Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios, School of Applies Scieces, Uiversity of Techology, Bagdad, Iraq How to cite this paper: Kider, JR ad Mousa, JI (206 Properties of Fuzzy Legth o Fuzzy Set Ope Access Library Joural, 3: e3068 http://dxdoiorg/04236/oalib03068 Received: September 5, 206 Accepted: November 3, 206 Published: November 7, 206 Copyright 206 by authors ad Ope Access Library Ic This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY 40 http://creativecommosorg/liceses/by/40/ Ope Access Abstract The defiitio of fuzzy legth space o fuzzy set i this research was itroduced after the studies ad discussio of may properties of this space were proved, ad the a example to illustrate this otio was give Also the defiitio of fuzzy covergece, fuzzy bouded fuzzy set, ad fuzzy dese fuzzy set space was itroduced, ad the the defiitio of fuzzy cotiuous operator was itroduced Subject Areas Fuzzy Mathematics Keywords Fuzzy Legth Space o Fuzzy Set, Fuzzy Covergece, Fuzzy Cauchy Sequece of Fuzzy Poit, Fuzzy Bouded Fuzzy Set ad Fuzzy Cotiuous Operator Itroductio Zadeh i 965 [] itroduced the theory of fuzzy sets May authors have itroduced the otio of fuzzy orm i differet ways [2]-[9] Cheg ad Mordeso i 994 [0] defied fuzzy orm o a liear space whose associated fuzzy metric is of Kramosil ad Mickalek type [] as follows: The order pair ( X, N is said to be a fuzzy ormed space if X is a liear space ad N is a fuzzy set o X [ 0, satisfyig the followig coditios for every xy, X ad st, [ 0, (i N( x,0 = 0, for all x X (ii For all t > 0, N( xt, = if ad oly if x = 0 t (iii N( xt, = N x,, for all 0 ad for all t > 0 (iv For all st>, 0, N( x+ y, t + s N( x, t N( y, s where a b = mi { ab, (v lim t N( xt, = DOI: 04236/oalib03068 November 7, 206
J R Kider, J I Mousa The defiitio of cotiuous t-orm was itroduced by George ad Veeramai i [2] Bag ad Samata i [2] modified the defiitio of Cheg ad Mordeso of fuzzy orm as follows: The triple ( X, N, is said to be a fuzzy ormed space if X is a liear space, is a cotiuous t-orm ad N is a fuzzy set o X [ 0, satisfyig the followig coditios for every xy, X ad st, [ 0, (i N( x,0 = 0, for all x X (ii For all t > 0, N( xt, = if ad oly if x = 0 t (iii N( xt, = N x, for all 0 (iv For all st>, 0, N( x, t N( y, s N( x+ y, t + s (v For x 0, N( x, : ( 0, [ 0,] is cotiuous (vi lim t N( xt, = The defiitio of fuzzy legth space is itroduced i this research as a modificatio of the otio of fuzzy ormed space due to Bag ad Samata I Sectio, we recall basic cocepts of fuzzy set ad the defiitio of cotiuous t-orm The i Sectio 2 we defie the fuzzy legth space o fuzzy set after we give a example; the we prove that every ordiary orm iduces a fuzzy legth, ad also the defiitio of fuzzy ope fuzzy ball, fuzzy coverget sequece, fuzzy ope fuzzy set, fuzzy Cauchy sequece, ad fuzzy bouded fuzzy set is itroduced I Sectio 3, we prove other properties of fuzzy legth space Fially i Sectio 4, we defie a fuzzy cotiuous operator betwee two fuzzy legth spaces Also we prove several properties for fuzzy cotiuous operator 2 Basic Cocept about Fuzzy Set Defiitio 2: Let X be a classical set of object, called the uiversal set, whose geeric elemets are deoted by x The membership i a classical subject A of X is ofte viewed as a characteritic fuctio µ A from X oto {0, such that µ A = if x A ad µ A = 0 if x A, {0, is called a valuatio set If a valuatio set is allowed to be real iterval [0, ] the A is called a fuzzy set which is deoted i this case by A ad µ A is the grade of membership of x i A Also, it is remarkable that the closer the value of µ ( x to, the more belog to A Clearly, A is a subset of X that has o sharp boudary The fuzzy set A is completely characterized by the set of pairs: = x, µ x : x X,0 µ x [] {( ( ( Defiitio 22: Suppose that D ad B are two fuzzy sets i Y The (i B µ ( y µ B ( y y Y (ii = B µ ( y = µ B ( y y Y (iii C = D B µ C ( y = µ ( y µ B ( y y Y E = B µ y = µ y µ y y Y (iv ( ( ( E D B OALib Joural 2/3
J R Kider, J I Mousa x = c D y y Y [4] Defiitio 23: Suppose that D ad B be two fuzzy sets i V ad W respectively the B is a fuzzy set whose membership is defied by: (v µ ( µ ( (, ( ( (, µ ab = µ a µ b ab V W B B [8] Defiitio 24: A fuzzy poit p i Y is a fuzzy set with sigle elemet ad is deoted by x or ( x, Two fuzzy poits x ad y are said to be differet if ad oly if x y [] Defiitio 25: Suppose that d is a fuzzy poit ad D is a fuzzy set i Y the d is said to belogs to D which is writte by d if µ ( x > [2] Defiitio 26: Suppose that h is a fuctio from the set V ito the set W Let D be a fuzzy set i W the h ( is a fuzzy set i V its membership is: µ d µ h d h E ( ( ( ( = h D D is a fuzzy set i W its membership is give by: for all d i V Also whe E is a fuzzy set i V the ( { µ µ : = E whe f ( y ad µ y = otherwise [5] ( ( 0 h E ( ( ( ( h E y d d f y Propositio 27: Suppose that h: V W is a fuctio The the image of the fuzzy poit d i V, is the fuzzy poit h( d i W with h( d = ( h( d, [] Defiitio 28: 2 A biary operatio : [ 0,] [ 0,] is said to be t-orm (or cotiuous triagular orm if pqtr,,, [ 0,] the coditios are satisfied: (i p q= p q (ii p = p (iii ( p q t = p ( q t (iv If p q ad t r the p t q r [2] Examples 29: Whe p q= p q ad p q= p q pq, [ 0,] the is a cotiuous t- orm [0] Remark 20: p> q, there is t such that p t q ad for every r, there is e such that r r e, where p, q, t, r ad e belogs to [0, ] [2] Defiitio 2: Let D be a fuzzy set i Z ad let τ be a collectio of all subset fuzzy set i D the ( D, τ is called a fuzzy topological space o the fuzzy set if (i D, φ (ii E i i for ay E = i, i =, 2,, (ii E i i, for = { E i : E i τ, i =, 2,, [0] Propositio 22: OALib Joural 3/3
Let T : be a arbitrary operator ad E A ad C The T E C E T C [9] ( if ad oly if ( 3 O Fuzzy Legth Space First we itroduce the mai defiitio i this paper Defiitio 3: J R Kider, J I Mousa Let X be a liear space over field ad let A be a fuzzy set i X let be a t- orm ad F be a fuzzy set from A to [0,] such that: (FL ( x > 0 for all x A x = if ad oly if 0 (FL 2 ( (FL 3 ( x = cx, = x, where 0 c c (FL 4 ( x + y ( x ( y (FL 5 F is a cotiuous fuzzy set for all x, y A ad, [ 0,] AF,, is called a fuzzy legth space o the fuzzy set The the triple ( Defiitio 32: Suppose that ( DF,, is a fuzzy legth space o the fuzzy set the is co- x, i D the F ( x, F x that tiuous fuzzy set if wheever {( is lim (, ( F x = F x x { ( Propositio 33: Let ( Y, be a ormed space, suppose that D be a fuzzy set i Y Put x = x The ( D,, is a fuzzy ormed space Let x, y D ad 0 c the x > 0 for all x D 2 x = 0 x = 0 x = 0 x = 0 3 ( cx, = cx = c x = c x 4 x + y = x+ y x + y = x + y Hece ( D,, is a fuzzy ormed space Example 34: Suppose that (, = for all pq, [ 0,] Defie F ( x p q p q The ( DF,, Y is a ormed space ad assume that D is a fuzzy set i Y Put = + x is a fuzzy legth space o the fuzzy set D, is called the fuzzy legth iduced by To prove ( DF,, is a fuzzy legth space o the fuzzy set D we must prove the five coditios of Defiitio 3: (FL Sice x > 0 for all x X so ( x > 0 for all x D, where > 0 (FL 2 It is clear that ( x = for each > 0 if ad oly if x = 0 (FL 3 If 0 c the for each x X, OALib Joural 4/3
J R Kider, J I Mousa F ( cx, = = = F x, + cx + c x c for each > 0 λ λ λ λ ( x, y = ( x+ y, λ = λ + x+ y λ + x + y λ + x λ + y (FL 4 > = ( x ( y + x + y, where λ mi {, (FL 5 F ( each x, y A x Hece ( DF,, = is cotiuous sice x is a cotiuous fuctio is a fuzzy legth space o the fuzzy set D Defiitio 35: Let A be a fuzzy set i X, ad assume that ( AF,, fuzzy set A Let B ( x, r = y : F ( y x > ( p So B ( x, p { for is a fuzzy legth space o the is said to be a fuzzy ope fuzzy ball of ceter x A ad radius r We omitted the proof of the ext result sice it is clear Propositio 36: I the fuzzy legth space ( AF,, o the fuzzy set, Let B ( x, r ad B ( x, r2 with x ad r, r2 ( 0, The either B ( x, r B ( x, r2 or B ( x, r2 B ( x, r Defiitio 37: The sequece {( x, i a fuzzy legth space ( AF,, o the fuzzy set is fuzzy coverges to a fuzzy poit x A if for a give ε, 0< ε <, the there exists a positive umber K such that ( x, x > ( ε K Defiitio 38: The sequece {( x, i a fuzzy legth space ( AF,, o the fuzzy set is fuzzy coverges to a fuzzy poit x A if lim ( x, x = Theorem 39: The two Defiitios 38 ad 37 are equivalet Let the sequece {( x, is fuzzy coverges to a fuzzy poit x A, the for a give 0< ε < the there is a umber K with ( x, x > ( r for all K, ad hece F ( x, x < r Therefore F ( x, x whe To prove the coverse, let F ( x, x whe Hece whe 0< ε <, there is K such that F ( x, x < r K So F ( x, x > ( r K Therefore { ( x, x by Defiitio 37 Lemma 30: Suppose that ( AF,, is a fuzzy legth space o the fuzzy set The ( x y = ( y x for ay x, y A Let x, y A x y = x y, t where t =, the ( (( OALib Joural 5/3
J R Kider, J I Mousa t = ( ( y x, t = ( y x, = ( ( y x, t = ( y x Lemma 3: AF,, is a fuzzy legth space The, If ( (a The operator (, (b The operator ( r, x x y x + y is cotiuous rx is cotiuous If {( x, x ad {( y, y as the ( ( x, + y, ( x + y ( x, x ( y, y = = Hece the operator additio is cotiuous fuctio Now if {( x, x, ad r r ad r 0 the lim r ( x, rx = lim ( rx, rx = lim ( rx, rx + rx rx = lim r( x x, + x ( r r lim x x, lim x, = r r r Ad this proves (b Defiitio 32: Suppose that ( AF,, is a fuzzy legth space ad the D is called fuzzy ope if for every y D there is B ( y, q A subset E A is called fuzzy c closed if E = E is fuzzy ope The proof of the followig theorem is easy ad so is omitted Theorem 33: Ay B ( y, q i a fuzzy legth space ( AF,, is a fuzzy ope Defiitio 34: AF,, is a fuzzy legth space, ad assume that D A The the Suppose that ( fuzzy closure of D is deoted by D or FC ( D ad is defied by D is the smallest fuzzy closed fuzzy set that cotais D Defiitio 35: Suppose that ( AF,, is a fuzzy legth space, ad assume that The D is said to be fuzzy dese i A if D = A FC D = A or ( Lemma 36: Suppose that ( AF,, is a fuzzy legth space, ad assume that, The d D if ad oly if we ca fid {( d, i D such that ( d, d Let d D, if d D the we take the sequece of fuzzy poits of the type d, d,, d If d D d, D as (, we costruct the sequece of fuzzy poits ( OALib Joural 6/3
J R Kider, J I Mousa follows: ( d, d > for each The fuzzy ball B d, cotais ( d, D ad ( d, d sice lim ( d, d = Coversely assume that ( d, D ad ( d, d the d D or the fuzzy ope fuzzy ball of d cotais ( d, with ( d, d, so d is a fuzzy limit fuzzy poit of D Hece d D 4 Other Properties of Fuzzy Legth o Fuzzy Set Theorem 4: Suppose that ( AF,, is a fuzzy legth space ad let that, the D is fuzzy dese i A if ad oly if for ay a A we ca fid d D with F a d ( ε for some 0 < ε < Let D be a fuzzy dese i A, ad a A so a D the usig 36 we ca fid {( d, D with F ( d, a > ( ε for all K Take d = ( d k, k, d a > ( ε To prove the coverse, we must prove that A D Let a A the there is ( d j, j D such that F ( d j, j a > j, where j Now take 0< ε < such that < ε for each j K Hece we have a sequece of fuzzy poits j {( d j, j D such that F ( d j, j a > > ( ε j for all j K that is ( d j, j so a D a Defiitio 42: Suppose that ( AF,, is a fuzzy legth space A sequece of fuzzy poits {( x, is said to be a fuzzy Cauchy if for ay give ε,0< ε <, there is a positive umber K such that F ( x, ( xm, m > ( ε for all m, K Theorem 43: If {( x, is a sequece i a fuzzy legth space ( AF,, with ( x, x the {( x, is fuzzy Cauchy If {( x, is a sequece i a fuzzy legth space ( AF,, with ( x, x So for ay ε,0< ε < there is a iteger K such that ( x, x > ( ε for all K Now by Remark 20 there is ( r ( 0, such that ( ε ( ε > ( r Now for each m, j K, we have ( m, m ( j, j ( m, m ( j, j x x x x x x > ( ε ( ε ( r OALib Joural 7/3
J R Kider, J I Mousa Hece {(, x is fuzzy Cauchy Defiitio 44: Suppose that {( x, is a sequece i a fuzzy legth space ( AF,, let ( be a k sequece of positive umbers such that < 2 < 3 < < k <, the the sequece, x, {( k k x of fuzzy poits is said to be a subsequece of {( Theorem 45: Suppose that {(, AF,, A the every subsequece {( xk, k of {(, coverges to x coverges to x x is a sequece i a fuzzy legth space ( Let {(, AF,, the lim F ( x, x = But { (, hece, lim F ( x, ( xm, m = as ad m Now (, (, (, (, x be a sequece of fuzzy poits i (, if it is fuzzy x is fuzzy fuzzy coverges to x x is fuzzy Cauchy by Theorem (43 xk k x xk k x x x Takig the limit to both sides as we get ( ( ( ( lim xk, k x lim xk, k x, lim x, x = Hece, {( xk, k fuzzy coverges to x, sice {( k, k subsequece of {( x, Therefore all subsequece {( xk, k of {(, fuzzy coverges to x Propositio 46: x was a arbitrary Let {( x, be a fuzzy Cauchy sequece i a fuzzy legth space ( AF,, tais a ( x, such that ( x, x, the {(, x { k k { k k x is co- x fuzzy coverges to Assume that {( x, is a fuzzy Cauchy sequece i ( AF,, < < there is a iteger K such that F( xm, m ( xj, j > ( ε K x, be a subsequece of {(, x, 0 ε m, j Let {( j j follows that ( j j So for all ε, wheever x ad ( j j It x, x wheever j Sice ( j is icreasig sequece of positive itegers Now (, (, (, (, m m m m > x ( x, ( ε x x x x x x Lettig m, we have Fx ( x, ( ε = ( ε Hece, the sequece {( x, it fuzzy coverges to x m m Defiitio 47: Suppose that ( AF,, is a fuzzy legth space ad The D is said to be fuzzy bouded if we ca fid q, 0 q F x > q x A < < such that, ( ( x OALib Joural 8/3
J R Kider, J I Mousa Lemma 48: is a fuzzy legth space If a sequece { ( x, with x the it is fuzzy bouded ad its fuzzy limit is uique Let ( x, x, that is for a give r > 0 the we ca fid K with F ( x, x > ( r for all K Put t = mi { F ( x, x, F ( x2, 2 x,, F ( xk, K x Usig Remark 20 we ca fid 0< ε < with t ( r > ( ε Now for K ( x, x ( x, ( xk, K ( xk, K x t r > ε Assume that ( AF,, ( x, ( ( Hece ( x, is fuzzy bouded i ( AF,, Suppose that ( x, x ad ( x, y F ( x x = ad ( (, (, Therefore lim, lim F x, y = Now F x y F x x F x y By takig the limit to both sides, as, F x y = So F x y =, hece x y = Defiitio 49: Suppose that ( AF,, is a fuzzy legth space o the fuzzy set The B [ x, q] = { y : x y ( q is said to be a fuzzy closed fuzzy ball with ceter x A ad radius q, 0< q < The proof of the followig lemma is clear ad hece is omitted Lemma 40: Ay B [ x, q] i a fuzzy legth space ( AF,, is a fuzzy closed fuzzy set Theorem 4: A fuzzy legth space ( AF,, is a fuzzy topological space Suppose that ( AF,, is a fuzzy legth space Put τ = { : x D if ad oly if there is 0< r < such that B ( x, r D We prove that τ F is a fuzzy topology o A (i Clear that φ ad A F (ii let, 2,, F ad put V = j= j We will show that V F, let a V the a D j for each j Hece there is 0 q j such that B ( a, qj D Put { i q = mi qj : j so q qj for all j, this implies that ( q ( qj for all j Therefore, B ( a, qj j j = V, thus = V (iii Let { D j : j I put F U = We will show that U j I j F Let b U, the b D j for some j I, sice D j F the there exists 0 < q < such that B ( b, p D, Hece B ( b j, p D D U j j I j = This prove that U F Hece, ( AF,, is a fuzzy topological space τ F is called the fuzzy topology iduced by F Defiitio 42: A fuzzy legth space ( AF,, is said to be a fuzzy Hausdorff space if for ay x, y A such that x y we ca fid B( x, r B y, t for some 0 < r < ad ( OALib Joural 9/3
J R Kider, J I Mousa ad 0 t = < < such that B( x, r B( y, t Theorem 43: Every fuzzy legth space ( AF,, is a fuzzy Hausdorff space Assume that ( AF,, is a fuzzy legth space ad let x,? y A with x y Put F x y = p, for some 0 < p < The for each s, 0< s <, we ca fid q such that q q s by Remark 20 Now cosider the two fuzzy ope fuzzy balls B ( x, q ad B ( y, q The B ( x, q B ( y, q = Sice if there z B x, q B y,q The exists δ ( ( [ ] p = x y x zδ zδ y q q s> p is a cotradictio, hece ( AF,, is a fuzzy Hausdorff space 5 Fuzzy Cotiuous Operators o Fuzzy Legth Spaces I this sectio, we will suppose A is fuzzy set of X ad D be fuzzy set of Y where X ad Y are vector space Defiitio 5: Let ( AF,, ad ( DF,, be two fuzzy legth space o fuzzy set ad respectively, let E A the The operator T : E D is said to be fuzzy cotiuous at a E, if for every 0< ε < there exist 0< δ < such that T( x T( a ( > ε wheever x E satisfyig ( x a ( > δ If T is fuzzy cotiuous at every fuzzy poit of E, the T it is said to be fuzzy cotiuous o E Theorem 52: Let ( AF,, ad ( DF,, be two fuzzy legth space, let E The operator T : E D is fuzzy cotiuous at a E if ad oly if wheever a sequece of fuzzy poits {( x, i E fuzzy coverge to a, the the sequece of fuzzy poits, T a {( T( x fuzzy coverges to ( Suppose that the operator T : E D is fuzzy cotiuous at a E ad let {( x, be a sequece i E fuzzy coverge to a Let 0< ε < be give By fuzzy cotiuity of T at a, the there exists 0< δ < such that wheever x E ad F ( x a ( > δ, implies F T ( x T ( a ( > ε Sice ( x, the we ca fid K with K such that ( x, a ( > δ Therefore whe K implies T( x, T( a ( > ε Thus T( x, T( a Coversely, assume that every sequece {( x, i E fuzzy covergig to a has the property that T( x, T( a Suppose that T is ot fuzzy cotiuous at a The there is 0< ε < ad for which o δ, 0< δ < ca satisfy the requiremet that x E ad x a ( > δ implies T( x T( a ( > ε This meas that for every δ,0< δ < there exists x E such that ( but F T( x T( a ( F x a > δ D ε For every K, there exist OALib Joural 0/3
J R Kider, J I Mousa ( x, E such that ( F x, a A, but T x does ot fuzzy coverge to ( a This cotradicts the as- F T( x, T( a ( ε The sequece { (, the sequece { (, sumptio that every sequece {(, property that T( x, ( x fuzzy coverges to a but x i E fuzzy covergig to a has the T a Therefore the assumptio that T is ot fuzzy cotiuous at a must be false Defiitio 53: ad let Let ( AF,, ad ( DF,, be two fuzzy legth spaces let E A : E D the fuzzy poit a is a fuzzy limit of E writte by ( T lim x a T x b = wheever b D if for every 0< ε < there is 0< δ < such that T( x b ( > ε whe x E ad x a ( > δ Propositio 54: Let ( AF,, ad ( DF,, be two fuzzy legth spaces, let E ad T : E D Suppose that a is fuzzy limit fuzzy poit of E The lim x a T( x b = if ad oly if for every sequece {(, ( x, ad ( x, a a x i E such that the argumet is similar that of Theorem (53 ad therefore is ot icluded Propositio 55: A operator T of the fuzzy legth space ( AF,, ito a fuzzy legth space ( DF,, is fuzzy cotiuous at the fuzzy poit a if ad oly if for every 0< ε < there exist 0< δ < such that B ( a, δ T B ( a, ε the operator T : is fuzzy cotiuous at a A if ad oly if for every 0< ε < there exist 0< δ < such that T( x T( a ( > ε for all x satisfyig F x a ( > δ ie x B ( a, δ implies T( x BT ( ( a, ε or TBa (, δ BT ( ( a, ε this is equivalet to the coditio Ba (, δ T BT ( ( a, ε Theorem 56: A operator T : A D is fuzzy cotiuous if ad oly if T ( G A for all fuzzy ope G of D where ( AF,, ad ( DF,, space is fuzzy ope i are fuzzy legth Suppose that T is fuzzy cotiuous operator ad let G be fuzzy ope i D we T G is fuzzy ope i A Sice ad A are fuzzy ope, we may T G T G show that ( suppose that ( ad ( Let x T ( G the T( x G Sice G is fuzzy ope, the there is 0< ε < such that BT ( ( x, ε G, sice T is fuzzy cotiuous at x by propositio 45 for this ε there exist 0< δ < such that Bx (, δ T BT ( ( x, ε T ( G Thus, every fuzzy poit x of T ( G is a iterior fuzzy poit ad so T ( G if a fuzzy ope i A Coversely, suppose that T ( G is fuzzy ope i A for ay fuzzy ope G of D Let x A for each 0< ε <, the fuzzy ball BT ( x, ε is fuzzy ope i A ( OALib Joural /3
J R Kider, J I Mousa sice x T BT ( ( x, ε Bx (, δ T BT ( ( x, ε it follows that there exist 0< δ < such that Theorem 57: A operator T : A D is fuzzy cotiuous o if ad oly if T ( E is fuzzy closed i A for all fuzzy closed E of D Let E be a fuzzy closed subset of D the E is fuzzy ope i D so that T D E is fuzzy ope i by theorem 46 but T E = T ( E so T ( E is fuzzy closed i A Coversely, suppose that T ( E is fuzzy closed i A for all fuzzy closed subset E of D But the empty fuzzy set ad the whole space A are fuzzy closed fuzzy set The A T ( E is fuzzy ope i ad T E = T ( E is fuzzy i A Sice every fuzzy ope subset of D is of the type E, where E is suitable fuzzy closed fuzzy set, it follows by usig theorem 56, that T is fuzzy cotiuous Theorem 58: Let ( AF,, ad ( DF,, ad ( Z, F, Z be three fuzzy legth spaces let T : ad S : Z be fuzzy cotiuous operator The the compositio SoT is a fuzzy cotiuous operator from A ito Z is a fuzzy ope sub- Let G be fuzzy ope subset of Z By theorem (56, S ( G set of D ad aother applicatio of the same theorem shows that T ( S ( G fuzzy ope subset of A, sice ( SoT ( G T ( ( = S G theorem agai that ( SoT is a fuzzy cotiuous Theorem 59: Let ( AF,, ad ( DF,, followig statemets are equivalet is a it follows from the same be two fuzzy legth spaces let T :, the the (i T is fuzzy cotiuous o A T E T E For all subset E of D (ii ( ( (iii T ( G T ( G (i (ii For all subset G of A Let E be a subset of D, sice E is a fuzzy closed subset of D, T ( E closed i A Moreover T ( ( E T E ad so T ( ( E T E [Recall that T ( E is the smallest fuzzy closed fuzzy set cotaiig T ( E (ii (iii is fuzzy ] ad Let G be a subset of A, the if E = T ( G, we have G T ( E T ( E thus ( ( T G T T E = E = T G G (iii (i ( ( Let S be a fuzzy closed fuzzy set i D ad fuzzy set ( T S = S By theorem OALib Joural 2/3
J R Kider, J I Mousa (47, it is sufficiet to show that S is a fuzzy closed i A, that is, S = S Now T S T T S S = S S T T S T S = S ( ( ( so that ( ( ( 6 Coclusio I this research the fuzzy legth space of fuzzy poits was defied as a geeralizatio of the defiitio of fuzzy orm o ordiary poits Based o the defied fuzzy legth, most of the properties of the ordiary orm were proved Refereces [] Zadeh, L (965 Fuzzy Sets Iformatio ad Cotrol, 8, 338-452 https:/doiorg/006/s009-9958(659024-x [2] Bag, T ad Samata, S (2003 Fiite Dimesioal Fuzzy Liear Spaces Fuzzy Mathematics,, 678-705 [3] Bag, T ad Samata, S (2006 Fixed Poit Theorems o Fuzzy Normed Spaces Iformatio Scieces, 76, 290-293 https:/doiorg/006/jis20050703 [4] Wag, CX ad Mig, M (993 Cotiuity ad Boudess Mappigs betwee Fuzzy Normed Spaces Fuzzy Mathematics,, 3-24 [5] Felbi, C (992 Fiite Dimesioal Fuzzy Normed Liear Spaces Fuzzy Sets ad Systems, 48, 239-248 https:/doiorg/006/065-04(9290338-5 [6] Golet, I (200 O Geeralized Fuzzy Normed Spaces ad Coicidece Theorems Fuzzy Sets ad Systems, 6, 38-44 https:/doiorg/006/jfss20090004 [7] Kider, J (202 New Fuzzy Normed Spaces Baghdad Sciece Joural, 9, 559-564 [8] Kider, JR (203 Completeess of the Cartesia Product of Two Complete Fuzzy Normed Spaces Egieerig ad Techology Joural, 3, 30-35 [9] Kider, JK ad Hussai, ZA (204 Cotiuous ad Uiform Cotiuous Mappig o Stadard Fuzzy Metric Spaces Egieerig ad Techology Joural, 32, -9 [0] Cheg, S ad Mordeso, J (994 Fuzzy Liear Operators ad Fuzzy Normed Liear Spaces Bulleti of Calcutta Mathematical Society, 86, 429-436 [] Kramosil, O ad Michalek, J (975 Fuzzy Metrics ad Statistical Metric Spaces Kyberetika,, 326-334 [2] George, A ad Veeramai, P (994 O Some Results i Fuzzy Metric Spaces Fuzzy Sets ad Systems, 64, 395-399 https:/doiorg/006/065-04(949062-7 OALib Joural 3/3
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