X

10 A

101 1011 1011 ABABABRRxRy;Rx yA=BAAAAB

1 A{01}B={ab} Rl={} R2={} AB R3{} R4{} A

2 X{123}XDxLx Dx={|xXyXxy} Lx{|xXyXxy} Dx Dx={} Lx Lx={}

3 AP(A)R1R2 R1={|xP(A)yP(A)xy} R2={|xP(A)^yP(A)^xy}

A{}P(A){{}}P(A)R1R2 R1{} R2{}

nn1012 nNn>1A1A2AnnA1A2AnA1Ann

10.1.2 A1013 A (1)AIA IA={|xA} (2)A()EA EA={|xA^yA} (3) A

4 A IA={} EA={}

1013 101. 4 ABR (1)Rdom(R) dom(R)={x|(y)(R)} (2)Rran(R) ran(R)={y|(x)(R)} (3)Rfld(R) fld(R)=dom(R)U ran(R)

5 A{abc}B{bcd}ABR={} dom(R)={ab} ran(R){bcd} fld(R)={abcd}

1011 ABRRxUURyUUR R{{x}{xy}}R{xy}R{xy}U RxyURxUUR,yUUR1012 ABR fld(R)=UU R

xxfld(R)xdom(R)xran(R)yRRxUUR ttUURR{{x},{xy}}u{{t},{tu}}R{{u},{ut}}Rtfld(R)

102

1021 1021 X{xlx2xm}Y{yly2yn}(1)RXYRmn(m,n)

(2)RXRmm(mm)

ABRABABmnM(R)m()n()mnM(R)ABM(R)rijRrij01

XYx11x22y1113r13=1R31r310 RXY

2 A{1234}A>>={}>

1022 1022 X{x1x2xm}Y{y1y2yn}(1)XYRG(R)VXYExiyjeijER(2)RXRG(R)VXExixjeijEReijR

3 1XYRG(R)1021XYx1x2y1,y2

4 2A>G(>)1022A

5 A={ab,c} R{,}G(R)1023aaeaaR

103

1031 1031 XYRYZS (1)RR-1YX R-1{|R} (2)RSSRXZ SR{|(z)(R^ S)} A

(4)ARR[A] R[A]{y|(x)(xA^R)}

RR-1RR-1RSRSSRSR1031RSSRRySySRxSR

XYRYZSSRRS(RS)SRXZSRRSSRR ARxAR AAYXYdom(R) AR ARR[A]R A

1 AR A={a{a}{{a}}} R={}

2 NRS R{} S={} R-1{} SR{} RS

1032 SRR-1M(R-1)RM(R)rijrji(ij)M(R-1)SR A|A|=nRSARSM(R)=(rij)M(S)(sij)nXnSR M(SR)(wij)() M(SR)=M(R)M(S)

wijk=1~n(rikskj)M(S)M(R)M(SR)wij^V{01}^1^1l0^1=1^00^00()V0V 001V0=0V1=1V11()

3 A{1,2,3,4}A R{,,} S{,,,}

1033 10.3.1 XYRYZS (1)dom(R-1)ran(R) (2)ran(R-1)=dom(R) (3)(R-1)-1R (4)(SR)-1=R-1S-1 (1)x xdom(R-1)(y)(R-1) (y)(R)xran(R) dom(R-1)ran(R) (2)(1)

10.3.2 XyQYZSZWR

SRRS

10.3.3 XYR2R3YZR1 (1)R1(R2UR3)R1R2UR1R3 (2)R1(R2R3)R1R2R1R3 XYR3YZR1R2 (3)(R1UR2) R3R1R3UR2R3 (4)(R1R2) R3 R1R3R2R3 () P167.

10.3.3 XYRA,B, (1)R[AUB]R[A]UR[B] (2)R[UA]=U{R[B] | B A} (3)R[AB] R[A] R[B] (4)R[A] {R[B] | B A }A =/= (5)R[A]-R[B] R[A-B] (2)(3)

4 ZRR{|xZ^yZ^yx2}A{12}B{Ol2}R[A]{14}R[B]{014}R[A]R[B]={14}ABR[AB]AB{12}R[AB]={14}R[A]R[B]=

104A

1041 1041 ARxAxRxRAxA RARA(x)(xAxRx)RA(x)(xA )

1 AIAEAB{x|xN^x0}DBLBAP(A)

2 AN

3 A{l23}R{}ARAM(R)1(rii1)G(R)RAM(R)0G(R)

1042 RAxyAxRyyRxRA(xRyyRx)(xy)RARA(x)(y)((xA^yA^xRy)yRx)A(x)(y)((xA^yA^xRy^yRx)xy)RA(x)(y)((xA^xA^xRy^xy) )

4 A5 B{x|xN^x0}6 A7 A{123}R={}67

RAM(R)(ij,rijrji)G(R)eijeji(eijeji)RAM(R)(ij,rij=1rji=0)G(R)(eijeji)

1043 RAxyzA(xRy^yRz)xRzRARA(x)(y)(z)((xA^yA^zA^xRy^yRz)xRz)

8 AB={x|xN^x0}9 A{123}R={}RRR

1042 10.4.1 R1,R2AR1-1R1R2R1UR2 xxA=>Rl^R2R1R2R1R2AR1-1R1UR2

1042 R1R2AR1-1 R1R2R1UR2A R1UR2R1VR2R1VR2R1UR2R1UR2AR1-1R1R2

1043 R1R2AR1-1 R1R2A R1-1 ^ R1-1R1^R1=>R1 R1-1 R1-1AR1R2^R1R2Rl^R2^R1^R2=>R1^R2R1R2R1R2A

R1UR210 A={123}R1{}R2={}AR1UR2{}A

1044 R1R2AR1-1R1R2A

R1UR211 A{123}R1={}R2{}AR1UR2{}A

1045 AR(1)RRR-1(2)RRR-1IA(1)RRRR-1RR-1R=R-1RR-1Rx

(2)R RR-1R^R-1 R^ R =>xy=>IA RR-1IA RR-1IA R^R R^R-1 RR-1 =>IA=>x=y R

105 ()

1051 1031051 ARnNRnRn (1)R0{|xA}=IA (2)Rn+1=RnR (n0) nRRnnAAn

1 A{abcd}RR{,,,}R0R1R2R3R4R5

1R2=R4=R6=R3=R5=R7=?1051 A|A|=nRAststRsRt iNRiAP(AA)|A|=n|AA|=n2|P(AA)|2(n2)RR1,R2,R3,,R2n2,ststRsRt(n+1n)

1052 ARAmn (1)RmRn=Rm+n (2)(Rm)nRmn (1)mn n=1RmR1=Rm+1 nk(k1)RmRk=Rm+knk1 RmRk+1=Rm (RkR)=(RmRk)R=Rm+kR =Rm+k+1

(2)mnn=1(Rm)1RmRm*1nk(k1)(Rm)kRmknk+1(Rm)k+1(Rm)k(Rm)=RmkRmRmk+mRm(k+1)

1053 ARAsts

(2)k k=0Rs+0+iRs+i knRs+np+iRs+ipt-skn+1Rs+(n+1)p+iRs+np+p+IRs+np+iRp=Rs+iRp=Rs+p+iRt+i=Rs+i

(3)q0kiq=s+kp+i0ip-1 Rq=Rs+kp+i=Rs+is+is+p-1t-1Rq=Rs+tB2 1RR2R4s2t4B{R0R1R2R3}R 4

1052 RARR'(RR')R'R'R

1052 ARAR'(1)R'()(2)RR'(3)A()RRRRRRR()r(R)(s(R)t(R))

r(R)s(R)t(R)r(R)Rs(R)Rt(R)R

3 1RRr(R)s(R)t(R)10.5.2

1053 1054 AR (1)Rr(R)R (2)Rs(R)R (3)Rt(R)R (1)RRRRRRRRr(R)r(R)R r(R)Rr(R)R (2)(3)

1055 AR1R2R1R2 (1)r(R1) r(R2) (2)s(R1) s(R2) (3)t(R1) t(R2)

1056 AR1R2 (1)r(R1)Ur(R2)r(R1UR2) (2)s(R1)Us(R2)s(R1UR2) (3)t(R1)Ut(R2) t(R1UR2)(1)r(R1)r(R2)Ar(R1)Ur(R2)AR1r(R1)R2r(R2)R1UR2r(R1)Ur(R2)r(R1)Ur(R2)R1UR2

r(R1UR2)r(R1)Ur(R2) R1R1UR2r(R1)r(R1UR2)r(R2)r(R1UR2)r(R1)Ur(R2)r(R1UR2) (2)(3) (3)

4 A{abc}R1R2R1={}R2{}t(R1)=R1={}t(R2)=R2={}t(R1)Ut(R2){}R1UR2{}t(R1UR2){}t(R1)U t(R2)t(R1UR2)

1054 R1057 ARr(R)=RUR0 xAR0RUR0RUR0ARRUR0RUR0RARRRRUR0RR0RRRRR0xyRRRRUR0RRUR0r(R)r(R)RUR0RxAR

1058 AR s(R)R UR-1 R U R-1RVR-1R-1VRRUR-1 RUR-1ARRUR-1

ARRRUR1RR1RRRRR1R,RRRR RUR-1RRUR-1s(R)s(R)RUR1RRR R

1059 ARt(R)RUR2UR3U RUR2UR3Ut(R)n1nNRnt(R) n n1R1Rt(R) nkRkt(R)nk1

Rk+1RkR (z)(R^Rk)=>(z)(t(R)^t(R))=>t(R) Rk+1t(R)nNn1Rnt(R)RUR2UR3Ut(R)

t(R) RUR2UR3URUR1UR2U^RUR2U (s)(Rs)^(t)(Rt) st =>(s)(t)(RtRs) (s)(t)(Rt+s) =>R U R2 UR3 U RUR2UR3UR t(R) RUR2 UR3 U t(R)RUR2UR3U

R+=Uk=1~RkRU R2UR3U R*Uk=1~Rk=R0URU R2UR3 U1053RR2t(R)=R+

10.5.10 A,|A|n,RAknt(R)R+=RUR2UURk R+p>0Rpx0=xx1x2xp-1xp=yRR...Rpp>np+1x0x1x2xp-1xpAnp+10t

R+R+t(R)=RUR2UR3UURn5 A{abc}R R{} r(R)=RUR0 {} s(R)RUR-1 {}

t(R)RUR2UR3 { }

AAR1962WarshallWarshall(B[ji]Bji) (1)BM(R) (2)i1n|A| (3)1jnB[ji]=11kn B[jk]B[jk]VB[ik] (4)i1 (5)in(3) M(R+)=B

6 AR

?10511 AR (1)Rs(R)t(R) (2)Rr(R)t(R) (3)Rr(R) (2) (2)r(R)xyAx=y r(R)r(R)xy r(R)RUR0=>R (xy)=>R (R)=>

t(R)RnRnn n1R1R n=k(k1)Rkn=k1 Rk+1RkR (z)(R^Rk) =>(z)(R^Rk) RRkRk+1 Rk+1nRn t(R)(n)(Rn) =>(n)(Rn)t(R)t(R)

10512 AR (1)rs(R)=sr(R) (2)rt(R)=tr(R) (3)st(R)=ts(R) rs(R)r(s(R)) (1)sr(R)s(RUR0) (RUR0)U(RUR0)-1 RUR0UR-1U(R0)-1RUR-1UR0 (RUR-1)U(RUR-1)0rs(R)

(2)(RUR0)nR0UR1UURnn n1(RUR0)1RUR0 =R0UR1 n=k(k1)(RUR0)kR0UR1UURkn=k1 (RUR0)k+1(R U R0)ko(RUR0) =(R0UR1U...URk)o(RUR0) =((R0UR1U...URk)oR)U((R0UR1U...URk)oR0) =(R1UR2U...URk+1)U(R0UR1U...URk) =R0UR1U...URk+1 tr(R)=t(RU R0) (RU R0)U(R UR0)2U(RUR0)3U... =(R0UR1)U(R0U R1UR2)U... =R0UR1 UR2UR3UR0U t(R) =t(R)U(t(R))0rt(R)

(3)Rs(R)t(R) ts(R)st(R)sts(R)ts(R) sts(R)=ts(R)st(R) ts(R) Rr(R)sr(R)tsr(R)tr(R)str(R)str(R)

106

1061 1061 ARRRA1 AIAEA

2 A{128}R{|xy(mod3)}xy(mod3)x-y3 xyzAx-x3x-y3y-x3x-yy-z3x-z=(x-y)+(y-z)3RRA

RA

9ABAA(a)AAAA(b)(c)RabcRcR

1062 AxA[x]R{y|yA^xRy}[x]RxRx[x]3 2R [1]R={147}[4]R[7]R [2]R={258}[5]R[8]R [3]R={36}[6]RA8A

1061 RAxyA (1)[x]R [x]RA (2)xRy [x]R=[y]R (3)x y[x]R[y]R (4)U{[x]R|xA}=A (1)xAxRxx[x]R[x]R[x]RA (2)x0[x]RxRx0x0Rx.xRyx0Ry,yRx0, x0[y]R.x0[y]Rx0[x]R[x]R[y]R

(3)[x]R[y]Rx0x0[x]Rx0[y]RxRx0yRx0x0RyxRy(4)xA[x]RAU{[x]R|xA} AxAx[x]RxU{[x]R|xA}AU{[x]R|xA}U{[x]R|xA}AAR

1063 ARRAAR AR={y|(x)(xA^y=[x]R)}4 2AR AR{[1]R[2]R[3]R} {{147}{258){36}}

1062 1064 A (1)(x)(xxA) (2) (3)UA (4)(x)(y)((x^y^xy)xy=)AAAA(P(A))AA

5 A{abcd}1{{a}{bc}{d}}2{{abcd}}A{a}{bc}{d}13={{ab}{c}{ad}} 4{{abd}}A

1062 ARAARARAR106310641061ARAAA

1063 AARR{|(z)(z^xz^yz)}RAA

10.6.4 AARRR RRxAxBByAyBxRy(xBR) [x]R=[y]R(R)yB'(xB'R') B=B'x' RR'xyAxRy[x]R[y]Rx[x]R^y[x]R xyxxR'y RR' AA

6 A{123}A

5R1={}R2={}R3={}R4={}R5={}

107 1071 1071 ARRRA1 A={catteachercolddeskknifeby}ARR={|xy}RR

,,,,1R,x1=cat,x2teacher,x3cold,x4=desk,x5=knife,x6by,

1072 ARCACxyxRyCRC={x|xA^(y)(yCxRy))2 1R{x1x2}{x3x4}{x6}{x2x4x5}{x1x2}x3{x1x2x3}

1073 ARCRCR(x)(y)((xCR^yCR)xRy) (x)(xA-CR(y)(yCR^x y))

3 1R{x1x2x3}{x2x3,x4}{x2x4x5}{x6}1071 ARCCRCCR A{a1a2...,an}C0C1C2

C0CCi+1Ci{aj}jajCiajCjR|A|nn-|C|CR.aA{a}CRCRA

1072 1074 A (1)(x)(xxA) (2) (3)UA A 1072 ARAACR(A)CR(A)

1073 A={A1A2An}RA1XA1UA2 XA2 UUAnXAn AARACR(A)AA

4 A={1234} 1={{123}{34}}2{{12}{23}{31}{34}} R{

108

1081 10.81 ARRRARxRyxyxy1 N-{0}AP(A)

1082 ARRRAR

1081 AAR RxAyAxyRRR1082 ARRUR0A1083 ARR-R0A

1083 AARAAR3 (N)

10.8.2

1084 xyAxyxyzAxzzyyxAcorAcovA{|xA^yA^yx}4 A{1234612}DAAAcovAcovA={}

AcovA(1)A(2)xyxyyx(3)covAxy

5 410.8.16 A{abc}1082

1083 1085 BA (1)(y)(yB^(x)(xByx))yB (2)(y)(yB^(x)(xBxy))yB (3)(y)(yB^(x)((xB^xy)xy))yB (4)(y)(yB^(x)((xB^yx)xy))yB

7 4

1086 BA (1)(y)(yA^(x)(xBxy))yB (2)(y)(yA^(x)(xByx))yB (3)C{y|yB}CB (4)C{y|yB}CB

8 A={234691218}ADA

B1{24}41242B2={469}B3={23}612186BBBA()()

1084 1087 xyAxyyxxy1088 xyAxyA9 NN-{0}AP(A)

1089 BA (1)xyBxyBAB (2)xyBxyBAB10 8{2412}{3618}{39}{18}{469}{1218}{49}AA

1084 AnAn nn=1A()n=knk+1Ak+1MAMA-MkA-MAMAk1

1085 Amn+1Am+1n+1

1085 10810 A11 ZZZZ

1086 xyA{xy}A.xyxyyx.

1087 A={a1a2an}BABBxyBxy

ZRabZ|a||b|aRb;a>0-aRa0R-1-1R11R-2-2R2Z0ZR

1088() ZornRR

10811 abRabab(1)[ab]{x|xR^axb)ab(2)(ab){x|xR^axb^xa^xb)ab(3)[ab){x|xR^axb^xb}(ab]={x|xR^axb^xa}ab(4) (-a]={x|xR^xa} (-a)={x|xR^xa^xa} [a){x|xR^ax} (a){x|xR^ax^xa} (-)=R