kã = λ [ ka λ, ] ka+ λ if k 0, = λ [ ka + λ, ka λ ] if k < 0. (4) ã b if a λ b λ ad a+ λ b+ λ for every λ (0, 1]. (5) ã < b if ã b ad there exists λ (0, 1] such that a λ < b λ or a + λ < b+ λ. (6) ã = b if ã b ad b ã. Defiitio 2.3. Let A F (R). (1) If there exists M F (R) such that ã M for every ã A, the A is said to have a upper boud M. (2) If there exists m F (R) such that m ã for every ã A, the A is said to have a lower boud m. (3) A is said to be bouded if A has both upper ad lower bouds. (4) A sequece {ã } F (R) is said to be bouded if the set {ã N} is bouded. Defiitio 2.4. A fuzzy distace ρ of two fuzzy umbers ã, b F (R) is a fuctio ρ : F (R) F (R) F (R) with the properties : (1) ρ(ã, b) 0, ρ(ã, b) = 0 iff ã = b. (2) ρ(ã, b) = ρ( b, ã). (3) Wheever c F (R), we have ρ(ã, b) ρ(ã, c) + ρ( c, b). If ρ is a fuzzy distace of fuzzy umbers, we call (F (R), ρ) a fuzzy metric space. We defie ρ(ã, b) = λ [ a 1 b 1, a η b η a + η b + η ] ( ) for ay ã, b F (R), where meas max.
4 120 Hee Cha Choi Theorem 2.1. [5,6,7] ρ defied by the above equality ( ) is a fuzzy distace of fuzzy umbers, that is, (F (R), ρ) is a fuzzy metric space. Defiitio 2.5. Let {ã } F (R), ã F (R). The the sequece {ã } is said to coverge to ã i fuzzy distace ρ, deoted by ( ρ) lim ã = ã if for ay give ε > 0 there exists a iteger N > 0 such that ρ(ã, ã) < ε for N. A sequece {ã } i F (R) is said to be a Cauchy sequece if for every ε > 0 there exists a iteger N > 0 such that ρ(ã, ã m ) < ε for, m > N. A fuzzy metric space (F (R), ρ) is called the fuzzy complete metric space if every Cauchy sequece i F (R) coverges. Theorem 2.2. [1,7] The sequece {ã } i F (R) is coverget i the metric ρ if ad oly if {ã } is a Cauchy sequece. Theorem 2.3. [7] The fuzzy metric space (F (R), ρ) is complete. 3. Mai theorem I this sectio, we prove that the space of coverget sequeces i F (R) is a fuzzy complete metric space with some fuzzy metric θ. Let C deote the set of all coverget sequeces of fuzzy umbers. MAIN THEOREM. (C, θ) is a fuzzy complete metric space with the fuzzy metric θ defied by θ(a i, A j ) = λ [ (a i ) 1 (a j) 1, (a i ) η (a j ) η (a i ) + η (a j ) + η ] where A i = {ã i } ad A j = {ã j } are coverget sequeces of fuzzy umbers with respect to the fuzzy metric ρ.
5 The completeess of coverget sequeces space 121 Proof. First we shall check that θ is a metric i C. It is easy to see that (i) θ(a i, A j ) 0, θ(ai, A j ) = 0 iff A i = A j, (ii) θ(a i, A j ) = θ(a j, A i ). Sice ρ is a fuzzy metric i F (R), the followig triagle iequality holds : ρ(ã i, ã j ) ρ(ã i, ã k ) + ρ(ã k, ã j ) where A i = {ã i }, A j = {ã j } ad A k = {ã k } are coverget sequeces with respect to ρ i F (R). Thus, we have (a i ) 1 (a j) 1 (a i) 1 (a k) 1 + (a k) 1 (a j) 1, (a i ) η (a j ) η (a i ) + η (a j ) + η (a i ) η (a k ) η (a i ) + η (a k ) + η + (a k ) η (a j ) η (a k ) + η (a j ) + η. Therefore, we have (a i ) 1 (a j) 1 + (a i ) 1 (a k) 1 + (a k ) 1 (a j) 1, (a i ) η (a j ) η (a i ) + η (a j ) + η (a i ) η (a k ) η (a i ) + η (a k ) + η (a k ) η (a j ) η (a k ) + η (a j ) + η. Hece, (iii) the triagle iequality θ(a i, A j ) θ(a i, A k ) + θ(a k, A j ) follows. Cosequetly, by (i), (ii) ad (iii), θ is a fuzzy metric i C. To show that C is complete i the fuzzy metric θ, let {A i } i=1 (where A i = {ã i } =1) be a Cauchy sequece i C. The, for ay ε > 0 there exists a iteger N such that ρ(ã i, ã j ) λ [ (a i ) 1 (a j) 1 (a i ) + η (a j ) + η ] = θ(a i, A j ) < ε 5, (a i ) η (a j ) η (1)
6 122 Hee Cha Choi for i, j > N. Thus, {ã i } i=1 is a Cauchy sequece i F (R) for each fixed. Sice (F (R), ρ) is complete from Theorem 2.3, by Theorem 2.2 ( ρ) lim i ã i = ã (say) for each. Hece, we have lim ρ(ã i, ã ) = 0 for each i lim λ [ (a i ) 1 (a i ) 1, (a i ) η (a ) η (a i ) + η (a ) + η ] = 0 for each lim λ [ (a i ) 1 (a ) 1 i (a i ) + η (a ) + η ] = 0 lim i θ(ai, {ã }) = 0 ( θ) lim i A i = {ã } =1., (a i ) η (a ) η So, the sequece {A i } i=1 coverges to {ã } =1, i.e., ( θ) lim i A i = {ã } =1 (This otatio meas that the sequece {A i } i=1 coverges to the sequece {ã } =1 i the fuzzy metric θ). We shall ow show that {ã } =1 C. I (1), takig the limit as j, we have ρ(ã i, ã ) < ε 5. Sice A i is coverget for each i, A i is a Cauchy sequece. Thus, for give ε > 0 there exists a iteger N i > 0 such that ρ(ã i, ã ik ) < ε 5 for, k > N i, for fixed i. Similarly, for give ε > 0 there exists N j > 0 such that ρ(ã j, ã jk ) < ε 5 for, k > N j, for fixed j. Let we put N = max{n, N i, N j }. The for give ε > 0 there exist ã ik, ã jk F (R) i coectio with (1) such that ρ(ã i, ã ik ) < ε 5, ρ(ã j, ã jk ) < ε 5, ρ(ã i, ã j ) < ε 5 for i, j, k, > N. Hece, we have ρ(ã ik, ã jk ) ρ(ã i, ã j ) + ρ(ã i, ã ik ) + ρ(ã j, ã jk ) ε 5 + ε 5 + ε 5 = 3 5 ε for i, j, k > N. Hece, {ã ik } i=1 is a Cauchy sequece i F (R), by the completeess of F (R), there exists ã k (say) F (R) such that ρ(ã ik, ã k )
7 3 5ε. Therefore, we have The completeess of coverget sequeces space 123 ρ(ã, ã k ) ρ(ã, ã i ) + ρ(ã i, ã ik ) + ρ(ã ik, ã k ) ε 5 + ε ε = ε for, k > N. Sice ε is arbitrary, {ã } is a Cauchy sequece ad hece {ã } coverges. Therefore, sice the sequece {ã i } i=1 has already coverged to the fuzzy umber ã (say) for each N by Theorem 2.2, {ã } C, this proves the completeess of C. Let B deote the set of all bouded sequeces i F (R). Let A i B, A i = {ã i } =1. Sice A i is bouded for each i, there exists a coverget subsequece {ã ik } k=1 of {ã i} =1 icludig ã i, ad hece {ã ik } k=1 is a Cauchy sequece. From this result ad mai theorem, we obtai Corollary 1. (B, θ) is a fuzzy complete metric space with the fuzzy metric θ defied by θ(a i, A j ) = λ [ (a i ) 1 (a j) 1, (a i ) η (a j ) η (a i ) + η (a j ) + η ] where A i = {ã i } =1 ad A j = {ã j } =1 are bouded sequeces i F (R). Corollary 2. (C, θ) (B, θ). Refereces 1. H. Choi, The completio of some fuzzy metric space, KagweoKyugki Math. J. 3 (1995), D. Dubois ad H. Prade, Operatios o fuzzy umbers, Iterat. J. of System Sci. 9 (1978), R. Goetschel ad W. Voxma, Topological properties of fuzzy umbers, Fuzzy Sets ad Systems 10 (1983), A. Kaufma ad M. Gupta, Itroductio to Fuzzy Arithmetic, Va Nostrad Reihold, New York, (1985).
8 124 Hee Cha Choi 5. G. Zhag, Fuzzy distace ad limit of fuzzy umbers, BUSEFAL 33 (1987), , Fuzzy limit theory of fuzzy umbers, Cyberetics ad Systems (1990), World Sci. Pub., , Fuzzy cotiuous fuctio ad its properties, Fuzzy Sets ad Systems 43 (1991), Departmet of Mathematics Kyug Hee Uiversity Yogi , Korea
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