O Smlarty ad Etropy of eutrosophc Sets Pak Majumdar, & S.K. Samata Departmet of Mathematcs, M.U.C Wome s College, urda (W..), Ida Departmet of Mathematcs, Vsva-harat, Satketa (W..), Ida Abstract: I ths paper e have troduced the oto of dstace betee to sgle valued eutrosophc sets ad studed ts propertes. We have also defed several smlarty measures betee them ad vestgated ther characterstcs. A measure of etropy of a sgle valued eutrosophc set has also bee troduced. Keyords: Sgle valued eutrosophc set; Hausdorff dstace; smlarty measure; eghts; etropy; cardalty. AMS Subject classfcato: 0399, 03E99. Itroducto I may practcal stuatos ad may complex systems lke bologcal, behavoral ad chemcal etc., e ecouter dfferet types of ucertates. Our classcal mathematcs does ot practce ay kd of ucertaty ts tools, excludg possbly the case of probablty, here t ca hadle a partcular kg of ucertaty called radomess. Therefore e techques ad modfcato of classcal tools are requred to model such ucerta pheomeo. I 965, L. A. Zadeh [4] coed hs remarkable theory of Fuzzy sets that deals th a kd of ucertaty ko as Fuzzess ad hch s due to partal membershp of a elemet a set. Later ths Fuzzess cocept leads to the hghly acclamed theory of Fuzzy Logc. After the veto of fuzzy sets may other hybrd cocepts begu to develop. I 983, K. Ataasov [] troduced the dea of Itutostc fuzzy sets, a set th each member havg a degree of beloggess as ell as a degree of o-beloggess. Ths s aga a geeralzato of fuzzy set theory. Although Fuzzy set theory s very successful hadlg ucertates arsg from vagueess or partal beloggess of a elemet a set, t caot model all sorts of ucertates prevalg dfferet real physcal problems such as problems volvg complete formato. Hece further geeralzatos of fuzzy ad tutostc fuzzy sets are requred. After that may theores have bee evolved hch are successful ther respectve domas. Recetly a e theory has bee troduced ad hch s ko as eutrosophc logc ad sets. The term eutro-sophy meas koledge of eutral thought ad ths eutral represets the ma dstcto betee fuzzy ad tutostc fuzzy logc ad set. eutrosophc logc as troduced by Floret Smaradache [0] 995. It s a logc hch each proposto s estmated to have a degree of truth (T), a degree of determacy (I) ad a degree of falsty (F). A eutrosophc set s a set here Correspodg author, E-Mal:pmajumdar@redffmal.com PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
each elemet of the uverse has a degree of truth, determacy ad falsty respectvely ad hch les betee ] 0, [, the o-stadard ut terval. Ulke tutostc fuzzy sets, here the corporated ucertaty s depedet of the degree of beloggess ad degree of o beloggess, here the ucertaty preset,.e. determacy factor, s depedet of truth ad falsty values. I 005, Wag et. al. [] troduced a stace of eutrosophc set ko as sgle valued eutrosophc sets (SVS) hch ere motvated from the practcal pot of ve ad that ca be used real scetfc ad egeerg applcatos. The sgle valued eutrosophc set s a geeralzato of classcal set, fuzzy set, tutostc fuzzy set ad paracosstet sets etc. Aga fuzzess s a feature of mperfect formato hch s due to partal beloggess of a elemet to a set. The term etropy as a measure of fuzzess as frst coed by Zadeh [5] 965. The etropy measure of fuzzy sets has may applcatos areas lke mage processg, optmzatos etc. [4, 9]. Actually a measure of mperfectess of formato represeted by a set s measured terms of etropy. So measurg of etropy of sgle valued eutrosophc sets ll be useful cases heever modelg of ucerta stuatos s doe through SVSs. O the other had smlarty s a key cocept a umber of felds such as lgustcs, psychology ad computatoal tellgece. I several problems e ofte eed to compare to sets. We are ofte terested to ko hether to patters or mages are detcal or approxmately detcal or at least to hat degree they are detcal. Smlar elemets are regarded from dfferet pots of ve by usg resemblaces, dstaces, closeess, proxmty, dssmlartes etc. So t s atural ad very useful to address the ssues smlarty measure for ths e set, vz. sgle valued eutrosophc set. The rest of the paper s costructed as follos: Some prelmary deftos ad results ere gve secto. I secto 3, dfferet types of dstaces betee to sgle valued eutrosophc sets have bee troduced. Dfferet measures of smlarty ad ther propertes have bee dscussed secto 4. I secto 5, the oto of etropy of a sgle valued eutrosophc set has bee gve. Secto 6 brefly compares the methods descrbed here th earler avalable methods. Secto 7 cocludes the paper.. Prelmares I ths secto e recall some deftos, operatos ad propertes regardg sgle valued eutrosophc sets (SVS short) from [], hch ll be used the rest of the paper. A sgle valued eutrosophc set has bee defed [] as follos: Defto. Let X be a uversal set. A eutrosophc set A X s characterzed by a truthmembershp fucto ta, a determacy-membershp fucto A ad a falsty-membershp fucto f A, here ta, A, f A : X [0,], are fuctos ad x X, x x( ta( x), A( x), f A( x)) A, s a sgle valued eutrosophc elemet of A. A sgle valued eutrosophc set A over a fte uverse X { x, x, x3,..., x } s represeted as PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
x A. t ( x ), ( x ), f ( x ) A A A I case of SVS the degree of truth membershp (T), determacy membershp (I) ad the falsty membershp(f) values les [0, ] stead of the o stadard ut terval ] 0, [ ordary eutrosophc sets. A example of SVS s gve belo: as the case of Example. Assume that X { x, x, x3}, here x s capacty, x s trustorthess ad, x3 s prce of a mache, be the uversal set. The values of x, x, x3 are [0, ]. They are obtaed from the questoare of some doma experts, ther opto could be a degree of good servce, a degree of determacy ad a degree of poor servce. A s a sgle valued eutrosophc set of X defed by A 0.3, 0.4, 0.5 0.5, 0., 0.3 0.7, 0., 0.. x x x 3 ext e state the deftos of complemet ad cotamet as follos: Defto.3 The complemet of a SVS A s deoted by t ( x) f ( x); ( x) ( x)& f ( x) t ( x) x X. c A c A c A A A A c A ad s defed by Defto.4 A SVS A s cotaed the other SVS, deoted as A, t ( x) t ( x); ( x) ( x) & f ( x) f ( x) x X. A A A To sets ll be equal,.e. A, ff A & A. f ad oly f Let us deote the collecto of all SVS X as ( X ). Several operatos lke uo ad tersecto has bee defed o SVS s ad they satsfy most of the commo algebrac propertes of ordary sets. Defto.5 The uo of to SVS A& s a SVS C, rtte as C A, hch s defed as follos: t ( x) max( t ( x), t ( x)); ( x) max( ( x), ( x)) & f ( x) m( f ( x), f ( x)) x X. C A C A C A Defto.6 The tersecto of to SVS A& s a SVS C, rtte as C A, hch s defed as follos: t ( x) m( t ( x), t ( x)); ( x) m( ( x), ( x)) & f ( x) max( f ( x), f ( x)) x X. C A C A C A For practcal purpose, throughout the rest of the paper, e have cosdered oly SVS a fte uverse. 3 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
3. Dstace betee to eutrosophc sets. I ths secto e troduce the oto of dstace betee to sgle valued eutrosophc sets A ad the uverse X { x, x, x3,..., x }. Defto 3. Let A x x ad t ( x ), ( x ), f ( x ) t ( x ), ( x ), f ( x ) be to sgle A A A valued eutrosophc sets X { x, x, x3,..., x }. The The Hammg dstace betee A ad s defed as follos: d ( A, ) { t ( x ) t ( x ) ( x ) ( x ) f ( x ) f ( x )}...() A A A The ormalzed Hammg dstace betee A ad s defed as follos: l ( A, ) { t ( x ) t ( x ) ( x ) ( x ) f ( x ) f ( x )}...() A A A 3 The Euclda dstace betee A ad s defed as follos: (, ) {( A( ) ( )) ( A( ) ( )) ( A( ) ( ))...(3) e A t x t x x x f x f x The ormalzed Euclda dstace betee A ad s defed as follos: q A t x t x x x f x f x (, ) {( A( ) ( )) ( A( ) ( )) ( A( ) ( ))...(4) 3 o for equatos () (4) the follog holds: ( ) 0 d ( A, ) 3...(5) ( ) 0 l ( A, )...( 6) ( ) 0 e ( A, ) 3...(7) ( v) 0 q ( A, )...(8) Example 3. Let X { a, b, c, d} be the uverse ad A ad be to sgle valued eutrosophc sets X defed as follos: 4 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
A a b c d {,,, }. 0.5, 0., 0.9 0.8, 0.4, 0. 0.3, 0.8, 0.7 0.6, 0.3, 0.5 a b c d {,,, }. 0.7, 0.4, 0. 0.5, 0.5, 0.3 0.,0., 0.3 0.8, 0., 0.6 The the dstace betee A, ll be as follos: d (, ) 3.3. A Smlarly the other three dstaces ll be 3.3 l ( A, ) 0.75, e ( A, ).5, q ( A, ) 0.33. The the follog result ca be easly proved. Proposto 3.3 The dstaces d, l, e, q defed above are metrc. Defto 3.4 (Cardalty) The mmum (or sure) cardalty of a SVS A s deoted as l l m cout( A) or c ad s defed as c ta( x ). The maxmum cardalty of A s deoted by u u max cout( A) or c ad s defed as c { t ( x ) ( ( x ))}. The cardalty of A s defed by the terval [ l u c, c ]. A A Example 3.5 For the SVS A gve example 3. e have the follog: l c t ( x ) 0.5 0.8 0.3 0.6. & A u c { t ( x ) ( ( x ))}.3.4 0.5.3 4.5. A A 4. Smlarty Measure betee to sgle valued eutrosophc sets I ths secto e defe the oto of smlarty betee to SVSs. We have adopted varous methods for calculatg ths smlarty. The frst method s based o dstaces defed the prevous secto. The secod oe s based o a matchg fucto ad the last oe s based o membershp grades. 5 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
I geeral a smlarty measure betee to SVSs s a fucto defed as satsfes the follog propertes: ( ) S( A, ) [0,], ( ) S( A, ) A, s : ( X ) [0,] hch ( ) S( A, ) S(, A), (9) ( v) A C S( A, C) S( A, ) S(, C) ut dvdual measures may satsfy more propertes addto to (9). o smlarty ca be calculated usg several techques. Here e have adopted three commo techques amely, the dstace based, the oe based o a matchg fucto ad lastly o membershp grade based. 4. Dstace based smlarty measure We ko that smlarty s versely proportoal th the dstace betee them. Usg the dstaces defed equato () (4) e defe measures of smlarty s betee to SV sets A ad as follos: s ( A, )...(0) d ( A, ) For example f e use Hammg dstace by s ad s defed by: d the the assocated measure of smlarty ll be deoted s A (, )...() d ( A, ) The follog example calculates the smlarty measure betee the to SVSs: Example 4.. The smlarty measure betee the to SVSs defed example 3. ll be s ( A, ) 0.33. 3.3 Proposto 4.. The dstace based smlarty measure follog propertes: s, betee to SVSs A ad satsfes the 6 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
( )0 s ( A, ) ( ) (, ) s A ff A ( ) s ( A, ) s (, A) ( v) A C S( A, C) S( A, ) S(, C) Proof. The results ()- () holds trvally from defto. We oly prove (v). Let A C. The e have t ( x) t ( x) t ( x); ( x) ( x) ( x)& f ( x) f ( x) f ( x) x U. o A c A c A c t ( x) t ( x) t ( x) t ( x) ad t ( x) t ( x) t ( x) t ( x) llhold. A A c C A C Smlarly, ( x) ( x) ( x) ( x) & ( x) ( x) ( x) ( x) ad A A c C A C f ( x) f ( x) f ( x) f ( x) & f ( x) f ( x) f ( x) f ( x) holds. A A c C A C Thus d A d A C s A s A C ad d C d A C s C s A C (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) s ( A, C) s ( A, ) s (, C). Ths s true for all the dstace fuctos defed equatos () to (4). Hece the result. 4. Smlarty measure based o membershp degrees Aother measure of smlarty s betee to SV sets A ad could be defed as follos: {m{ ta( x ), t ( x )} m{ A( x ), ( x )} m{ f A( x ), f ( x )}} s ( A, )...() {max{ t ( x ), t ( x )} max{ ( x ), ( x )} max{ f ( x ), f ( x )} A A A Example 4.. Here the smlarty measure betee the to SV sets defed example 3. ll be 3.8 s ( A, ) 0.535. 7. 7 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
Proposto 4.. The dstace based smlarty measure follog propertes: ( )0 s ( A, ) ( ) (, ) s A ff A ( ) s ( A, ) s (, A) s, betee to SVS A ad satsfes the 3 3 3 ( v) A C s ( A, C) s ( A, ) s (, C). Proof. Propertes () ad () follos readly from defto. () It s clear that f Coversely, let A s ( A, ). {m{ ta( x ), t ( x )} m{ A( x ), ( x )} m{ f A( x ), f ( x )}} s ( A, ) {max{ t ( x ), t ( x )} max{ ( x ), ( x )} max{ f ( x ), f ( x )} A A A {m{ t ( x ), t ( x )} m{ ( x ), ( x )} m{ f ( x ), f ( x )}} x A A A {max{ t ( x ), t ( x )} max{ ( x ), ( x )} max{ f ( x ), f ( x )} x A A A {[m{ t ( x ), t ( x )} max{ t ( x ), t ( x )}] [m{ ( x ), ( x )} max{ ( x ), ( x )}] x A A A A m[{ f ( x ), f ( x )} max{ f ( x ), f ( x )}] 0 Thus for each x, A A m{ t ( x ), t ( x )} max{ t ( x ), t ( x )} 0, m{ ( x ), ( x )} max{ ( x ), ( x )} 0 & A A A A m{ f ( x ), f ( x )} max{ f ( x ), f ( x )} 0 holds. A A Thus t ( x) t ( x), ( x) ( x) & f ( x) f ( x) A. A A A (v)o e prove the last result. Let A C. The e have t ( x) t ( x) t ( x); ( x) ( x) ( x)& f ( x) f ( x) f ( x) x U. o A c A c A c 8 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
t ( x) ( x) f ( x) t ( x) ( x) f ( x) ad A A A A C t ( x) ( x) f ( x) t ( x) ( x) f ( x) A C C A ta( x) A( x) f ( x) ta( x) A( x) fc ( x) s ( A, ) s ( A, C) t ( x) ( x) f ( x) t ( x) ( x) f ( x) Aga smlarly e have: A C C A t ( x) ( x) f ( x) t ( x) ( x) f ( x) & C A A C t ( x) ( x) f ( x) t ( x) ( x) f ( x) C C A C C t ( x) ( x) fc ( x) ta( x) A( x) fc ( x) s (, C) s ( A, C) t ( x) ( x) f ( x) t ( x) ( x) f ( x) C C C C A s ( A, C) s ( A, ) s (, C). Hece the proof of ths proposto s complete. 4.3 Smlarty measure based o a matchg fucto ext cosder a uverse here each elemet x has a eght. The e requre a e measure of smlarty dfferet from those dscussed earler. Ofte eghts are assocated th each elemet of a uverse to gve a order of mportace amog the elemets. To llustrate the stuato e gve a example: Suppose that there s a system to detect a dsease based o several symptoms assocated th t. o each symptoms s characterzed by three thgs amely a degree of truth, a degree of determacy ad a degree of falsty. So for each dsease e ca have a correspodg sgle valued eutrosophc set th symptoms as ts elemets. These SVSs ll act as our koledgebase. Wheever a patet comes th some health problem the system ll geerate hs correspodg SVS. The measure of smlarty, as dscussed prevous sectos, betee these SVSs ca detect the possble dsease. ut e ko that may dseases have several symptoms commo. So for a specfc dsease all the symptoms are ot equally mportat. Here e ca assg eghts to each symptom correspodg to a partcular dsease. So e have a uverse of symptoms here each symptom has ts correspodg eghts. I ths case e caot use smlarty measures descrbed 4. ad 4.. Aga a sutable fucto s ofte used to measure the smlarty betee to sets hch s called a matchg fucto. Che [, 3] frst troduced the oto of matchg fucto. Here a eghted smlarty measure s betee A ad has bee defed usg a matchg fucto as follos: s ( ta( x ) t ( x ) A( x ) ( x ) f A( x ) f ( x )) ( A, )...(3) {( ta( x ) A( x ) f A( x ) ).( t ( x ) ( x ) f ( x ) )} 9 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
Example 4.3. Cosder the to SV sets example 3.. Further let the elemets a, b, c, d of the uverse X have eghts 0.,0.3, 0.5, 0. respectvely. The the eghted smlarty measure betee the to SV sets ll be s 0.0.37 0.30.4356 0.5 0.6 0. 0.656 0.379 ( A, ) 0.84 0.0.759 0.3 0.4956 0.5 0.708 0. 0.707 0.4538 Proposto 4.3. The eghted smlarty measure s, follog propertes: ( )0 s ( A, ) ( ) s ( A, ) f A ( ) s ( A, ) s (, A) Proof. Trvally follos from defto ad Cauchy-Schaarz equalty. betee to SVS A ad satsfes the ote that here the result () ad (v) (.e. mootoacty la) of equato (9) ll ot hold due to the effects of the eghts. 5. Etropy of a Sgle Valued eutrosophc Set Etropy ca be cosdered as a measure of ucertaty volved a set, hether fuzzy or tutostc fuzzy or vague etc. Here the SVS are also capable of hadlg ucerta data, therefore as a atural cosequece e are also terested fdg the etropy of a sgle valued eutrosophc set. Etropy as a measure of fuzzess as frst metoed by Zadeh [5] 965. Later De Luca-Term [5] axomatzed the o-probablstc etropy. Accordg to them the etropy E of a fuzzy set A should satsfy the follog axoms: ( DT) E( A) 0ff A ( DT ) E( A) ff ( x) 0.5, x X A X ( DT3) E( A) E( ) ff As less fuzzy tha,. e. f A( x) ( x) 0.5 x X...(4) Seve or f A( x) ( x) 0.5, x X. ral c ( DT 4) E( A ) E( A). othe r authors have vestgated the oto of etropy. Kaufma [6] proposed a dstace based measure of soft etropy; Yager [3] gave aother ve of degree of fuzzess of ay fuzzy set terms of lack of dstcto betee the fuzzy set ad ts complemet. Kosko [7] vestgated the fuzzy etropy relato to a measure of subset hood. Szmdt & Kacprzyk [] studed the etropy of tutostc fuzzy sets. Majumdar ad Samata [8] vestgated the etropy of soft sets etc. 0 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
Defto 5. Smlarly here case of SVS also e troduce the etropy as a fucto E : ( X ) [0,] hch satsfes the follog axoms: ( ) E ( A) 0 f A s a crsp set ( ) E ( A) f ( t ( x), ( x), f ( x)) (0.5,0.5,0.5) x X A A A ( ) E ( A) E ( ) f A more ucerta tha,... (5). e. t ( x) f ( x) t ( x) f ( x) ad ( x) ( x) ( x) ( x) A A A c A c ( v) E ( A) E ( A ) A( X ). o otce that a SVS the presece of ucertaty s due to to factors, frstly due to the partal beloggess ad partal o-beloggess ad secodly due to the determacy factor. Cosderg these to factors e propose a etropy measure E of a sgle valued eutrosophc sets A as follos: E ( A) ( t ( x ) f ( x )) ( x ) ( x )...(6) x X A A A c A c Proposto 5. E satsfes all the axoms gve defto 5.. Proof () For a crsp set A ad ( x) 0 x X. Hece E ( A) 0 holds. A () If A be such that ( t ( x), ( x), f ( x)) (0.5,0.5,0.5) x X, A A A the ta( x) f A( x) ad A( x) ( x) 0.5 0.5 0 x X E ( A) A c () It holds from defto. (v) E ( A) E ( A c ) holds obvously from defto. Thus E s a etropy fucto defed o ( X ). Example 5.3 Let X { a, b, c, d} be the uverse ad A be a sgle valued eutrosophc set X defed as follos: A a b c d {,,, }. 0.5, 0., 0.9 0.8, 0.4, 0. 0.3, 0.8, 0.7 0.6, 0.3, 0.5 The the etropy of A ll be E ( A) 0.5 0.48. PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
6. Dscussos The techques of smlarty ad etropy descrbed here are totally e cocepts for sgle valued eutrosophc sets. For the case of etropy o earler methods of measurg etropy ll be able to determe the etropy of a SVSs, as here the sum of the degrees of truth, determacy ad falsty s ot ecessarly bouded. A fuzzy set F ca be cosdered as a SVS th degree of determacy ad degree of falsty zero, the ts etropy ll be E ( F) ( ta( x )). For crsp set ths value s, x X hch s coformable th earler ko result. The smlarty measuremets descrbed here are aga totally e cocepts for SVSs. These measures ca also be appled to measure the smlarty of tutostc fuzzy sets (IFS) here the determacy factor should be replaced by t f case of IFS. 7. Cocluso The recetly proposed oto of eutrosophc sets s a geeral formal frameork for studyg ucertates arsg due to determacy factors. From the phlosophcal pot of ve, t has bee sho that a eutrosophc set geeralzes a classcal set, fuzzy set, terval valued fuzzy set, tutostc fuzzy set etc. A sgle valued eutrosophc set s a stace of eutrosophc set hch ca be used real scetfc ad egeerg applcatos. Therefore the study of sgle valued eutrosophc sets ad ts propertes have a cosderable sgfcace the sese of applcatos as ell as uderstadg the fudametals of ucertaty. I ths paper e have troduced some measures of smlarty ad etropy of sgle valued eutrosophc sets for the frst tme. These measures are cosstet th smlar cosderatos for other sets lke fuzzy sets ad tutostc fuzzy sets etc. Ackoledgemet: The authors scerely thakful to the aoymous reveers for ther helpful commets hch has helped to rerte the paper ts preset form. Refereces [] Ataasov, K, Itutostc Fuzzy Sets. Fuzzy Sets & Systems, 0 (986) pp 87-96 [] Che, S.M., A e approach to hadlg fuzzy decso makg problems, IEEE Trasactos o Systems, Ma ad Cyberet, 8(988)0-06 [3] Che, S.M., Hsao, P.H., A Comparso of smlarty measures of fuzzy values, Fuzzy Sets ad Systems, PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com
7(995), 79-89 [4] Cheg, C. C., Lao, K. H., Parameter optmzato based o etropy eght ad tragular fuzzy umber, It. J. of Egeerg ad Idustres, () (0) 6-75 [5 ] De Luca, A., Term, S., A defto of a o-probablstc etropy the settg of fuzzy sets theory, Iformato & Cotrol, 0 (97), 30-3 [6] Kaufma, A., Itroducto to the theory of Fuzzy Subsets-Vol : Fudametal Theoretcal Elemets, Academc Press, e York (975). [7] Kosoko,., Fuzzy etropy ad codtog, Iformato Scece, 40 () (986) 65-74 [8] Majumdar, P., Samata, S., Softess of a soft set: Soft set etropy, accepted Aals of fuzzy math. ad formatcs, 0. [9] Pasha, E., Fuzzy etropy as cost fucto mage processg, Proc. of the d IMT-GT regoal cof. o math., Stat. ad appl., Uverst Sas Malaysa, Peag, 006. [0] Smaradache, F., A Ufyg Feld Logcs. eutrosophy: eutrosophc Probablty, Set & Logc. Rehoboth: Amerca Research Press, (999). [] Szmdt, E., Kacprzyk, J., Etropy for tutostc fuzzy sets, Fuzzy Sets & systems, 8 (00) 467-477. [] Wag, H. et al., Sgle valued eutrosophc sets, Proc. Of 0 th It. cof. o Fuzzy Theory ad Techology, Salt Lake Cty, Utah, July -6 (005). [3] Yagar, R. R., O the measure of fuzzess ad egato, Part I: Membershp the ut terval, Iteratoal Joural of Geeral Systems, 5(979) 89-00. [4] Zadeh, L., Fuzzy sets. Iformato ad Cotrol. 8 (965), pp 87-96. [5] Zadeh, L., Fuzzy sets ad systems, : Proc. Symp. o Systems Theory, Polytechc Isttute of rookly, e York, 965. 3 PDF geerated by deskpdf Creator Tral - Get t at http://.docudesk.com