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“”“”“”“”“”“”1.{0,1}2.ABABAB={(x,y)|xA,yB}(2-1)AxBy(x,y)‘’(x,y)AB3.(1)AA=AAA=A(2)AB=BAAB=BA(3)(AB)C=A(BC)(AB)C=A(BC)(4)A(BC)=(AB)(AC)A(BC)=(AB)(AC)(5)A(AB)=AA(AB)=A(6)AU=UAU=A()A=AA=(7)(Ac)c=A(8)AAc=UAAc=(9)(AB)c=AcBc(AB)c=AcBc4.(1)(2)(3)(4)(5)()1.UU[0,1]ÃÃ:U[0,1]uÃ(u)UÃÃÃ(u)uÃÃ(u)(2-2)2.Zedeh()1.ÃUUuÃÃ2.B~)()(~~uuBAµµ≥B~BA~~⊇BA~~⊇AB~~⊇B~BA~~=))(),(max()()(~~~~~~uuuuBABABAµµµµµ=∨=∪))(),(min()()(~~~~~~uuuuBABABAµµµµµ=∧=∩)(1~~uAAcµµ−=3.-BABA~~~~µµµ•=⋅BABABA~~~~~ˆ~⋅+−+=µµµµ1)(~~~~∧+=⊕BABAµµµ0)(~~~ˆ~∨−=−BABAµµµ0)1(~~~~∨−+=⊗BABAµµµ∞−+−=λµµµλ111~~~AAAc()A~A~A~A~A~~==IUAA~B~B~A~A~B~B~A~IIUU==)C~B~(A~C~)B~A~()C~B~(A~C~)B~A~(IIIIUUUU==)C~A~()B~A~()C~B~(A~)C~A~()B~A~()C~B~(A~UIUIUIUIUI==A~)B~A~(A~A~)B~A~(A~==IUUIA~UA~UUA~==IUAA~~=φUφφ=IA~~)~(AAcc=cccccBABAB~~)~~(~A~)B~A~(cUIIU==1A~U10≤≤λλµ≥)(~uA})(|{~~λµλ≥=uuAAA~λλµ)(~uA})(|{~~λµλ=uuAAA~λA~UλA~λλA~U]1,0[~~∈=λλλAAλλA~λA~⎩⎨⎧∉∈=λλλλµλAxAxxA~0~)(~(2-3)(2-4)(2-5)YXf→:)~(~:~AfAf=f~fA~f~)(~Aff~f1.2.(1)A~)(~xAµbxa))(),(min()(~~~baxAAAµµµ≥A~(2-6)(2)()1.ni≤mj≤]1,0[∈ijrmnijrR×=)(mnijijsrSR×∨=)(UmnijijsrSR×∧=)(ImnijcrR×−=)1(3.(1)(2)(3)(4)(5)(6)RRRRR==IURRSSRRSSRIIUU==T)(SRTS)(RT)(SRTS)(RIIIIUUUU==T)(RS)(RT)(SRT)(RS)(RT)(SRUIUIUIUIUI==SS)(RSSS)(RS==IUUI)(RRcc=cccccSRSRSUIIU==)(RS)(RcRREOROERERROERO====⊆⊆IIUU,,,,OERSRSSRSR=⇔=⇔⊆IU2211,SRSR⊆⊆)()(),()(21212121SSRRSSRRIIUU⊆⊆ccSRSR⊇⇔⊆()mnR×∈µ]1,0[∈λ)(ijrRλλ=⎩⎨⎧≥=λλλijijijrrr01λRRλ(2-8)(2-7)(1)(2)(3)(2)SR⊆]1,0[∈λλλSR⊆λλλSRSRUU=)(λλλSRSRII=)(()1.(2-9)lmjkmnijrRqQ××==)(,)(RQonlSSikiksQiRklknirqsjkijmiik≤≤≤≤∧∨==1,1),(12.(1)(2)(3)(4)(5))()(RSQRSQoooo=mnnmnmnmRRRRR==+)(,o)()()(SRSQSRQoUooU=)()()(RSQSRQSoUoUo=RIRRIOORRO====oooo,RQ⊆SRSQoo⊆RSQSoo⊆2121,RRQQ⊆⊆2211RQRQoo⊆()()1.(2-10)BA,},|),{(BbAabaBA∈∈=×R~ABBAR⎯→⎯~2.(1)(2)(3)(4)(5)(6)),(),(~~~~babaSRSRµµ∨⇔U),(),(~~~~babaSRSRµµ∧⇔I),(1),(~~~babaRRRccµµ−=⇔),(),(~~~~babaSRSRµµ≤⇔⊆),(),(~~~~babaSRSRµµ=⇔=),(),(~~~abbaRRRTµµ=⇔(7)(8)(9)AI~⎩⎨⎧≠==⇔bababaII01),(~~µI~ABA×O~BAbabaOO×∈∀=⇔),(,0),(~~µO~BA×BA×BA×E~E~BAbabaEE×∈∀=⇔),(,1),(~~µ3.(1)(2)(3)(4)(5)(6)(7)RRcc~)~(=RRTT~)~(=REREER~~~~~~==IUOORROR~~~~~~==IUERO~~~⊆⊆IIUUniTiTniiniTiTniiRRRR1111~)~(~)~(======SR~~⊇ccSR~~⊆()()1.WVU,,Q~UVR~VWQ~R~RQ~~oUW)),(),((),(~~~~wvvuwuRQVvRQµµµ∧∨=∈oWVU,,(2-11)TSRTSR~)~~()~~(~oooo=)~~()~~(~)~~(),~~()~~()~~(~TQTSTQSQRSRQSRoUooUoUoUo==)~~()~~(~)~~(),~~()~~()~~(~TQTSTQSQRSRQSRoIooIoIoIo⊂⊂QRSRQS~~~~~~oo⊂⇒⊂RIRRIOORRO~~~~~~~~~~====oooonmnmRRR+=~~~o43421Loo43421LoonnmmRRRRRRRR~~~~,~~~~==mnnmRR~)~(=1.(2-12)),,2,1(niiL=]1,0[∈ia),,,(21naaaaL=},,,{21nuuuUL=A~niuaiAi,,2,1)(~L==µ23mlnlba××∈∈µµ,babaTo=×nlba×∈µ,)(1iiniTbababa∧∨==•=oba,)(1iinibaba∨∧==ab()12(1)[0,1](2)“”“”[0,1]3.e-MorganDe-Morgan·(1)“”“”A∨∧αααααα=∨=∧,αββααββα∨=∨∧=∧,)()()()(γβαγβαγβαγβα∨∨=∨∨∧∧=∧∧βββαβββα=∧∨=∨∧)(,)(A),,(∧∨=AA)()()()()()(γβγαγβαγβγαγβα∧∨∧=∧∨∨∧∨=∨∧),,(∧∨AA000111=∧=∨=∧=∨ααααααcαα=cc)(1,0=∨=∧ccαααα),,,(cA∧∨()De-MorganDe-MorganDe-Morganccccccβαβαβαβα∧=∨∨=∧)()(4.),,2,1(nixiL=10≤≤ix•),,,(21nxxxFLffffixFGfGF∧GF∨fFffcFff℘fFffffF∈℘Fn),,2,1](1,0[nixiL=∈),,,(21nxxxFLn},max{),(212121xxxxxxf=∨=},min{),(212121xxxxxxf=∧=xxxf−==1)(∑∏===pinjijixF11∏∑===pinjijixF11(2-13)(2-14)12“”“”“”“”“”“”3()λλ)](~[))(~(uAuAH=)(~uAUλHλ1λλH1λλH(2-15)(2-16)))(),(~())(~())((vAvuEuAEuFAUv∧∨==∈oE~U),(+∞−∞=U⎪⎩⎪⎨⎧≥−−=−−δδ||0||),(~2)(vuvuevuEvuδ(2-17))](~[))((uAduAPaa=aPad⎪⎩⎪⎨⎧−≤−≤≤=axaaxaaxxda112/1012/10)((2-18)(2-19)4.R“*”“”,“”,“”“”RSR~,~SR~*~))()(()(~~*~*~yxzSRzyxSRµµµ∧∨==(2-20)(2)“”“”xxr)(rrxxrs)()(sr→2.“”3.)()(sr→x)(~))(~1()~~()))(()((xSxRSRxsrc∨−=−=→xxR~yS~'~Ry)~~(~~''SRRS→=o4.(1)Ifthenelse()R~S~]~~[]~~[~TRSRTc×+×=N~XYR~XR~N~NRY~~o=(3)“…”“…”1~R1~S2~R2~SmR~mS~1~R1~S2~R2~SmR~mS~()()(1)(2)(3)(4)()-()1.(1,(22:(1(2(33MamdaniLarsenTakagi-SugenoTsukamoto4()Mamdani(MISO)Mamdaniifis,andis,and…andisthenisifis,andis,and…andisthenis…ifis,andis,and…andisthenis1R1z11A2z12Apz1pAu1BRAAABpoL)(''2'1'×××=2R1z1z21A2z2z22Apzpz2pAuu2BmBmRmA1mA2mpAppZZZzzz×××∈LL2121),,,(()T-ST-STakagiSugeno1985,ifis,andis,and…andisthenisifis,andis,and…andisthenis…ifis,andis,and…andisthenis1R1z11A2z12Apz1pAu2R1z1z21A2z2z22Apzpz2pAuumRmA1mA2mpA),,,(211pzzzfuL=),,,(21pmzzzfuL=),,,(212pzzzfuL=∑∑==⋅=mjjmjpjjwzzzfwu1121'),,,(L(1)/(2)(3)(1)T-S…RXYo=XXYYoR1x1A2x2AkxkA)(xfy=)(xfixkkxpxppy+++=L110(2-21)(2-22)T-S)(xf)(xf),(21kkC21,rr21112221002121),(),(kkrkrkxxkkCxxf∑∑===(2-23)T-S(1)(2)(3)MISOniipnnnppppnnnppinnininyiiipypyyyyAisnkuiskuAisnkuAiskuAisnkyAiskyAiskyifR+++++++++++−−−−−−−−−−LLLLLL1111)(,,A)(,,)(,,)(,)(,,)2(,)1(:i111111121ττττ)()2()1()(210yiniiiinkypkypkyppkytheny−++−+−+=L)()()(111111111pppipnnnppinnnnninkupkupnkuppypyy−−++−++−−+++++++++++−τττLLLL∑∑===niiniiiyy11ˆµµ(2-24)(2-25)(1)(2)(3))()()(),(),()()()(tXCtYtutXgtXftuBtXAtXjjj=+++=&),,(1tuXgxfβα,uβ||||||||||),,(||||||||),(||),,,(),(11uXtuXgXtXftuXgButXguxfjββα+≤≤=(2-26)(2-27)(2-28)QP1,)()()()(maxmin−=−+−ufjjTjjQQQKBAPPKBAβλλα21uuu+=)1)](()([4;;maxmin221ufxTjPQPXBuKXuβλαλβγγ−−−=−=(2-29)(2-30)(2-31)(2-32)(2-33))(ˆ)(ˆ)](ˆ)([)()(ˆ)(ˆtXCtYtYtYLtuBtXAtXiiii=−++=&(2-34))(ˆ)()()(1tXKtututuiimiii==∑=η(2-35)