《點(diǎn)集拓?fù)渲v義》第四版(熊金城)課后習(xí)題答案

§

$1? fi? 1

§1.1 $3…………1

§1.2 $#$…………1

§1.3 §fi 5

§1.4 §'§fi 6

§1.5 $‰………………………6

§1.6 *t%#$ 9

§1.7 a.,fla., 9

§1.8 ???3………………10

$2? @fl67fl?:$‰ 12

§2.1 <67fl?:$‰

§2.2 @fl67fl?:$‰

…………………12

…………………15

§2.3 =>fl=>fi 16

§2.4 fl,‰,‰￠ 16

§2.5 ?C,4E 18

§2.6 fl? 21

§2.7 @fl67?§fi 23

$3? ?67,($)L67,M67 26

§3.1 ?67 26

§3.2 ($)L67 28

§3.3 M67 31

$4? ?N? 35

§4.1 ?N67 35

§4.2 ?N?$‰RST$ 38

§4.3 ?Ny@ 40

§4.4 C?N67

§4.5 ‰Z?N67

…………41

…………42

$5? $§a.??3 44

§5.1 $—fl$=a.??3 44

§5.2 ay67 46

§5.3 Lindelo¨f67 47

$6? y$??3 49

§6.1 T,T,Hauzdorf67 49

§6.2 fl^,flfi,T,T67 51

§6.3 Uryzohnfl3aTietzeflfl?3 53

I

§6.4

yKfl^67,Tyshonof67 ………

54

§6.5

y$??3fl?67,($)L67aM67……

55

§6.6

a<fl67…………

57

$7?

hi?…………………

58

§7.1

hi67………………

58

§7.2

hi?fly$??3…………………

59

§7.3

n‰fi?67R?hi?………

60

§7.4

fi?hi??t%7§fi…………

61

§7.5

<67?hi?…………………

63

§7.6

Chi67,fihi67……………

64

$8?

yq<67…………

66

§8.1

<67yqfl……………………

66

§8.2

<67yq?flhi?,Baire?3……………

67

$9?

L67…………………

69

§9.1

*$?"L………………………

69

§9.2

L67…………………

69

§9.3

aL@fl?u………………………

72

§9.4

TyshonofvL?3……………………

74

§9.5

@fl67flfifl?z$……………

75

$10?

$‰67……………

76

§10.1

Afi~@fl…………

76

§10.2

—i~<a—i~@fl………

76

§10.2

hifl@fl……………

76

?fl

………………76

#

?1? fi÷

§1.1 $

1.1 ?flfifi???:‰‰%6?‰‰%???‰‰7$￠?§fi?‰

‰7$??§fi?

(1) ?={s|scZ,?%flflycZ,?s=2y};(2) B={2};

(3) C={s|sc?,%s=1};

(4) D={s|sc?,%sfi?.};(5) E={s|scQ,%s=2}.

?:B=DC?CZ,C=E=?,C,E%B??,B,D%???.

1.2 ??fi?fi§fififl?fl??.

(1) ?={?}; (2) ?c{?};

(3) ?C{?}; (4) ?={?};(5) ?c{{?}}.

?:(2),(3)fifl?.

1.3 ??,?,…,??%,%?n“1.??:?$

?C?C…C?C?C?

^

?=?=…=?

?:?fi?C?,i=1,2,…,n–1,?C?,$§?,?C?C?C?,%?C?,?

??=?,i=1,2,…,n–1.

??=?=…=?.

1.4 ?X={a,b,c}.fi{X ￥(X).

?:￥(X)={?,{a},,{c},{a,b},{a,c},{b,c},{a,b,c}}.

1.5 ?Xfi$nKAfl??y?fifi .X ￥(X)?$:?KAfl??y??

?:￥(X)}y?$1K6,CKSA,CK?$QKy? ,…,CK?$n–1K

y? ,CK?$nKy? (X$?),?￥(X)?$1+C+C+…+C=2KAfl??

y?.

§1.2 $$

2.1 ?,B,C?%,??:

?C?UB,?t?3?fiB;

@?CB,^$?UCCBUCa?fiCCBfiC;(3) @?CB,^B–(B–?)=?.

?:(1)?3,(2)?fi?CB,???=?fiB,B=?UB,flfiBUC=(?UB)UC=BU(?UC)3?UC

·1·

?fiC=(?fiB)fiC=?fi(BfiC)CBfiC

?fi?CB,^scB–(B–?)fi%?fiscB%s?B–?fi%?fisc?,?B–(B

–?)=?.

2.2 ?Xfi?,?,B§?fi%?,??:

?–B=?fiB′;

?–B=(?UB)–B=?–(?fiB);

(3) @?UB=X,?%?fiB=?,^$?′=B,B′=?;

(4) (?–B)fi(?–B)=(?fi?)–(BUB).

?:(1)sc?–B,fi%?fisc?,s?Bfi%?fisc?,scB′,fi%?fisc?fiB′,??

?–B=?fiB′.

(2)(?UB)–B=(?UB)fiB′=(?fiB′)U(BfiB′)=?fiB′=?–B

?–(?fiB)=?fi(?fiB)′=?fi(?′UB′)==(?fi?′)U(?fiB′)=?fiB′=?–B??,?–B=(?UB)–B=?–(?fiB)

?fi?fiB=?,??B–?=B,?′=X–?=(?UB)–?=B–?=B,?fl?′=B,

flfiB′=(?′)′=?.

(4)(?–B)fi(?–B)=(?fiB′)fi(?fiB′)=(?fi?)fi(B′fiB′)=(?fi

?)fi(BUB)′=(?fi?)–(BUB).

2.3 ??,?,…,?,B?%,%?nfifl‰..??:

y?

Bfi(U?)=U(Bfi?)

BU(fi?)=fi(BU?)

DeMorgan

B–(U?)=fi(B–?)

B–(fi?)=U(B–?)

?:(1) scBfi(U?)escB%scU?eflfli,?scB%sc?eflfli,?

scBfi?escU(Bfi?)

??Bfi(U?)=U(Bfi?)

scBU(fi?)escB?scfi?escB?sc?(i=1,2,…,n)e?k?i,?s

cB?sc?e?k?i,scBU?escfi(BU?)

??BU(fi?)=fi(BU?)

(2) scB–(U?)escB%s?U?escB%s??(i=1,2,…,n)e?k?i,s

cB%s??e?k?i,scB–?escfi(B–?)

??B–(U?)e=fi(B–?)

·2·

scB–(fi?)escB%s?fi?escB%flfli,s??eflfli,scB%s??e

flfli,scB–?escU(B–?)

??B–(fi?)e=U(B–?)

2.4 ??,?,…,??%,%?nfifl‰..??:

U?–(fi?)=U(?–?) (?=?)

?:@?X=U?fi?.÷?=?,U?–(fi?)=U(?–(fi?))=U(U(?

–?))3U(?–?).

fl—fi?,???–?C(?–?)U(?–?)

??k,@sc?–?,%sc?,s??,^fisc?,sc?–?,fis??,sc?

–?,??$:sc(?–?)U(?–?),flfi:

?–?C(?–?)U(?–?)C(?–?)U(?–?)U(?–?)C…C(?

–?)U(?–?)U…U(?–?)CU(?–?)

??,U(?–?)CU(?–?)

?U?–(fi?)=U(?–?) (?=?)

2.5 ??aB%QK.?$?flB??}?=B

?=B=(?–B)U(B–?)

??:??}#$?fl??‰?3,%:?$?,BaC?%,^

?=B=B=?;

?=?=?;

~ ~

flfl—K ?, ??=?=?;

(4) (?=B)=C=?=(B=C).

?:(1),(2),?3.

~ ~

(3)$?=?,^?=?=(?–?)U(?–?)=?

(4)(?=B)=C=((?–B)U(B–?))=C=(((?–B)U(B–?))–C)U(C–((?

–B)U(B–?)))=(((?–B)–C)U((B–?)–C))U(C–((?UB)–(?fiB)))=

(?–(BUC))U(B–(?UC))U(C–(?UB))U(?fiBfiC)=(B–(?UC))U(C

–(?UB))U(?–(BUC))U(?fiBfiC)=(B=C)=?=?=(B=C).

2.6 ??nK?,?,…,??‰?,?,}÷?#$,?:?flfi2KAfl??

;?%??$nK,y??‰?,?,}÷?#$??flfi2KAfl??.[?:?fl??nKQQfi??‰?,?,}÷?#$?:?flfi2KAfl??.%,fl?,?,…,??‰÷?#$flfi?fi{n=2–1KQQfi?B,B,…,B,?%??

—??a$B,B,…,B?‰÷?#$${.?%?,?,…,??‰÷?#$flfi *flB,B\

–2,…,B?‰÷?#$flfi *??.k‰QK??fl?%?$?$—Cy??.N

·3·

‰?k??‰?yfl,‰{?fl$?$=Cy?????fl?q?f.]

?:yQ???

$—?,??nKQQfi? ?‰?,?,}÷?#$?:?flfi2KAfl?? .

?fi,N‰?#$,nKQQfi? ?:?flfi

C

+C

+…+C=2–1

KAfl?? ,fi?#$?§fl6,}#$fifl$§fl?fl? ,‰?yfif$—???.

$=?,??knK?‰?,?,}÷?#$?:?flfi2–1KAfl?? .

fiE=fi?

E=?fi?fi…fi?fi…fi? i=1,2,…,n

E =?fi?fi…fi?fi…fi?fi…fi?

i<i

E =?fi?fi…fi?fi…fi?fi…fi?fi…fi?

i<i<i

……

E =?

j?i,i,…,i

%??$?$?,EKY:fi1,E

KY:fiC,E

KY:fiC,…,E

KY:fiC,?y?KY:fi

%yfiU?

1+C

+C

+…+C

=2–1

÷B=E,B

=E

–B,B

=E–B,…,B

=E

–B,

B =E

–(UB),B

=E

–(UB),…,B

=E

–(UB),

……

B=?–(U

B),B=?–(U

B),…,B=?–(U

B),

B,B,…,B

Y:$2–1KQQfi?,%—?a$B,B,…,B

‰,?,

}÷?#$$$,%??,?,…,?‰,?,}÷?#$flfi )flB,B,…,B‰‰÷?#$flfi ),?.

$$—?12B,B,…,B

‰,?,}÷?#$flfi fl? 4 KY:fi2,fl

fi?,?,…,?‰,?,}÷?#$Y:7flfi2KAfl,? 4.

9,fi?={(e,…,1,…,e):e=0;1,k?i},i=1,…,n,E=fi?={(1,1,…,1)}

E=?fi?fi…fi?fi…fi?={(1,…,e,…,1):e=0;1} i=1,2,…,n

E =?fi?fi…fi?fi…fi?fi…fi?={(1,…,e,…,e,…,1):e=0;1} i

<i

E =?fi?fi…fi?fi…fi?fi…fi?fi…fi?={(1,…,e,…,e,…,e,…,

1):e=0;1} i<i<i

……

·4·

E

=?={(e,…,1,…,e

):e=0;1} j?i,i,…,i

fiB=E={(1,1,…,1)},

B=E

–B

={(0,1,…,1)},B

=E

–B

={(1,0,1,…,1)},

…,B=E–B={(1,1,…,1,0)},

B =E–(UB)={(0,0,1,…,1)},

B =E

–(UB)={(0,1,0,1,…,1)},…,

B

=E

–(UB)={(1,1,…,1,0,0)},

……

B=?–(U

B)={(1,0,…,0)},

B=?–(U

B)={(0,1,0,…,0)},…,

B=?–(U

B)={(0,…,0,1)}

‰fi=fi2–1KAfl,??AB,B,…,B ,q?{$‰2–1K?AE‰,?,}

#$FGflfi2KAfl,? 4.

Hfl,J$nK4,y‰,?,}÷?#$F7flfi2KAfl,? 4.

§1.3 fi

KX={a,b},Y={c,d,e}.Lfi‰XxY%$fiO.P:XxY={a,b}x{c,d,e}={(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)}

KXaYR%4.?U:??Xk???,BaYk??C,D,$:(?UB)x(CUD)=(?xC)U(?xD)U(BxC)U(BxD)

(?fiB)x(CfiD)=(?xC)fi(BxD)(XxY)–(?xC)=((X–?)xY)U(Xx(Y–C))

(?–B)xC=(?xC)–(BxC)

3.3 KX={a,b,c},Y={d,e,f,g};R={{a,d},{a,e},{b,f}}.÷?={a,c},B={d,e,

g}.L[R(?),R(B),R\],aR?$].

P:R(?)={d,e},R(B)={a}

RangeR={d,e,f},DomainR={a,b}

3.4 KRfifl4Xfi4Y?§fi.?U:??kc?,BCX,§fiR(?)fiR(B)=R(?

fiB)fif}yijkfl%:??kcs,ycX,@s?y,R({s})fiR({y})=?.

?:}y?:KxcR(?)fiR(B),%xcR(?),xcR(B),flflsc?,ycB,sRx,yRx,@

s?y,R({s})fiR({y})=?,xcR({s})fiR({y})$$.?s=y,flfixcR(?fiB),

%R(?)fiR(B)CR(?fiB),?w1.3.2(2)$R(?)fiR(B)3R(?fiB).Hfl$R(?)fi

R(B)=R(?fiB)

ij?:Ks,ycX,s?y,$R({s})fiR({y})=R({s}fi{y})=R(?)=?3.5 KX,X,X%÷K4.fl§?$XxXxX=XxXxX?

P:(1)fiX(i=1,2,3)?Y}$—Kfi XxXxX=XxXxX=?.

·5·

(2)?k?i=1,2,3,X??,fiX=X=XXxXxX=XxXxX.

§1.4 fi

4.1 ‰{~flt?,??,§÷???Qkfifl~fl$÷k§fi9?.

P:9(1) KX={a,b,c},R={(a,a),(a,b),(b,a),(b,b)},?3R~fl???,§

?,fl~flt??,Hfi(c,c).

(2)fl?R??$§fiR={(s,y):s,ycR,sy“0},?3R~flt??,???,R

fl~fl§?.??k,$s>0,y=0,x<0,(s,y),(y,x)cR,(s,x)?R.(3)2fiO“￠?§fi”C%t?,§ ,fl%??.

KRfi4X???,§ §fi,?URfi§?§fifi%?fiDomainR=X.

?:@Rfi§?§fi,R?$t??,%kcscX,sRs,%?DomainR=X,t$,@DomainR=X,kcscX,$ycX,sRy,@R?$???t§?,%?yRs,flfisRs,%Rfi?$t??,?R%X?§?§fi.

Lfi{R?§?§fiR,y=

R?R={{scR:s“0},{scR:s<0}}.

P:÷F(s)=1+zgns–|zgns|,R?§?§fiRfi

R={(s,y):s,ycR,F(s)F(y)=1}

R?§fi?$fi

?URfi§?§fi.

R={(s,y):s,ycR,s–ycZ},

?:?3§fiR~flt??,???,KsRy,yRx,%s–y=ncZ,y–x=ncZ,s–x=(s

–y)+(y–x)=n+ncZ,%?sRx,%R~fl§?,?RfiR§?§fi.

KR,Rfi4X?QK§?§fi,?UR○Rflfi§?§fifi%?fiR○R=R

○R.

?:HR,R%§?§fi,@R○R%§?§fi,R○R=(R○R)=R○R=R

○R;t$,@R○R=R○R,R○R=R○R=(R○R)=(R○R),kcscX,

sRs,sRs,%?sR○Rs,%O(X)CR○R;@(R○R)○(R○R)=R○(R○R)○R=

R○(R○R)○R=(R○R)○(R○R)CR○R,?R○R%§?§fi.

§1.5 ‰

Kf:X?Y.?U:

??kc?CX,?Cf(f(?)).

??kcBCY,B3f(f(B)).

ffiflk$‰fi%?fi??—BCY,f(f(B))=B.

?:(1),(2)q?.

(3)Kffiflk$‰,?kcycBCY,flflscf(B),f(s)=y,%?y=f(s)c

f(f(B)),%BCf(f(B)),$(2)a,B3f(f(B)),?B=f(f(B)).

t$,K?—BCY,f(f(B))=B,??fl,$B=Y,$f(f(Y))=Y,fif(Y)=X,?

ffiflk$‰.

·6·

Kf:X?Y.?Ufifi?kfl§?:

(1) f%——$‰.

??kc?,BCX,f(?fiB)=f(?)fif(B).

??kc?CX,?=f(f(?)).

??kc?CX,f(X~?)=f(X)~f(?).

?:(1)→(3) ÷f:X?f(X)=kcscX,f(s)=f(s),f:s?f(s)%flk——$‰,

?kc?CX,$fi?5.1(3)a:

?=f((f)(?))=f(f(?))=f(f(?))

(3)→(2)?$f:X?f(X)?k,f%flk,?kc?CX,$fi?5.1(3)$f(?fiB)

=f(f(f(?))fif(f(B))=f(f(f(?)fif(B)))=f(f(f(?)fif(B))=f(?)fif(B)

=f(?)fif(B)

(2)→(4)HX=?U(X~?),f(X)=f(?)Uf(X~?)

f(X)~f(?)=f(X)fi(f(X)~f(?))=(f(?)Uf(X~?))fi(f(X)~f(?))=f(X~?)fi(f(X)~f(?))=f(X~?)fif(X)~f(X~?)fif(?)=f(X~?)

(4)→(1)@f(s)fl%——,flfls,scX,s?s,f(s)=f(s),f(s)cf(X–

{s})=f(X)~f({s})=f(X)~f({s})$$,?f%——$‰.

KXaY%QK4,f:X?Y.?Ufifikfl§?:(1) f%——$‰;

(2) f%~‰;

(3) f○f=iaf○f=i,

%?iaiy?%XaY??$‰.

?:(1)→(2)$?w1.5.3f=.

(2)→(3)Kf%~‰,kcscX,$fi?5.1(3)$f((f)({s}))={s},%

f(f({s}))={s},Hflf○f=i@f:Y?Xfi$‰,f:X?Yfi~‰,?k?=f○f=i.

(3)→(1)Hfif○f=i,%?f%——$‰,@f○f=i,%?f%~‰,Hflf%~—

—$‰.

5.4 KX,Xfi4,p:XxX?Xfi$iK?‰,i=1,2.

(1) fl§??fip%flk?fl§??fip%——?

(2) @scX,fi{4p({a}?.

P:(1)fis??,(j?i),P%flk,fisfi?A(j?i),P%——.

(2)P({a})={a}xX,P({a})=Xx{a}

KX,Yfi4,?$O:X?XxX,=??kc,scX,O(s)=(s,s).?U:(1) O%——$‰.

(2) P○O=i,(i=1,2).

(3) O(X)%?$1.4.1????.

?:(1)fis,ycX,s?y,$(s,s)?(y,y).%?O%——$‰.

(2)kcscX,(P○O)(s)=P((s,s))=s,(i=1,2).%?P○O=i,(i=1,2).

(3)HfiO(X)={O(s):scX}={(s,s):scX},%?O(s)%?$1.4.1????.

KX,Yfi4,acX,bcY.?$$‰k:X?XxY,=??kcscX,k(s)=

(s,b);k:Y?XxY,=??kcycY,k(y)=(a,y).?U:

·7·

(1) k,kR%——$‰.

(2) k(X)=Xx,k(Y)={a}xY.

(3) p○ki,p○k=i.

(4) p○k:Y?Xfi$￠\a$‰,p○k:X?Yfi$￠\b$‰,%?p%XxX

$iK?‰,i=1,2.

?:(1)Ks,scX,%s?s,k(s)=(s,b)?(s,b)=k(s),%?k%——

$‰;?wk%——$‰.

(2)k(X)={k(s),scX}={(s,b):scX}=Xx,%?k(X)=Xx;?wk(Y)={a}xY

(3)kcscX,(p○k)(s)=p(k(s))=p((s,b))=s,%?p○k=i;?wP○

k=i.

(4)kcycY,(p○k)(y)=p(k(y))=p((a,y))=a,%?p○k:Y?Xfi$￠

\a$‰;?w,p○k:X?Yfi$￠\b$‰.

5.7 X,Xfi4.÷

M(X,X)={s:{1,2}?XUX:s(1)cX,s(2)cX}

?$j:M(X,X)?XxX,=scM(X,X),j(s)=(s(1),s(2))c(X,X).?Uj%flk——$‰.

?:K(s,s)cXxX,?$s:{1,2}?XUX,=s(1)=s,s(2)=s,scM(X,

X),%j(s)=(s,s),%j%flk$‰;@Ks,scM(X,X),s?s,%(s(1),s(2))?

(s(1),s(2)),%?j(s)?j(s),j%——$‰.

3k%‰,j%flk——$‰.

5.8 Kf,g,hR%$‰.?U:

ffifflfl(fi).

@ffigflfl(fi),gfihflfl(fi),ffihflfl(fi).

@ffigflfl(fi),%gfifflfl(fi),f=g.

?:(1)Hf|X=f,%?ffifflfl(fi).

@ffigflfl,gfihflfl,fl?Kf:X?Y,g:??Y,h:B?Y,%?X3?3B,f

|?=g,g|B=h,%?f|B=g|B=h,%ffihflfl.

?wa?,@ffig fi,gfih fi,ffih fi.

@ffigflfl,%gfifflfl,fl?Kf:X?Y,g:X?Y,X3X,X3X,?

X=X,flfif=f|X=f|X=g,%f=g.

?wa?,@ffig fi,%gfif fi,f=g.

5.9 KXaY%QK4,f:X?Y,g:Y?X.?U:?$f○g=i,g%—K?‰,f%—K

~‰.

?:?k?y,ycY,Kg(y)=g(y)=scX,

f○g(y)=f(g(y))=f(s)cY

f○g(y)=f(g(y))=f(s)cY

Hf○g=i,%?,y=f○f(y)=f(s)=f○g(y)=y,%g%?‰

?k?ycY,y=f○g(y)=f(g(y)).Hg:Y?X,%?g(y)cX,?%f%~‰.

·8·

§1.6 §%$

6.1 K{?},{?}fikcQK),?U:

(1) (U?)U(U?)=U?

(fi?)fi(fi?)=fi?

(2) (U?)fi(U?)=U(?fi?)(fi?)U(fi?)=fi(?U?).

?:(1)Hfi(U?)C(U?),(U?)C(U?),%?(U?)U(U?)C(U?);fl—fi?,KscU?,flfl?cTUT',sc?,?cT;

$?cT',%?sc(U?)U(U?),%U?C(U?)U(U?),?(U?)U(U?)=(U?)

Hfifi?Cfi?,fi?Cfi?,%?fi?C(fi?)fi(fi?);flfi?,Ksc(fi?)fi(fi?),%kcacT,tcT',$sc?,tsc?,

%?scfi?,%(fi?)fi(fi?)Cfi?,?(fi?)fi(fi?)

=fi?.

(2)(U?)fi(U?) =U(?fi(U?)) =UU(?fi?)

=U(?fi?)

(fi?)U(fi?)=fifi(?U?)=fi(?U?)

6.2 @{T}fi—),%??—acT,??f—K){?} ?U:

U?=U(U?)fi?=fi(fi?).

?:Hfi?—acT,R$U?3U?,%?U?3U(U?);flfi?,KscU?,flfltcUT,sc?,flfiflflacT,tcT,%scU?CU(U?),%?U?CU(U?)

?U?=U(U?).

Hfifi?Cfi?,kcacT,%?fi?Cfi(fi?);flfi?,Kscfi(fi?),?kcacT$scfi?,flfi?kcacTttcT,R$sc?,

%scfi?,%?fi?3U(U?),?fi?=fi(fi?).

§1.7 ,fl,

7.1 ?UKfl$wQfia.

?:HfikcrcQ,ra?—$fi?yp?q,p,qcZ,q>0,??

Q={p?q:p,qcZA?,q>0}={(p,q):p,qA?,q>0}“ZxZ,fiZxZ%a,?

Q%a.

7.2 ?U?4R??,@￠?$—fl??,?=R.

?:K?(CR)￠?fl??E,EC?CR,HflE“?“R,??E=R,$Contor–Bernztein?w1.7.9a?=R.

7.3 ?UR=R.

?:fl??fiKflE=R,fiB={(s,s,…,s,…):0<s<1,n=1,2,…},B

·9·

=E.

??k,kcsc(0,1),(s,s,…)cB,Hfl,(0,1)“B;t$,kcs=(s,s,…,s,…)c

B,s????fifi$$:

s=0.ss…s…

s=0.ss…s…

…………

s=0.ss…s…

fl$‰p:B?(0,1),p(s)=0.sss…ss…s…,?3p%——$‰,?B

“(0,1).

$Contor–Bernztein?w1.7.9a,B=(0,1)=R.HflE=R.

$?R=R,HfiR={(s,s):s,scR},%?R={(s,s,0,…,0,…):s,scR}

“E=R,%R“R,@?3R“R,$Contor–Bernztein?w1.7.9aR=R.

7.4 @Xfifi a.?U2fifla.2?%$$?fifi?)fia

.

?:$=?fi$$??(0,1)?,@—K$Q?$$,fi?$1‰fi —?,fi=

?fiKflfi?,?=(0,1).HfiX%fia,X?y?$?3?aq,q,…,q,…,fl

$‰p:2??,=kcDc2,p(D)=0.tt…t…,fiqcD,t=1;fiq?D,t=0.

??,p%flk——$‰,?2=?=(0,1),%2%fla.

KX={q,q,…,q,…},fiX%X?nKy?%$?fifi?),Xa(??

kfi2–1K),%UX={2?%$$?}.Hfl2?%$$?fifi?)fia

.

7.5 KXfi4.?U2={0,1}.

?:fl$‰F:2?{0,1},=?—?c2,

%?scX

F(?)(s)={1, sc?

0, s??

?3F%——$‰,@?kcfc{0,1},fi?={scX:f(s)=1}?c2,%F(?)=

f,%?F%flk$‰.

?F%flk——$‰,%2={0,1}.

§1.8 üz

8.1 ?U?w1.8.2a1.8.3R§?????w.

?:(1)?w1.8.2§?????w.

~ ~

KX% 4,fiXfi%$X? fifi ),X.$?w1.8.2a,flfl$‰

~ ~

y:X?U~?=X, =k—?cX,y(?)c?,%y%X???,Hfl???wfif.14?

w1.8.2?Ua,?w1.8.2fl???w§?.

(2)?w1.8.3§?????w.

~ ~ ~

KX%4,fiXfi%$X?fifi ),X.÷?={?x{?}:?cX},

?%.%?fiOQQfi?.$Zermelo??(?w1.8.3)flfl4C,=??—

·10·

~

?x{?}c?,(?x{?})fiC={(h,?)}fi?A,hc?,?$$‰s:X?X,=—?c

~

X,s(?)=h,s%X???,????wflfl,14?w1.8.3?U%a,?w1.8.3fl

???w§?.

8.2 KX,Yfi4,?UY“Xfi%?fiflflflXfiYk$‰.

?:KY“X,%flflYfiX?——$‰f,?$g:X?Y,

g(s)={f(s), fiscf(Y)

y, fis?f(Y)

%?yfiY?—??y,g%flXfiYk$‰.t$,@flflflXfiYk$‰g,fi

?={?:ycY,g(y)=?}

?%X?),%??fiOQQfi?,$Zermelo??flfl4CCX,=??—?c

?,?fiC%?A,%?flflCfiYk——$‰,%C=Y,@C“X,?Y“X.

·11·

?2? flflfi3‰

§2.1 ?flfi3‰

1.1 ?$o,o':R×R?R=??kcs,ycR,$o(s,y)=(s–y),o'(s,y)=|s–

y|.?Uoao'Rfl%R ?.

?:$x=

s+y

,%?s,ycR,s?y,o(s,x)=(

2

(s–y)

s–y

),o(x,y)=(

2

s–y

2)

,o(s,x)+

o(x,y)=

2

<o(s,y),?ofl%R ?

fio'(s,y)=0,%s–y=0,s=±y,?o'hfl%R ?.

1.2 ?U:??$KA ??R%$? ??.

?:K(X,p)fi??,%Xfi$ ,?y?UX—?R%fl.HfiX%$

)={s},%X?%$?A

,fis=min{p(s,y):s,ycX,s?y},?X?kc—As,B(s,s

2

fifl.%?X—?R%fl.

1.3 K(X,p)%—K$? ??.?U:

X—?R%fl;

(2)?$Yh%??,k?$‰f:X?YR%??.

?:(1)K?fiXk—? ,kcsc?,$s=1,B(s,s)={s}C?,%??fifl .

4

Kf:X?Yfik—$‰,UfiY?k—fl,$(1)a,f(U)fiXfl,%?ffi??$

‰.

1.4 4XQK?pap?fi§?,?$X??%??(X,p)?flfi

%?fi?%??(X,p)fl.

Kpap%4XQK§? ?,Y%—K??,f:X?Y.?Uf,???pfi?

%??fi%?fif,???pfi?%??.

?:K?%Y?k—fl,$f,???pfi????=f(?)%??(X,p)?

fl,Hp,p§?=f(?)%??(X,p)?fl,%f,???pfi???.

ij????U.

1.5 ?$p,p:R×R?R=??k?s=(s,s),y=(y,y)cR,

p(s,y)=max{|s–y|,|s–y|}

p(s,y)=|s–y|+|s–y|

?U:

(1)p,pR%R ?.

(2)??(R,p),(R,p),(R,p)(p?$?92.1.2)$yK,?fl(c%—

4??$—?fi?%fl,??fl—?fi?h%fl).

(3)Kf:R?Rfi—$‰,@f??R ?p,p,p$—fi?fi??$‰,f??R

?p,p,p$fl—fi?h%??$‰.

·12·

?:(1)p,p~fl?kfl1),2)%?3,fi???yh~fl÷?fl§fi.

Ks=(s,s),y=(y,y),x=(x,x)cR,

p(s,y)=max{|s–y|,|s–y|}“max{|s–x|+|x–y|,|s–x|+|x–y|}“max{|s–x|,|s–x|}+max{|x–y|,|x–y|}=p(s,x)+p(x,y).

p(s,y)=|s–y|+|s–y|“|s–x|+|x–y|+|s–x|+|x–y|=(|s–x

|+|s–x|)+(|x–y|+|x–y|)=p(s,x)+p(x,y).

?p,pR%R ?.

(2)Hfip(s,y)“p(s,y)“2p(s,y) g

KUfi(R,p)?fl,%kcscU,flfls>0,B(s,s)CU,%?B(s,s)$$?

?(R,p)?s???].

$g@4fl§fi,B(s,s?2)CB(s,s)CU,%U%(R,p)fl.t$,KU%(R,p)

?fl,%kcscU,flfls>0,B(s,s)CU,$g}4fl§fi,B(s,s)CB(s,s)C

U,%U%(R,p)fl.

Hfl(R,p)fl(R,p)$,?fl.

@p(s,y)“p(s,y)“2p(s,y) g

fi$flfl§fi,fika?(R,p)fl(R,p)$,?fl,Hfl(R,p),(R,p),(R,p)$

yK,?fl.

fl?Kf?R ?pfi?fi??$‰,KUfiR?(?R? ?fi?)k—fl,

f(U)fi(R,p)?fl,$(2),f(U)h%(R,p),(R,p)?fl.Hfl,f??R

?p,p$—fi?,h%??$‰.

%yQ??????U.

1.6 flfi???Rfi??R $‰n,?:R?R?$fi:??k?s=(s,s),

n(s)=max{s,s}

?(s)=s+s

?U:na?R%??$‰.($:y?$R ?pap(??fi?5).)

?:fl?n%??$‰.Ks=(s,s)cR%kc—A,kcs>0,?y=(y,y)cR,

Hfi

p(s,y)=max{|s–y|,|s–y|}

“|max{s,s}–max{y,y}|=|n(s)–n(y)|

(%?p%fi?1.5??$R ?),?n(B(s,s))CB(n(s),s).%nflscR??R?pfi?%??,$?scR%kc,flfin??R ?pfi???,$fi?1.5(3)

a,n??R ?pfi???.

???%??$‰.Ks=(s,s)cR%kc—A,kcs>0,?y=(y,y)cR,Hfi

p(s,y)=|s–y|+|s–y|

“|(s+s)–(y+y)|=|?(s)–?(y)|,

(%?p%fi?1.5??$R ?),??(B(s,s))CB(?).%?flscR??R?p

fi?%??,$?scR%kc,????R ?pfi???,flfi??R ?fi??

?.

·13·

1.7 K(X,p)fi??.p',p":XxX?Ry??$fi??kcs,ycX,

p'(s,y)=p(s,y)

1+p(s,y)

p"(s,y)={p(s,y), fip(s,y)“1,

1, fip(s,y)>1,

?U:

(1)p',p"R%X ?.

(2)??(X,p),(X,p'),(X,p")$yK,?fl.

(3)Kf:X?Yfi—$‰,%?Yfi??.@f??X ?p,p',p"$—fi?fi??$‰,

f??X ?p,p',p"$fl—fi?h%??$‰.

?:(1)p',p"?3~fl?kfl(1),(2),fi??Up',p"~fl÷?fl§fi,?kcs,y,xc

X,Hfi

p'(s,y)=1– 1

1+p(s,y)

= p(s,x)

“1– 1

1+p(s,x)+p(x,y)

+ p(x,y)

1+p(s,x)+p(x,y)

“p'(s,x)+p'(x,y)

1+p(s,x)+p(x,y)

@p"fl~fl÷?fl§fi,%flfls,y,xcX,p"(s,y)>p"(s,x)+p"(x,y),$p"?$,p"(s,y)

“p(s,y),%p"(s,y)“1,flfip"(s,x)<1,p"(x,y)<1,?p"(s,x)=p(s,x),p"(x,y)=p(x,

y),?%,p(s,x)+p(x,y)<p(s,y),flp%?$$.

Hflp',p"R%X ?.

(2)?kcs,ycX,Hfip'(s,y)“p(s,y),%?B(s,s)CB(s,s)(s>0),flfi(X,p')

?fl%(X,p)?fl,t$,@V%(X,p)?fl,kcscV,$0<s<1?2,B(s,s)

CV,??ycB(s,s?2),Hfip(s,y)=p'(s,y)

s?2

“

=2s<s.HflB(s,s?2)C

1–p'(s,y) 1–1?4 3

B(s,s)CV,%V%(X,p')?fl,?(X,p),(X,p')$yK,?fl.

@p"(s,y)“p(s,y),%?(X,p")?fl%(X,p)?fl;flfi?,fip(s,y)“1,

p"(s,y)=p(s,y)“p'(s,y);fip(s,y)>1,p"(s,y)=1>p'(s,y),Hfl?$p"(s,y)“p'(s,y),flfi(X,p')?flh%(X,p")?fl,?fi(X,p)?flh%(X,p")?fl,?(X,p),(X,p")$yK,?fl.

3k%‰,(X,p),(X,p'),(X,p")$yK,?fl.

(3)flfi?1.5(3)?U??.

1.8 fi?1.5afi?1.6kfla12‰?fin‰fi??R,??.

:fi1.5 ‰?:p,p:R×R?R?$fiflRxRfiR$‰.??ks=(s,

s,…,s),y=(y,y,…,y)cR,

p(s,y)=max{|s–y|,|s–y|,…,|s–y|}p(s,y)=|s–y|+|s–y|+…+|s–y|,

$

(1)p,p%R .

(2)(R,p),(R,p),(R,p)(%?p%#2.1.2??$)$yK'?fl.

·14·

(3)?:R?Rfi—$‰,@???R p,p,p$—fi/fi?1$‰,???R

p,p,p$fl—fi/h%?1$‰.

?

fi1.6‰?:fln‰fi?Rfi78R$‰n,?:R Ry:?$fi:??k

s=(s,s,…,s)cR,n(s)=max{s,s,…,s},?(s)=s+s+…+s,n,?fi?1$‰.

?flfi1.5a1.6=>.

§2.2 flflfi3‰

2.1 ?#2.2.5.

?:(1)Xc”,?fiX'=?,@3%X—Ka8?,flK, ?$$?c”.

(2)?,Bc”,?$?aB$?$—K%,?fiB=?c”.J??aBfl%,

‰(?fiB)'=?'UB'%X—Ka8?,N??fiBc”.

(3)”C”,÷”=”–{?},@3$U?=U?,?$”=?,U?=U?

=?c”,”??,k?$?c”,‰(U?)'=(U?)'=fi?'C?'%X—Ka8?,N?U?c”.3kN‰,”%X—K@fl.

2.2 NfiW38,÷?={n,n+1,…},n=1,2,….÷”={?,?,?,…}

?”fiN@fl.

fi{1cNN$flZ[.

?:@3?,N=?c”,@?fi?=?c”,n=1,2,…,k”C”,U?=?

c”,?fl”fiN@fl.

(2)1cN^—flZ[fi?=N.

2.3 _n=2,3,4{:

abnKA d—?$:gK@fl?

abnKA@fl—?$:gK?h§j=?

:fin=2,$4K@fl,3K?h§j=.

fin=3,$29K@fl,9K?h§j=.X={a,b,c},lfl?fi:$1=1K:pq@fl.$2=1K:$s@fl.$3=3K:{{a},?,X}§.$4=6K:{{a},{a,b},?,X},{{a},{a,

c},?,X}§.$5=3K:{{a},{b,c},?,X}§.$6=3K:{{a},{a,b},{a,c},?,X}§.$7

=3K:{{a},,{a,b},?,X}§.$8=6K:{{a},,{a,b},{a,c},?,X},{{a},,{a,

b},{b,c},?,X}§.$9=3K:{{a,b},?,X}§.

fin=4,(t).

2.4 y:u?$fiaa8fi?%afl.

:$fi(X,”)fiX%$ ”=￥(X).?a?$p(X,p)fi$s .

a8fi(X,”)fiX%a8 ”=￥(X).?a?$p(X,p)fi$s .

2.5 ?:|—K$s%afl.

?:@(X,”)%$s,%”=2,??Xk$sp,$§2.1fi3(1)X|—?%(X,p)fl,?flX@fl”%$p‰fl{?,%(X,”)%afl

.

2.6 (X,p)%—K ?:flfi@flX%—K$s,fi%?fip%—K$

s.

?:(}y?) (X,p)%$s ,X|—K??fifl,?%”=￥(X),%(X,

·15·

”)%—K$s.

(???) (X,”)%$s,”=￥(X).N?{s}%fl,$(X,p)?fl ?$,fl

fls>0,B(s,s){s},?k?ycX,y?s$y?{s},@3y?B(s,s).$p(s,y)“s,$0<&<s,?k?ycX,y?sp(s,y)>&fi?.$$s?$p%—K$s.

2.7 ”a”%dXQK@fl.?:”fi”h%X@fl.‰#?”U”a?fl

%X@fl.

?:@”,”%X@fl,$??,Xc”,”,N??,Xc”fi”;k?,Bc”fi”,%

?,Bc”,”,N??fiBc”fi”,k”'C”fi”,%”'C”,”,U?c”,”,N?U?c”fi”,?fl”fi”%X@fl.

#:X={a,b,c},”={{a},{b,c},{a,b,c},?},”={,{a,c},{a,b,c},?},??

”,”%X@fl,”U”={{a},,{a,c},{b,c},{a,b,c},?},fi{a},c”U”,

{a,b}={a}U?”U”,?fl”U”fl%X@fl.

2.8 {”}%$dX—‰@flfifi—K?,%??T?.?:fi”

%X—K@fl.

?:fifi2.7a?.

2.9 (X,”)%—K@fl,%??%k?—Kfl??Xy?.÷X=XU{?},”

=”U{X}.?(X,”)%—K@fl.

?:@3?,Xc”;k?,Bc”,@?,B?$—KfiX,@3?fiBc”;@?,Bc”,

?fiBc”C”,??$?fiBc”;k”C”,@Xc”,U ?=Xc”;@X

?”,%”C”,h$U ?c”C”,??$U ?c”,N?(X,”)fi@fl.

2.10 ?:

(1)fl@flfipqk?$‰%?1$‰;

(2)fl$sfi@flk?$‰%?1$‰.

?:(1)?:X?Yfifl@flXfipqY$‰,?fi?(?)=?,?(Y)=X,fiYfipq,N?Y?k—fl %Xfl,%?fi?1$‰.

(2)?:X?Yfifl$sXfik—@flY$‰,?Y?|flU,?fiXfi$s

,N??(U)%Xfl,%?%?1$‰.

2.11 ‰#?:@fl$?1——$‰‰$‰a?fl%?1.?$??N?t@fl%afl,?‰?‰{‰fi#???

#:Rfi78,i%fl$@fl(R,”)fi@fl(R,”')?$‰,3i%

——?$‰,?R?k??c”,(i)(?)=i(?)=??”',%ifl?.

2.12 "XaY%QK?&@fl.?(:?$X%a-fl,/Yh%a-fl.

?:"”a”y2%XaYQK@fl,p%X—K-,/”=”.@"?%XaY

—K?&$‰,?—fla,bcY,÷

p(a,b)=p(?(a),?(b))

a??(?%(X,”)fi(Y,”)?&$‰.

k$Uc”→?(U)c”=”→U=?(?(U))c”.

k$Uc”→?(U)c”=”→U=?(?(U))c”.

?”=”.

·16·

§2.4 ?,,§

4.1 9:fla‰￠.

(1)"?%$fiX?—Kfi?.9?fla‰￠;

"?%aBfiX?—KflaB?.9?fla