Complement of an Extended Fuzzy Set

Transcription

1 Internatonal Journal of Computer pplatons ( ) Complement of an Extended Fuzzy Set Trdv Jyot Neog Researh Sholar epartment of Mathemats CMJ Unversty, Shllong, Meghalaya usmanta Kumar Sut ssstant Professor, epartment of Mathemats, Jorhat Insttute of Sene Tehnology, Jorhat, ssam BSTRCT It has been aepted that for a fuzzy set ts omplement, nether s the null set, nor s the unversal set Whereas the operatons of unon nterseton of two rsp sets are ndeed speal ases of the orrespondng operatons of two fuzzy sets, they end up wth peular results whle defnng In ths regard, H K Baruah proposed that n the urrent defnton of the omplement of a fuzzy set, fuzzy membershp funton fuzzy membershp value had been taken to be the same, whh led to the onluson that the fuzzy sets do not follow the set theoret axoms of exluson ontradton H K Baruah has put forward an extended defnton of fuzzy set redefned the omplement of a fuzzy set aordngly In ths paper, we are tryng to mprove the noton of unon nterseton of fuzzy sets proposed by Baruah generalze the onept of omplement of a fuzzy set when the fuzzy referene funton s not zero We support our defnton of omplement of an extended fuzzy set wth examples show that ndeed our defnton satsfes all those propertes that omplement of a set really does n lassal sense Keywords Fuzzy set, fuzzy membershp funton, fuzzy referene funton, fuzzy membershp value, omplement of an extended fuzzy set 1 INTROUCTION Fuzzy Set Theory was ntrodued by Loft Zadeh [5] n 1965 t was spefally desgned to mathematally represent unertanty vagueness wth formalzed logal tools for dealng wth the mpreson nherent n many real world problems Fuzzy sets are sets wth boundares that are not prese The membershp n a fuzzy set s not a matter of affrmaton or denal, but rather a matter of a degree Zadeh s fuzzy set theory hallenged not only probablty theory as the sole agent for unertanty, but the very foundatons upon whh probablty theory s based: rstotelan two valued log Out of several hgher order fuzzy sets, Intutonst Fuzzy Sets (IFS) ntrodued by tanassov [1, ] s of great mportane lthough IFS are defned wth the help of membershp funtons, these are not neessarly fuzzy sets Fuzzy sets, on the other h, are Intutonst Fuzzy Sets Researh on the theory of fuzzy sets has been growng steadly sne the nepton of the theory n the md 1960 sthe body of onepts results pertanng to the theory s now qute mpressve Researh on a broad varety of applatons has also been very atve has produed results that are perhaps even more mpressve Fuzzy set theory proposed by Professor L Zadeh [5] s assumed as a generalzaton of lassal or rsp sets The theory of fuzzy sets should atually have been a generalzaton of the lassal theory of sets n the sense that the theory of sets should have been a speal ase of the theory of fuzzy sets Unfortunately, ths s not the ase It has been aepted that for a fuzzy set ts omplement, nether s the null set, nor s the unversal set Whereas the operatons of unon nterseton of two rsp sets are ndeed speal ases of the orrespondng operatons of two fuzzy sets, they end up gvng peular results whle defnng In ths regard H K Baruah, [3, ] has forwarded an extended defnton of fuzzy sets whh enables us to defne omplement of fuzzy set n a way that gve us the null set the unversal set We agree wth hm as ths new defnton satsfes all the propertes regardng omplement of a fuzzy set In ths artle, we put forward a defnton of omplement of an extended fuzzy set where the fuzzy referene funton s not always zero The defnton of omplement of a fuzzy set proposed by Baruah [3, ] an be seen as a partular ase of what we are gvng We support our defnton of omplement of an extended fuzzy set wth examples show that ndeed our defnton satsfes all those propertes that omplement of a set really does n lassal sense We mprove the noton of fuzzy unon nterseton proposed by Baruah [3, ] prove emorgan Laws n Fuzzy Set Theory We further put forward an extended defnton of subset of a fuzzy set from our st pont Fnally we defne unon nterseton for an arbtrary olleton of fuzzy sets over the same unverse prove emorgan Laws for an arbtrary olleton of fuzzy sets over the same unverse PRELIMINRIES H K Baruah [3, ] gave an extended defnton of fuzzy set n the followng manner ordng to hm, to defne a fuzzy set, two funtons namely fuzzy membershp funton fuzzy referene funton are neessary Fuzzy membershp value s the dfferene between fuzzy membershp funton referene funton Fuzzy membershp funton fuzzy membershp value are two dfferent thngs In the Zadehan defnton of omplementaton, these two thngs have been taken to be the same, that s where the error les Let µ 1 ( µ ( be two funtons, 0 µ ( µ 1 ( 1 For a fuzzy number denoted by {x, µ 1 (, µ ( x ε U, we would all µ 1 ( the fuzzy membershp funton, µ ( a referene funton, suh that {µ 1 ( µ ( s the fuzzy membershp value for any x In the defnton of omplement of a fuzzy set, the fuzzy membershp value the fuzzy membershp funton 39

2 Internatonal Journal of Computer pplatons ( ) have to be dfferent, n the sense that for a usual fuzzy set the membershp value the membershp funton are of ourse equvalent Let (, µ ) { x,, µ 1 1( µ ( (, µ ) { x,, B µ 3 3 ( µ ( be two fuzzy sets defned over the same unverse U Then the operatons nterseton unon are defned as ( µ, µ ) B( µ ) { x, mn(, ),max(, ) 1( 3( ( ( ( µ, µ ) B( µ ) 1 3, µ { x, max(, ),mn(, ) 1( 3( ( ( Two fuzzy sets { x ( C,µ C : { x (,µ : n the usual defnton would be expressed as ( C, 0) { x, µ C (,0 ( µ, 0) { x, µ (,0 C µ ordngly, we have ( µ 0) (,0) C C, µ { x, mn( µ (, µ ( ),max( 0,0) C { x, mn( µ C (, µ ( { x, µ ( µ ( x C whh n the usual defnton s nothng but we have ( µ 0) (,0) C C, µ { x, max( µ (, µ ( ),mn( 0,0) C { x, max( µ (, µ ( C { x, µ ( µ ( x C C Smlarly, whh n the usual defnton s nothng but C Thus we have seen that for unon nterseton of two fuzzy sets, the extended defnton leads to the unon nterseton under the stard defnton These new defntons lead to the onluson that for usual fuzzy sets ( µ, 0) { x, µ (,0 B( 1, µ ) { x,1, µ ( defned over the same unverse U we have ( µ, 0) B( 1, µ ) { x, mn( µ (,1 ),max( 0, µ ( ) { x, (, µ ( x µ, whh s nothng but the null setϕ ( µ, 0) B( 1, µ ) { x, max( µ (,1 ),mn( 0, µ ( ) { x 1,0,, whh s nothng but the unversal set U Ths means f we defne a fuzzy set ( (, 0) ) { x,1, µ ( of ( µ, 0) { x, µ (,0 µ, t s nothng but the omplement Thus t an be onluded that ( µ 0) ( ( µ, 0) ) φ set ( µ, 0) ( ( µ, 0) ) U, the unversal set Example 1,, the null Let U { a, be the unversal set We take two fuzzy sets B as ( 1, µ ) {( a,01,0)(, 0,01 )(,,0,0) (, µ ) {( a,09,03 )(, 05,03 )(,,05,03) µ B µ 3 Then( B)( µ 5, µ ) { x,, µ x 5( 6 ( {( a, 01,03 )(, 0,03 )(,,0,03) 6 We have seen that µ 5( a ) < µ 6( a), µ 5( b ) < µ 6 ( b) whh s gong aganst our assumpton that µ 5 ( µ 6( gan( B)( µ 7, µ ) { x,, µ x 7 ( 8( {( a, 09,0)(, 05,01 )(,,05,0) 8, whh s not true To avod suh degenerate ases, we mprove these defntons of unon nterseton as follows efnton 1 Let ( µ, µ ) { x,, 1 1( µ ( B( µ, µ ) { x,, 3 3 ( µ ( be two fuzzy sets defned over the same unverse U To avod degenerate ases we assume that (, ) max(, ) mn µ 1( 3( ( µ ( Then the operaton nterseton s defned as ( µ, µ ) B( µ ) { x, mn(, ),max(, ) 1( 3( ( ( If for some U, mn( µ 1( x ), µ 3( ) < max( µ (, µ ( ), then our onluson s that B ϕ 0

3 Internatonal Journal of Computer pplatons ( ) If for some U, mn( µ 1( x ), µ 3( ) max( µ (, µ ( ), then also B ϕ Further, we defne the operaton unon, wth mn ( µ 1(, µ 3( ) max( µ (, µ ( ) as ( µ, µ ) B( µ ) 1 3, µ { x, max(, ),mn(, ) 1( 3( ( ( lso, our another onluson s, that f for some U, mn( µ 1( x ), µ 3( ) < max( µ (, µ ( ), then the unon of the fuzzy sets B annot be expressed as one sngle fuzzy set The unon, however, an be expressed n one sngle fuzzy set f mn( µ 1( x ), µ 3( ) max( µ (, µ ( ) bove example makes ths lear For usual fuzzy sets wth referene funton 0, t s qute obvous to see that the above ondtons for defnng nterseton unon hold good efnton ( µ, µ ) { x,, Let 1 1( µ ( be a fuzzy set defned over the unverse U Then the omplement of the extended fuzzy ( µ, ) set 1 µ s defned as ( ( µ, )) 1 µ { x,, µ x 1 ( ( { x,,0 { x,1, µ x ( 1( ( ( )) µ, Membershp value of x n 1 µ µ ( x ) + ( 1 µ 1( ) 1+ µ ( µ 1( µ ( ) 0 If x, then membershp value of x s 1+ 0 µ 1( x ) 1 µ 1( s gven by For U mn µ ( x ),1 < max 0, µ 1(, so the unon of these two fuzzy sets annot be expressed as one sngle fuzzy set, ( ) ( ) Remark 1 µ (, If 1 µ (, ths defnton gves ( ( µ, )) { x,, 1 µ 1 1 ( 1 ( { x,,0 { x,1, µ x 1( 1( { x, max(,1 ),mn( 0, ) 1( 1( { x,1,0 U Thus U ϕ Example Let U { a, be the unversal set We take a fuzzy null set ϕ {( a, 01,01 )(, 0,0)(,,0,0) ϕ {( a, 01,0)(, 0,0)(,,0,0) { ( a,1,01 )(, 1,0)(,,1,0) {( a, max(01,1), mn(0,01) )(, max(0,1), mn(0,0) ), (, max(0,1), mn(0,0) ) {( a, 1,0)(, 1,0)(,,1,0) U Remark Let us see what happens when we take fuzzy referene funton 0 n ths defnton ( ( µ 0) ) { x,µ,0 1, 1 ( { x 0,0 { x,1, x, µ 1(, (Here 0 mn( 0,1) < max( 0, )) µ 1( x µ 1( ) { x, 1, µ x ϕ 1( { x, 1, µ x 1( Whh s what Baruah [3, ] has defned Thus we have seen that our defnton of omplement of an extended fuzzy set yelds Baruah s defnton [3, ] when we take fuzzy referene funton 0 Next we show that our defnton of omplement of an extended fuzzy set satsfes the set theoret axoms of ontradton exluson Thus we put forward the followng two propostons Proposton 1 For a fuzzy set ( µ 1, µ ) 1 (, µ ) ( ( µ )), we have µ 1 1, µ ϕ (Contradton) (, µ ) ( ( µ )) µ 1 1, µ U (Exluson) Proof 1 ( µ, µ ) ( ( µ )) 1 1, µ { x, µ 1 (, µ ( { x, µ 1 (, µ ( x { x, µ 1(, µ ( { x, µ (,0 { x,1, µ 1( x { x, µ 1(, µ ( { x, µ (,0 { x, µ 1(, µ ( { x,1, µ 1( { x, mn( µ 1(, µ ( ), max( µ (,0) { x, mn( µ 1(,1 ),max( µ (, µ 1( { x,, { x,, µ x ( ( 1( 1( ϕ ϕ ϕ Thus (, µ ) ( ( µ )) µ 1 1, µ ϕ (Contradton) ( µ 1, µ ) ( ( µ 1, µ )) { x µ (, µ ( { x, µ (, (, 1 1 µ 1

4 Internatonal Journal of Computer pplatons ( ) { x, µ 1 (, µ ( x { x, µ (,0 { x,1, µ 1( { x, µ 1 (, µ ( { x, µ (,0 { x, 1, µ 1( { x, max( µ 1 (, µ ( ),mn( µ (,0) { x,1, µ 1( x { x, µ 1 (,0 { x,1, µ 1( x { x, max( µ 1 (,1 ),mn( 0, µ 1( ) { x, 1,0 U Thus (, µ ) ( ( µ )) µ 1 1, µ U (Exluson) Proposton (emorgan Laws) Let ( µ, µ ) { x,, 1 1( µ ( (, µ ) { x,, B µ 3 3 ( µ ( be two fuzzy sets defned over the same unverse U To avod degenerate ase we assume that mn ( µ 1(, µ 3( ) max( µ (, µ ( ) Then ( ( µ µ ) B( µ µ )) ( ( µ µ )) ( B( µ )) 1 1, 3, 1, 3, µ ( ( µ ) B( )) ( ) 1, µ µ 3, µ µ 1, µ B µ 3, µ Proof 1 ( ( µ, µ ) B( µ )) ( ) ( ( )) { x,, { x,, µ x 1 ( ( 3 ( ( { x, max( µ 1 (, µ 3 ( ),mn( µ (, µ ( ) { x, mn(, { x,1, max(, ) ( ( 1( 3( gan ( ( µ, µ )) ( B( µ µ )) 1 3, { x, µ 1 (, µ ( { x, µ 3 (, µ ( x { x µ (,0 { x,1, ( x, µ 1 { x µ (,0 { x,1, ( x, µ 3 [{ x µ (,0 { x,1, µ ( { x, (,0 ], 1 µ [{ x µ (,0 U { x,1, µ ( { x,1, ( ], 1 µ 3 [, µ U { x µ (,0 { x, (,0 { x 1, µ ( { x,1, ( x ], 1 µ 3 [, 1 µ U { x 1, µ ( { x, (,0 { x µ (,0 U { x,1, ( ], µ 3 [{ x mn( µ (, µ ( ϕ], U [ ϕ { x 1, max( µ 1 (, µ ( ], 3 { x mn( µ (, (, µ { x 1, max( µ (, (, 1 µ 3 Thus ( ( µ, µ ) B( µ, µ )) ( ( µ, µ )) ( B( µ µ )) , We have assumed that (, ) max(, ) mn µ 1( 3( ( µ (, so ( ) max(,0) x µ ( mn 1, ( 1( µ 1( ) as suh { x, 1, { x,,0 ϕ 1( Smlarly ( (,1) max( 0, ) x µ ( mn ( 3( µ 3( ) hene { x,,0 { x,1, ϕ ( ( ( µ, µ ) B( µ )) { x,, { x,, µ x 1 ( ( 3 ( ( { x, mn( µ 1 (, µ 3 ( ),max( µ (, µ ( ) { x max( µ (, ( 3( U, µ { x 1,mn( µ (, ( gan, 1 µ 3 ( ( µ, µ )) ( B( µ )) { x, µ 1 (, µ ( { x, µ 3 (, µ ( x { x,,0 { x,1, µ x ( 1( { x,,0 { x,1, µ x ( 3( { x,,0 { x,,0 ( ( { x, 1, { x,1, µ x 1( 3( { x, max(, ( ( { x, 1, mn(, ) 1( 3( Thus ( ( µ, µ ) B( µ µ )) ( ( µ µ )) ( B( µ )) 1 3, 1, 3, µ emorgan laws for usual fuzzy sets wth referene funton 0 an be obtaned from the above emorgan laws by takng µ ( µ ( 0 Proposton 3 For a fuzzy set ( µ 1, µ ) ( ( µ, µ )) ( µ ) 1 1, µ, we have

5 Internatonal Journal of Computer pplatons ( ) Proof ( ( µ, )) 1 µ { x,, x 1 ( µ ( { x,,0 { x,1, µ x ( 1 ( { x,,0 { x,1, µ x ( 1 ( { x, 1, { x,,0 ( 1 ( { x, mn( 1, ),max(,0) 1( ( { x,, µ x 1( ( ( µ ( 1, µ )) (Involuton) Thus ( ( µ, )) µ ( 1, µ ) 1 µ Example 3 Let U { a, be the unversal set We take a fuzzy set a, 01,0, 0,01,,0,0 {( )( )( ) {( a, 0,0),( 01,0),(,0,0) { ( a,1,01 ),( 1,0),(,1,0) Thus ({( a, 0,0)(, 01,0)(,,0,0) { ( a,1,01 )(, 1,0)(,,1,0 ) ) {( a, 0,0)(, 01,0)(,,0,0) { ( a,1,01 )(, 1,0)(,,1,0 ) {( a, 1,0)(, 1,01 )(,,1,0) { ( a,01,0)(, 0,0)(,,00) {( a, 01,0)(, 0,01 )(,,0,0) We now proeed to defne subset of a fuzzy set from our st pont efnton 3 Let (, µ ) { x,, µ 1 1( µ ( B( 3, µ ) { x, µ 3 (, µ ( over the same unverse U The fuzzy set ( µ 1, µ ) of the fuzzy set B( µ 3, µ ) µ be two fuzzy sets defned f, µ 1( x ) µ 3 ( µ ( x ) µ ( Two fuzzy sets s a subset C { x,µ C ( : { x,µ ( : n the usual defnton would be expressed as C( µ C, 0) { x, µ C (,0 ( µ, 0) { x, µ (,0 ordngly, we have C ( µ 0) ( µ,0) C, f, µ C ( µ (, Whh an be obtaned by puttng µ ( x ) µ ( 0 n our new defnton Proposton For fuzzy sets ( µ, µ ), B( µ µ ), C( µ ) 1 3, 5, µ 6 over the same unverse U, the followng propostons are vald To avod degenerate ases we assume that (, ) max(, ) mn µ 1( 3( ( µ ( 1 ( µ, µ ) B( µ µ ), B( µ µ ) C( µ ) 1 3, 3, 5, µ 6 ( µ, µ ) C( µ ) 1 5, µ 6 µ 1 µ B µ 3, µ µ 1, µ (, ) ( ) ( ), ( µ, µ ) B( µ µ ) B( µ ) 1 3, 3, µ µ 1 µ µ 1, µ B µ 3, µ 3 (, ) ( ) ( ), B ( µ, µ ) ( µ µ ) B( µ ) 3 1, 3, µ ( µ, µ ) B( µ ) ( µ, µ ) B( µ µ ) ( µ ) 1 3, 1, µ 5 ( µ, µ ) B( µ ) ( µ, µ ) B( µ µ ) B( µ ) 1 3, 3, µ Proof Let ( µ, µ ) { x,, 1 1( µ ( (, µ ) { x,, B µ 3 3 ( µ ( C ( µ, µ ) { x,, ( µ 6 ( be three fuzzy sets over the same unverse U 1 ( µ, µ ) B( µ ) x U, µ 1( 3(, µ ( ( B ( µ, µ ) C( µ ) 3 5, µ 6 x U, µ 3( 5(, µ 6( ( Thus x U, µ 1 ( 5 (, µ 6 ( ( hene ( µ, µ ) C( µ ) 1 5, µ 6 ( µ, µ ) B( µ ) { x, mn(, ),max(, ) 1( 3( ( ( It s lear that ( µ (, µ ( ) (, µ ( max( µ (, ( )) x U, mn µ x Thus ( µ, µ ) B( µ µ ) ( µ ) 1 3, 1, µ The seond result an be smlarly found out, 3

6 Internatonal Journal of Computer pplatons ( ) 3 ( µ, µ ) B( µ ) { x, max(, ),mn(, ) 1( 3( ( ( It s lear that x U, µ 1 ( max( µ 1 (, µ 3 ( ),mn( µ (, µ ( ) ( Thus ( µ, µ ) ( µ µ ) B( µ ) 1 1, 3, µ The seond result an be smlarly found out ( µ, µ ) B( µ ) x U, µ 1( 3(, µ ( ( Now, ( µ, µ ) B( µ ) { x, mn(, ),max(, ) 1( 3( ( ( { x,, µ x 1( ( (, ) µ 1 µ 5 ( µ, µ ) B( µ ) x U, µ 1( 3(, µ ( ( Now, ( µ 1, µ ) B( µ 3, µ ) { x, max(, ),mn(, ) 1( 3( ( ( { x,, µ x 3 ( ( B ( µ 3, µ ) Takngµ ( x ) µ ( µ 6 ( 0, we obtan the same results for usual fuzzy sets We now proeed to defne arbtrary fuzzy unon nterseton usng the extended defnton of fuzzy sets gven by Baruah [3, ] efnton Let I { (, ) I µ 1 µ be a famly of fuzzy sets over the same unverse U To avod degenerate ases we assume that mn ( µ 1( ) max( µ ( ) Then the unon of fuzzy sets n I s a fuzzy set gven by (, µ ) { x,max( ),mn( ) µ 1 1 ( µ ( nd the nterseton of fuzzy sets n I s a fuzzy set gven by ( µ, µ ) { x,mn( ),max( ) 1 1 ( µ ( Example Let U { a, 1( µ 11, µ 1) {( a,05,01 )(, 06,0)(,,1,03), ( µ µ ) {( a,06,0)(, 08,0)(,,09,01) 1, ( µ ) {( a,1,0)(, 09,01 )(,,0,0) 3 µ 31, 3 be three fuzzy sets over U Then 1 ( µ 11, µ 1) ( µ 1, µ ) 3( µ 31, µ 3) {( a, max(05,06,1), mn(01,0,0) ), ( b, max(06,08,09), mn(0,0,01) ), (, max(1,09,0), mn(03,01,0) ) {( a, 1,0)(, 09,0)(,,1,01) 1 ( µ 11, µ 1 ) ( µ 1, µ ) 3 ( µ 31, µ 3 ) {( a, mn(05,06,1), max(01,0,0) ), ( b, mn(06,08,09), max(0,0,01) ), (, mn(1,09,0), max(03,01,0) ) a, 05,0, 06,0,,0,03 {( )( )( ) Proposton 5 1 ( ) ( ) I µ 1, µ µ 1, µ µ 1 µ µ 1, µ (, ) ( ) I Proof 1 ( µ, µ ) 1 { x, max( ),mn( ) 1( ( It s qute obvous that µ ( max( )) mn ( ) 1 µ 1( x ( ( Thus by our extended defnton of subset of a fuzzy set, ( µ 1, µ ) ( µ 1, µ ) I ( µ, µ ) 1 { x, mn( ),max( ) 1( ( It s qute obvous that mn ( ) ( max( ) 1( µ 1( µ µ ( Thus by our extended defnton of subset of a fuzzy set, ( µ 1, µ ) ( µ 1, µ ) I Proposton 6 (emorgan Laws) Let I { (, ) I µ 1 µ be a famly of fuzzy sets over the same unverse U To avod degenerate ases we assume that mn ( µ 1( ) max( µ ( ) Then ( ) { ( ) 1, 1 µ µ µ 1, µ ( ) { ( ) 1, µ µ µ 1, µ

7 Internatonal Journal of Computer pplatons ( ) Proof 1 µ 1 ( ) µ, { x, max( ),mn( ) 1 ( ( { x, mn( { x,1, max( ) ( 1( { ( µ, ) 1 µ x, µ (,0 x,1, µ 1 ( { { { x, µ (,0 { x,1, µ 1( { x, mn( µ ( { x,1, max( µ 1( Thus ( µ 1, µ ) { ( µ 1, µ ) In a smlar way, the result () an be establshed 3 CONCLUSION We have seen that f we use the extended defnton of fuzzy set omplement of an extended fuzzy set, we arrve at the onluson that the fuzzy sets, too, follow the set theoret axoms of exluson ontradton, even when the fuzzy referene funton s not zero We have defned arbtrary fuzzy unon nterseton fnally proved emorgan Laws for an arbtrary olleton of fuzzy sets over the same unverse U We hope that our fndngs wll help enhanng ths study on fuzzy sets CKNOWLEGEMENTS Our speal thanks to the referee the edtor of ths journal for ther valuable omments suggestons whh have mproved ths paper 5 REFERENCES [1] tanassov K, Intutonst Fuzzy Sets, Fuzzy Sets Systems, vol 0, pp 87-96, 1986 [] tanassov K, Intutonst Fuzzy Sets Theory pplatons, Physa Verlag, Sprnger Verlag Company, New-York (1999) [3] Baruah H K, The Theory of Fuzzy Sets: Belefs Realtes, Internatonal Journal of Energy, Informaton Communatons, Vol, Issue, pp 1-, May 011 [] Baruah H K, Towards Formng Feld Of Fuzzy Sets, Internatonal Journal of Energy, Informaton Communatons, Vol, Issue 1, pp 16-0, February 011 [5] Zadeh L, Fuzzy Sets, Informaton Control, 8, pp ,