离散数学(第五版)清华大学出版社第6章习题解答

6

6.1A:;B:;C:;D:;E:

,

5.3

.

.

1

S

,<S,>

.

1S

2

3

4?xS,x-1S.

12

123

2

S

*

<S,,*>

,

,

,,

.

:

1<S,>S

2<S,*>

3*

,

,

,

,

4*

5*

6*

,

7|S|?2,xS,x0,x-1S*

1,23;

.

1,2,34;

1,2,35;

<S,

1-6;

7

.

3

<S,o,*>

.

73

<S,

,

?x,yyxy

S.

S,

<S,,>

.<S,o,*>

*

<S,o,*>

6.3

2

2n

1

n+n=2n?S1

S1+

22*2=4?S2

S2*

3S3

n=6

o,*

S3

n

o

n*

*o

<S3,o,*>

2*3=3*2=0.S6

n

23

n

Sn

Sn

4S4

,xy=min{x,y},xy=max{x,y}

23

xy,xyS4

5

S4

S5

62A:;B:;C:;D:;E:

G

Z4.Zn

n

n

0

<G,>

22

1

34

G

|G|=n?xG,|x|n

74

n

r

xr=e

r

x

|G|=44

124

21=20.

22=22=0.

|2|=2

31=,32=33=2,33=33=1,34=3333=0,

|3=|3

6.32A:;B:;C:;D:;E:

1

UI

DM

?xL,0VX=X,xV0=x,xV1=1,1Vx=10

<L,,,',0,1>

V

1V

01

{0,1}

2

,

.

,

(ab)(abc)(bc)

=(ab)(bc)

=(ba)(bc)

=b(ac)

b(ac)

6.4

Z

x,y,zZ

(xoy)oz=(x+y-2)+z-2=x+y+x-4,

xo(yoz)=xo(y+z-2)=x+(y+z-2)-2=x+y+z-4,

75

2

?xZ,4-xx

<Z>

6.5

G

abcd

aabcd

bbadc

ccdba

ddcab

a

b2=a,c4=cc22=b2=ad4=d2d2=b2=a.

|b|=2,|c|=|d|=4.|G|G

{a},{a,b},G

x

G=<x>

G4

G

S={{a},{a,b},G}<S,?>

6.4

1G=<a>

G

aa-1G

ak

ak

k

01

<ak>={e,ak,a-k,a2k,a-2k,L},

k

{ak}<al>

G=<a>n

G={e,a,L,an-1}G

(n)

(n)

n

n

n

n

.

r,ar

Gd

.

.|D|=4,4

13.G=<c>

76

c1=c,c3=d.

.4

1,2,4,G

3,

4

14

<c>=<c>=<a>={a}.1

4

22

<c>=<c>={b,a}.2

4

<c1>=<c>=G.4

6.4

.

6.6Z[i]

,

,

,

,

.

0,?a+biZ[i],-a-bia+bi

.Z[i]

.

,

.

6.7(1)

6.8

,(2),(3)(4)

x,yS,S

.

xy=(xy)(x''y)S,

S

,

<S,>.

.

0,

?xS

x0=(x0)(x''0)=(x1)(x'0)=x0=x.

0x=x.?xS

xx=(xx)(x''x)=00=0.

6.9(1)X={1,4,5}

(2)<B>={B,B2}={1,4,?5}}.,

G

,a,bG.

y=ba-1.

ax=bG

x=a-1b.

,

ya=bG

X={1,3}-1{3,4,5}={1,3}{3,4,5}={1,4,5}

BP(A)BB=?,

77

n?Bn

B=?

??n

<B>

B

,

,B?.

6.10(1)=(124)(356),=(1634)(25).

(2)

=(15423)(356),=(15462),

-1=(15423)(563)(421)=(1256)(34).

,

,

,(1).

ik(ik)=1.

:1,i1,i2,L,ik,

1,

(1).

(1)=1,

(1)=i1,i11,

,{1,2,L,n}

(i1)=i2.

,

(1)=i1,(i1)=i2,L,)=ik(ik,(ik)=1.

-1

(1,i1,i2Lik).

{1,2,L,n}

,

,

=(1,i1,i2Lik);

{1,2,L,n}

.

.

.(1)=2,(2)=4,(4)=4,(4)=1,

(3)=6,(6)=5,(5)=3,

(124).

{3,5,6}

3,

,

(365).

,

,

=(124)(365).

1

.,=(13)(2)(46)(5)

(2)(5)

,

1-,

(13)(46).

.

,

,

.,=(124)(365)

n

=(365)(124)=(124)(365).

-1,-1?

,

78

(1),(2),,(n)

.

,(1)=((1))=(6)=5.

(2)=3,

(3)=1,

(4)=2,

(5)=4,

(6)=6,

=(15423)(6),

=(15423).

.=

L

,

-1=-1,L-1-1,

-1=(iLii),j=1,2,L,k.

12kk21j

12lj12l

-1=(563)(421).

-1=(

(i,iLi),

,=(124)(365),

)-1=(15423)(563)(421).

-1(1)=

(4)=2,,

(5)=4,

-1(2)=

-1(4)=

-1(6)=

(1)=5,

(2)=3,

(3)=1,

-1(3)=

-1(5)=

(6)=6,

,

(15423)

6.11(1)

,?

-1=(1256)(34).

6,

-1=(5)

(6).

(6)=(6).

.G={e}

,

.G

,

.

(2)

,

,

,

.

(3)

.

6.12(1)

(2)

6.5

.

.?x,yA,xyxy

,xyxy

.A

.

,

,

,

,,

.?x,yA,xyxy

,xyxy

.

.A

,

,

79

x

,

,,

.?xA,110x

,

.

.

6.13(1)6.1(3),(4),(5),(8)

.(3)

{f,g}

;(4)

{a,e}

;(5)

{d,e}

;(8)

{d,e}

.

(2)6.1

,

(1),(2)

,

,

;(6)

;(7)

.

6.1(1)(2)

6.3

,

.

6.1(6)(7)

6.3(2)

.

,6.1(6){a,b,c,d,f}(7)