搜索
QVWSR1.?\ A∪(B∩C)=(A∪B)∩(A∪C).XUux∈(A∪(B∪C)).px∈A,9x∈A∪B,x∈A∪C, )x∈(A∪B)∩(A∪C).px∈B∩C,9 !0x∈A∪Bdx∈A∪C, x∈(A∪B)∩(A∪C),* A∪(B∩C)⊂(A∪B)∩(A∪C).V$-[ ux∈(A∪B)∩(A∪C).px∈A, m0x∈A∪(B∩C).px6∈A,/x∈A∪Bdx∈A∪C,Ox∈Bdx∈C, %x∈B∩C, !0x∈A∪(B∩C),* (A∪B)∩(A∪C)⊂A∪(B∩C). %A∪(B∩C)=(A∪B)∩(A∪C).2.?\ (1)A−B=A−(A∩B)=(A∪B)−B;(2)A∩(B−C)=(A∩B)−(A∩C);(3)(A−B)−C=A−(B∪C);(4)A−(B−C)=(A−B)∪(A∩C);(5)(A−B)∩(C−D)=(A∩C)−(B∪D);(6)A−(A−B)=A∩B.XU(1)A−(A∩B)=A∩∁s(A∩B)=A∩(∁sA∪∁sB)=(A∩∁sA)∪(A∩∁sB)=A−B;(A∪B)−B=(A∪B)∩∁sB=(A∩∁sB)∪(B∩∁sB)=A−B;(2)(A∩B)−(A∩C)=(A∩B)∩∁s(A∩C)=(A∩B)∩(∁sA∪∁sC)=(A∩B∩∁sA)∪(A∩B∩∁sC)=A∩(B∩∁sC)=A∩(B−C);(3)(A−B)−C=(A∩∁sB)∩∁sC=A∩∁s(B∪C)=A−(B∪C);(4)A−(B−C)=A−(B∩∁sC)=A∩∁s(B∩∁sC)=A∩(∁sB∪C)=(A∩∁sB)∪(A∩C)=(A−B)∪(A∩C);(5)(A−B)∩(C−D)=(A∩∁sB)∩(C∩∁sD)=(A∩C)∩∁s(B∪D)=(A∩C)−(B∪D);(6)A−(A−B)=A∩∁s(A∩∁sB)=A∩(∁sA∪B)=A∩B.3.?\ (A∪B)−C=(A−C)∪(B−C);A−(B∪C)=(A−B)∩(A−C).XU(A∪B)−C=(A∪B)∩∁sC=(A∩∁sC)∪(B∩∁sC)=(A−C)∪(B−C);(A−B)∩(A−C)=(A∩∁sB)∩(A∩∁sC)=A∩∁sB∩∁sC=A∩∁s(B∪C)=A−(B∪C).4.?\ ∁s(∞Si=1Ai)=∞Ti=1∁sAi.XUux∈∁s(∞Si=1Ai),9x∈S, x6∈∞Si=1Ai,* &n(i,x6∈Ai, %x∈∁sAi,*)1x∈∞Ti=1∁sAi.ux∈∞Ti=1∁sAi,9&n(i,x∈∁sAi,x∈S,x6∈Ai,* x∈S, x6∈∞Si=1Ai, x∈∁s(∞Si=1Ai). %∁s(∞Si=1Ai)=∞Ti=1∁sAi.5.?\ (1)(Sα∈ΛAα)−B=Sα∈Λ(Aα−B);(2)(Tα∈ΛAα)−B=Tα∈Λ(Aα−B).XU(1)Sα∈ΛAα−B=(Sα∈ΛAα)∩∁sB=Sα∈Λ(Aα∩∁sB)=Sα∈Λ(Aα−B);(2)Tα∈ΛAα−B=(Tα∈ΛAα)∩∁sB=Tα∈Λ(Aα∩∁sB)=Tα∈Λ(Aα−B).6.u{An}}$U8 OB1=A1,Bn=An−(n−1Sν=1Aν),n1.?\{Bn}}$U: G )dnSν=1Aν=nSν=1Bν,1≤n≤∞.XUpi6=j, .uij. mBi⊂Ai(1≤i≤n).Bi∩Bj⊂Ai∩(Aj−j−1[n=1An)=Ai∩Aj∩∁sA1∩∁sA2∩···∩∁sAi∩···∩∁sAj−1=∅./Bi⊂Ai(16=i6=n) nSi=1Bi⊂nSi=1Ai.ux∈nSi=1Ai,px∈A1,9x∈B1⊂nSi=1Bi.px6∈A1,Win}M Jm yx∈Ain,x6∈in−1Si=1Ai)x∈Ain.!x∈Ain−in−1Si=1Ai=Bin⊂nSi=1Bi. %nSi=1Ai=nSi=1Bi.7.uA2n−1= 0,1n,A2n=(0,n),n=1,2,···,h U{An} r 7 Tlimn→∞An=(0,∞);ux∈(0,∞),9 8N,yxN,* nNw 0xn,x∈A2n, %x 2 N $_D )x 2 (An, x∈limn→∞An,1 mlimn→∞An⊂(0,∞), %limn→∞An=(0,∞).limn→∞An=∅;p0x∈limn→∞An=∅,9 8N,yn(nN,0x∈An.* p2n−1Nw x∈A2n−1,0x1n.Wn→∞ 0x≤0, O^ %limn→∞An=∅.8.?\limn→∞An=∞Sn=1∞Tm=nAm.XUux∈limn→∞An,9 8N,y$nN,x∈An, %x∈∞Tm=n+1Am⊂∞Sn=1∞Tm=nAm, %limn→∞An⊂∞Sn=1∞Tm=nAm.ux∈∞Sn=1∞Tm=nAm,90n,yx∈∞Tm=nAm,&n(m≥n,0x∈An, %x∈limn→∞An.* limn→∞An=∞Sn=1∞Tm=nAm.29.O $0(−1,1)7(−∞,+∞) $$&+ $$&+ H z Tϕ:(−1,1)→(−∞,+∞).&n(x∈(−1,1),ϕ(x)=tanπ2x.ϕK}(−1,1)7(−∞,∞) $$&+ 10.?\ Fg[k$!%9 3 ! Æ 87=0a[r ! Æ 8}& XUE?\g[S:x2+y2+(z−12)2=(12)2k(0,0,1)!94xOya[M& O O/g= ,PN &n((x,y,z)∈S\(0,0,1),ϕ(x,y,z)=x1−z,y1−z∈M.&?ϕ}$$ -r * S4M}& 11.?\ /B r℄ : G MiCOÆA 5 9AF(ÆO XUuG={△z|△z}B r: G MiC},8Y$△zGnj$!0Q rz,y△z4rz&+ *Æ△z}: G * 0&+3 }$&$ /20Q }O )G40Q IERT$$ &+3 %GF(}O 12.?\ 0 Æ0Q ( zLÆ$O XUuAn}n 0Q ( z l n=1,2,···,9A=∞Sn=0An.An/n+10%R B6 L# n ( z n+100Q bG Oj 0% $0Q b Oj$0Q * Y0B6%R ‘ $0O * /§4#Q6,An=a,1/§4#Q4,A=a.13.uA}a[r%0Q!(P $}0Q )ÆG 0Q Æ J 6 l 9A}O XUn(AG 6 /q0%RB6L#:(x,y,r).bG(x,y)}6 P r}6 J x,y1J‘ 0Q )r‘ 20 0Q *)$}O %A=a.14.?\ :5 S !M(E}O (0 XUuf}(−∞,∞)r :5 B S !l ÆE,/ / O (1)n(x∈(−∞,∞),lim△x→0+f(x+△x)=f(x+0)?lim△x→0−f(x+△x)=f(x−0)$ 8 (2)x∈E / DÆf(x+0)f(x−0).(3)n(x1,x2∈E,px1x2,9f(x1−0)f(x1+0)≤f(x2−0)f(x2+0),* Y$x∈E,&+2B r MiC(f(x−0),f(x+0)),d/(3)OEG!x&+ ! MiC}: G * /11 OF(}O 15.~; y(0,1)7[0,1]AC$$&+ $H-* 3TB(0,1)G0Q l R={r1,r2,···},Wϕ(0)=r1,ϕ(1)=r2,ϕ(rn)=rn+2,n=1,2,···ϕ(x)=x,x∈((0,1)\R),9ϕ}[0,1] (0,1)r $$-t 16.uA}$O 8 9A 00 IOÆ 8’ O XUuA={x1,x2,···},A 0 I l ÆeA.An={x1,x2,···,xn},An I l ÆeAn.&A eAnG202n05 )eA=∞Sn=1eAn,* eAF(ÆO 1AG$05 LÆ 8}O *)eA}O 17.?\ [0,1]r l Q OÆ 8b Æc.XUB[0,1]r Q l ÆA,[0,1]r0Q l Æ{r1,r2,···}, mB=(√22,√23,···,√2n,···)⊂AWϕ(√22n)=√2n+1,n=1,2,···ϕ(√22n+1)=rn,n=1,2,···ϕ(x)=x,x6∈B.9ϕ}A [0,1] $$&+ /[0,1] Æc,OA #}c.18.pAGY05 /: %R O 0D $&$L# A={ax1x2x3···},)Y0xij $0 Æc 9A #}c.XUuxi∈Ai,Ai=c,i=1,2,···.*)0Ai x R $$-tϕi.Wϕ}A E∞ $-t &n(ax1x2x3···∈A.ϕ(ax1x2x3···)=(ϕ1(x1),ϕ2(x2),ϕ3(x3),···). [?\ϕ}$$-t pϕ(ax1x2x3···)=ϕ(ax′1x′2x′3···),9&n(i,ϕi(xi)=ϕi(x′i)./2ϕi}$&$ * xi=x′i, %ax1x2x3···=ax′1x′2x′3···.&n((a1,a2,a3,···)∈E∞,ai∈R,i=1,2,···,*Æϕi}-r 0xi∈Ai,yϕi(xi)=ai. %0ax1x2x3···∈A,yϕ(ax1x2···)=(ϕ1(x1),ϕ2(x2),···)=(a1,a2,···),ϕ}$$-t %A4E∞ 2c.19.p∞Sn=1An Æc,?\ 8n0,yAn0 #}c.XU/2E∞=c, Z .u∞Sn=1An=E∞..+?* pAnc,n=1,2,···.uPiÆE∞ RGo #) -t px=(x1,x2,···,xn,···)∈E∞,9Pi(x)=xi.WA∗i=Pi(Ai),i=1,2,···,49A∗iAic,i=1,2,···. %&Y0i, 8ξi∈R\A∗i,2}ξ=(ξi,ξ2,···,ξn,···)∈E∞. ?ξ6∈∞Sn=1An.|xr pξ∈∞Sn=1An,9 8i,y∈Ai,2}ξi=Pi(ξ)∈Pi(Ai)=A∗i,4ξ∈R\A∗iX’ %ξ6∈∞Sn=1An=E∞,14ξ∈E∞X’ * Fs 8℄0i0,yAi0=c.20.BY jCÆ0;1 Ul Æ 8ÆT,h?T Æc.XUuT={{ξ1,ξ2,···}|ξi=0or1,i=1,2,···}.OT E∞ -tϕ:{ξ1,ξ2,···}→{ξ2,ξ3,···},9ϕ}T E∞ Iϕ(T) $$-t %A≤E∞=c,+A (0,1]iC42I f 4 {$$&+ %Y0x∈(0,1]$O $ Æx=0.ξ1ξ2···,bGY0ξiÆ0;1,Wf(x)={ξ1,ξ2,···},9f}(0,1] T If((0,1])r $$-t *)T≥(0,1]=c.Kr A=c.5 fu- - +E Q -x 9 DFNEIP EoBMKG E′BMCH ¯EBMA 1.2℄ P0∈E′ I|%m!8’P0 V,U(P,δ)(Æ P0 7 )7 =’)P0 P1~)E({xq 1 P1’ g’2),(P0∈Eo I/|’8’P0 V,U(P,δ)( Æ P0 7 ) . yU(P,δ)⊂E.OLoP0∈E′,/%m 8P0 V,U(P,δ), ’P0 7 V,U(P0)⊂U(P,δ), .P1∈E∩U(P0)⊂E∩U(P,δ)dP16=P,Dm;8’P0 V,78’ P1)P0~)E.,3 om 8’P0V,’)P0 P1~)E, l%m P0 V,U(P0)7 ’)P0 P1~)E, P0∈E′.oP0∈Eo,/’U(P0)⊂E.,3 oP0∈U(P,δ)⊂E,