4.1 4.2 4.3

4.1 14.1.1 ABRABRABA=BRA

RRxRyRxRyR=RABR=ABRABA=BAR=AAAR={|xA}AIA

nS AiS AinA1=A2==AnSAn

4.1.2 RABD(R)={x|(y)(xRy)}R(R)={y|(x)(xRy)}F(R)=D(R)+R(R)D(R)R(R)F(R)RD(R)AR(R)B

RSRSRSR-SRSRx(RS)yxRyxSyx(RS)yxRyxSyx(R-S)yxRyxSyx(RS)yxRyxSyxR yxRy

24.1.3 A={a1,a2,,am}B={b1,b2,,bn}RAB 1 aiRbjrij= 0 MR=(rij)mnR

RMRMRRARAaiajaiajRaiajaiajRaiRGR

34.1.4 RAAAxxRxRAR

U=Z

4.1.5 RAAAxxRxRAR

4.1.4

4.1.6 RAAAxyxRyyRxRAR(x)(y)(x,yAxRyyRx)R

UZ=

4.1.7 RAAAxyxRyyRxx=yRAR(x)(y)(x,yAxRyyRxx=y)Rx=yRx=y

U=Z=

4.1.8 RAAAx, y, zxRyyRxxRzRAR(x)(y)(z)(x,y,zAxRyyRzxRz)R

=

ARMR0GRARMRGR

ARMR1GRARMRrij=rjk=1rik=1GRMRGRR

=Z

4.1.1 RAARR

4.2 R, SABRSRSRR-SAB

14.2.1 RABSBCRSACRSRSRSRSRSRS={|(b)(bBaRbbSc)}

4.2.1 RAARIA=IAR=R4.2.2 RABS,TBCWCD R(ST)=RSRT R(ST) RSRT (ST)W=SWTW (ST)W SWTW

4.2.3 RABSBCTCD(RS)T=R(ST)

24.2.2 RAnNRnRn(1) R0=IA(2) Rn+1=RnR4.2.4 RAAm, nN(1) RmRn=Rm+n(2) (Rm)n=Rmn

4.2.5 RAA|A|=nijRi=Rj0i

34.2.3 RABRBAR-1R-1RR-1RR-1={|R} xRyy R-1x-1=(AB)-1=BA

4.2.7 RABSBC(RS)-1=S-1R-14.2.8 RSAB (R-1)-1=R D(R-1)=R(R)R(R-1)=D(R) (RS)-1= R-1S-1 (RS)-1= R-1S-1 (R-S)-1= R-1-S-1 RSR-1S-1

4.2.9 RAARR=R-1

4

4.2.4 RARR1 R1 RR1 R2RR2R1R2Rr(R)s(R)t(R)

4.2.10 RAA Riff r(R)=R R iff s(R)=R Riff t(R)=R

R4.2.4RR1r(R)=Rr(R)=R4.2.4RRR

4.2.11 RAA r(R)=RIA s(R)=RR-1 t(R)=

4.2.12 RAA|A|=nt(R)= 4.2.13 RAA rs(R)=sr(R) rt(R)=tr(R) st(R)ts(R)

5RSABTBCMR=(aij)mnMS=(bij)mnMT=(cij)npdij

MRS=MRMSdij=aijbij 1im1jn MRS=MRMS dij=aijbij 1im1jn dij=aji 1im1jn

MR-S=MR dij=aij(bij) 1im1jn =MRT dij=aij 1im1jn MRT=MRMT dij= (aikckj) 1im1jn

4.3 14.3.1 RARRARaRbababRabbaab

mZ

4.3.2 ma, bZma-b=kmabmab(mod m)4.3.1 mAZR

4.3.3 RAaA[a]R={x|xAaRx}[a]RaRa[a][a]Ra[a]R

4.3.2 RA a, bAaRb[a]=[b] a, bAaRb[a][b]= =AA

4.3.4 RAA/R={[a]R|aA}A/RARA/RR

4.3.5 AB={A1,A2,,An}AiAiAj=ij =ABABAA/RAAB={A1,A2,,An}AR

R= R={|(Ai)(AiBa, bAi)}RAA/RBAA

24.3.6 RARRARRaRbab

RAR-1ARR-1abababABABAB363|636

4.3.7 RA a, bAa

A={1,2,3}A1

4.3.8 a,bAa

4.3.9 BAbB (x)(xBbx)bB (x)(xBxb)bB (x)(bxxb)bB (x)(bxbx)bB

BBBBBB

4.3.10 BAbA (x)(xBbx)bB (x)(xBxb)bB

bBbbbbBglb bBbbbbBlub

BBBBBBBB

abbaab

4.3.11 a,bAaba,bAabbaA

4.3.12 AA

34.3.13 RARRA

4.3.14 AB={A1,A2,,Am} =ABAAAA

4.3.15 RACACa,baRbCRAaRA

4.3.16 RACRA-CCCRCR

4.3.17 RAACR(A)R

1

1

4.3.3 A{A1,A2,,Am}R=

4.3.4 ARCR(A) ARCR(A)4.3.3CR(A)ARCR(A)

44.3.18 RARRA

4.3.5 RARIA

4.3.6 RA Rr(R)=RIA RR-IA