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1~511,“”,()“”“”;“”“”,“”12345,,,,,ABC,,,,ABCFN,,,abcAx2“”,.xAxxA∈xA∉A3:“”——A={},{1,2,,,}n=B=….,F={}.“”——()px{()},xpx,x2{10}.xx−=210x−=22{(,)1}Cxyxy=+=4{1,2,,,}n=22{(,)1}Cxyxy=+=210x+=∅,,ABF.={}.,“”.F210x+=5,,{1,2,,}n=—{0,1,2,}=±±——(,)=−∞+∞—{i,,i=1}abab=+∈−——∅——————nn×nn×nn61{1};xx∈≥R“1”“1”xA∉AxxA∈,,2{,,},{,,},{,,}abcbcacba3.1.1.1,AB1xAxB∀∈⇒∈AB⊂BA⊃ABBAAB,{}{,}NR2,AB⊂BA⊂AB2{10}{1,1}.xx−==−AB=.AB≠3.AB⊂AB≠AB⊂≠AB.∅AA∅⊂:22{(,)1}Cxyxy=+={1,2,,,}n=⊂2.“”:AA⊂1AC⊂,AB⊂BC⊂2..1XX,;3,ABX⊂21.2.XAA1—BB{}xBBxxAA≡∈∈∩AB{}xBBxxAA≡∈∈∪2—ABACAA\XABA{\}xxxBAAB≡∈∉3—.B3(1),,ABAABB∩⊂∩⊂,,AABBAB⊂∪⊂∪\,\;ABAABB⊂⊄(2),,AAA∩∅=∅∪∅=,;AXAAXX∩=∪=(3),,(),CCCCAAAAXAA∩=∅∪==,.CCXX=∅∅=(4);CCABAB⊂⇔⊃(5)\;CABAB=∩(6)AB∩=∅,.CCABBA⊂⊂4,,ABCX⊂1.1X1,AAAAAA∩=∪=,;ABBAABBA∩=∩∪=∪2()(),ABCABC∩∩=∩∩()();ABCABC∪∪=∪∪3()()();ABCABAC∪∩=∪∩∪()()(),ABCABAC∩∪=∩∪∩4(),CCCABAB∩=∪().CCCABAB∪=∩5—DeMorgan112,bb123,,aaa3211(,),ab21(,),ab22(,),ab32(,).ab12(,),ab31(,),abAB×=12(,),ab31(,),ab11{(,),ab21(,),ab22(,),ab32(,)}.ab123{,,}Aaaa=12{,}Bbb=,AB×ABABAB×AB“”1n,AB,aAbB∀∈∀∈(,)aba(,)abb(,)ab12112212(,)(,).aaababbb==⇔=iiaA∈(,)xy(,,)xyz12,,,nAAAnn12(,,,)naaa(1,2,,).in=,{(,),}abaAbB∈∈(,)ab,AB31.3Descartes.AB×ABABAB×12(,)xy[0,1]y∈[1,3][0,1]×[1,3]x∈1.1,[1,3][0,1]{(,)13,01}.xyxy×=≤≤≤≤5ABBA×≠×12,.AA×∅=∅∅×=∅3n12,,,nAAAn1121{{1,2,,},}niniiAAAAxinxA==∩∩∩≡∀∈∈∩iA——121{{1,2,,},}niniiAAAAxinxA==∪∪∪≡∃∈∈∪iA——12121{(,,,),1,2,,}ninniiiAAAAxxxxAin==×××≡∈=∏12,,,nAAAn——.nA1niiA=∏12nAAAA====,,nn=×××RRRR.nn=×××n2DeMorgan11(),()nniiiiBABA==∪=∪∩∩11(),.......nnCCiiiiAA===∪∩2.1f21.4,,,XYXXxYyf:fXY→Y:fxyyxf()yfx=yfx=Xf()fD()fX=DY{(),}yyfxxX=∈Xf()fR.fYfX,XY()fAff{(),}yyfxxA=∈YAX⊂.A()fX1()fy−fy1({})fy−f{}y1()fB−BBX{()}xXfxB∈∈XBY⊂.f1(){()}fyxXfxy−=∈=1()fy−1()fB−:,,1()fy−1()fB−1fabcdXY2345()fb1(1)f−1(3)f−1({2,3})f−()fR({,})fad:fXY→XYXf1.2..()3,({,}){2,3};fbfad==111({2,3}){,,},(3){,},(1).fabdfbdf−−−===∅3.()fgfg=()().,()()fgXfgxXfxgx===⇔∀∈=DD4.1.5:fXY→1()fY=R,,s.t.(),yYxXyfx∀∈∃∈=1,()yYfy−∀∈≠∅ff12()()fxfx≠12xx≠(),,s.t.()1fxXfxy∀∈∃∈=yR2ff3YX3:()fxxx∈R1.2()Ixx=()xX∀∈:IXX→XIXI1.3.12,,,,.naaa{}:na,naA∈,n∀∈NA1.5A()nfna=:fA→N{}na:{}nfan→∈NN{}na.1:f−().fX→Rf(())XfD()fRxX∈()fxy=(),f∀∈yR:fXY→1.1.6,,,XI1:fYX−→:fXY→:fXY→:fXY→1XXII−=.:.:,:fXYgYZ→→XZ(())xgfxgf:gfXZ→,()()(())xXgfxgfx∀∈=21.7.31;YffI−=1,XffI−=1:fYX−→:fXY→1.31,:,:fXYgYZ→→:gfXZ→2,:,:fXYgYZ→→(3),,:gfXZ→111().gffg−−−=.1.81.AAAIE:21fnn−,2.1.6AA12(,,,,)().nijAaaaiaa=≠≠j,:nfna3.12,,,,nAAA1{,};nnnAxnxA∞=≡∀∈∈∩N1{,s.t.}.nnnAxnxA∞=≡∃∈∈∪N41.4123P9—10.,AB1ABAB∩AB∪2ABAB∪1.8Q{0},+−=∪∪QQQ:frr−r+∀∈Q+Q−Q{0}+Q1nnA∞+==∪Q1.4(2),+nA312,,,{}nAnnn=5.()(0,1)(0,1),,(0,1)R(,)abP6.D“”),X:fAαα()XDα⊂∀∈{}ADαα∈D{}.DAαα∈{}DAαα∈{,};DAxDxAαααα∈≡∀∈∈∩{,}.DAxDxAαααα∈≡∃∈∈∪D={}nnA∈{}nnA∈,Arcsinyfxx=={Arcsin}xAyyx==[11]x∀∈−,[11]()xxfA∈−=∪R[11]{}xxA∈−.1.1.9KK,ab∀∈K,ab±∈K,ab∈K(0)abb∈≠KK,,+NRZ01,,QRCK.2.A⊂R10MxA∀∈xM≤A2,,L∈RxL≤LxA∀∈AAA3l∈RxA∀∈xl≥lAA,AA“”“”A3.1.10.A⊂RA≠∅1(),µ∃∈R,,0,,s.t.,xAxxAxεεµεµε∀∈≤∀∃∈−µAsupAµ=2(),.ν∃∈R,,0,,s.t.,xAxxAxεενενε∀∈≥∀∃∈+νAinfAν=4.1.,supAinfAA(inf)AA∈supmax(infmin)AAAA==supAA∈25supAinfAsupAinfA3supAmaxAinfAminA+∞4.AAsupA=+∞AinfA=−∞5.AB⊂⊂RinfinfsupsupBAAB≤≤≤1.8;(0,1)A=sup1A=inf0A=,[1,2){e}B=−∪supe,B=inf1;B=−,11{1,,,}23C=−−−sup0,inf1;CC==−inf1,sup==+∞NNinf,sup.=−∞=+∞RR5.7“”“Cauchy”61.5{}nxlimsup{}nnnnxx→∞∈=Nliminf{}nnnnxx→∞∈=.NP12[],inf{}nxν={}nx,0ε∀N∃∈NnN,nxνε−.nxνενε−+[].{}nxinf{}nnxν∈=N,,,0ε∀N∃∈N{}Nnxxn∈∈N.Nxνε+{}nxnN,nNvxxεννε−≤≤+,nxνε−,liminf{}.nnnnxx→∞∈=N3“”“”“”“”“”3R.13R3,,∀∈Rabc,λµ∀∈R+abλ⋅a1“+”“”:⋅+abλ⋅a3∈R3∈R2“+”:+ab+=ba()()+=+ab+ca+bca+0=a()−=a+a0⋅()()λµλµa=a()λλλa+b=a+b()λµλµ+a=a+a3“”:1⋅a=a.12434“+”“”⋅3\“”3R3R“+”“”33×RR3×\R3R3R⋅3R21.11(XK=KR=^K).:“”(“”+XXX×→(,)xyxyX+∈6Xxy+x+y“”“”,).XX×→K(,)xxXλλ⋅∈6⋅Xxλ⋅λxxλ⋅“+”(1),xyX∀∈xyyx+=+(2),,xyzX∀∈()();xyzxyz++=++(3),s.t.(0X∃∈xX∀∈0xx+=0Xx(4),s.t.(),uX∃∈xX∀∈0xu+=ux−()0xx+−=“”⋅(5),λµ∀∈KxX∀∈()()xxλµλµ⋅⋅=⋅X(6),λ∀∈K,,xyX∀∈()xyxyλλλ+=+(7),λµ∀∈K,xX∀∈()xxxλµλµ+=+(8)xX∀∈1xx⋅=“+”“”(,),X⋅K(,,,)X+⋅KX=KR=KC31200,00,()()()xxxxλλλλ==−=−=−3,xX∀∈4000.xxλλ=⇒==4.1.9.nRnC1212(,,,),(,,,)()TTnnnnxaaaybbb∀==∈RC()λ∀∈RC1122(,,,)Tnnxyababab+=+++12(,,,)Tnxaaaλλλλ=.()nnxy+∈RC()nnxλ∈RC125678()nnRC0(0,0,,0)()Tnn=∈RC()nnRC12(,,,)Tnxaaa=().nn∈RC12(,,,)Tnxaaa−=−−−n\n^1.10.mn×Rmn×C,()mnmnijijAaBb××∀==∈RC()λ∀∈RC,ijijijmnmnABabAaλλ××+=+=()mnmnAB××+∈RC().mnmnAλ××∈RC()mnmn××RC()mnmn××RCmnO×()mnmn××RC()mnmnijAa××=∈RC()()mnmnmnmnijAa××××−=−∈,RCRC()nnnn××CRn1.11.[,]Cab,[,],fgCab∀∈,,fgfλλ∀∈+R()()()()fgxfxgx+=+[,]xab∀∈()()()fxfxλλ=[,]xab∀∈1[,],[,]fgCabfCabλ+∈∈2“+”O[,],()0xabOx∈=[,]Cab3“”⋅[,]Cab1.13.l∞—l∞1212(,,,,),(,,,,)kkxylξξξηηη∞∀==∈()λ∀∈RC,xyxλ+11(,,,)kkxyξηξη+=++1(,,,)kxλλξλξ=,,xyxλ+,.xylxlλ∞∞+∈∈(1)(2)(5)(6)(7)(8),0=(0,0,…,0,…)l∞l∞12(,,,,)kxξξξ=12(,,,,).kxξξξ−=−−−()l∞(),(0c0cl∞l∞22121(,,,,){}kkklxξξξξ∞===+∞∑,,,XX1.14,abab⊕=kkaa=:,⊕:+R,,abk+∀∈∀∈RR+R,abka++⊕∈∈:RR,⊕::(1);ababbaba⊕===⊕(2)()()()();abcabcabcabc⊕⊕===⊕⊕1;+R+R(3)1,,a+∀∈R11,aaa⊕=⋅=(4),a+∀∈R111()aaaaa⊕=⋅=1a(5),,,akl+∀∈∀∈RR()()()();llkklklakaaakla====::::(6),,,abk+∀∈∀∈RR()()()kkabkabab⊕==::;kkkkababkakb==⊕=⊕::(7),,,akl+∀∈∀∈RR();klklklklaaaaaakala++===⊕=⊕:::1(8)1.a