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►Содержание►
3.31. C6.
3.33. This is the group O(2) we meet in Chapter 5.
3.15. Group.
3.27. D2.

310 ANSWERS TO THE ODD-NUMBERED EXERCISES

2.35. (A

 

B)

 

(A

 

B)

 

(A

 

B ); A

 

B; A

 

B.

2.39.

 

 

 

 

 

 

 

 

 

 

 

 

 

D

 

 

 

 

 

 

 

U

 

 

 

 

 

 

D′

 

 

 

 

 

 

 

U′

2.41. (A (B C)) (B C).

 

 

2.43. (A B) (A B ) C.

2.45. A C D.

 

 

 

 

 

 

 

 

 

 

2.47.Orange: (A B ((C D) (C D ))) (((A B) (A B )) C D ). Green: A B C D.

2.49.Let the result of multiplying AB by CD be EF. Then the circuit for E is (A B C) (A ((B C ) D) (B C D )), and the circuit for F is ((A C) (B D)) (A B C D ).

2.51. (A B) (A B ); (A B) (A B ) (A B ).

2.53.

2.55.

 

2.57.

60

12

 

8

 

 

 

 

 

4

6

4

 

 

 

 

 

30

2

 

 

 

 

2

6

10

 

 

Boolean algebra

 

 

 

 

 

 

 

 

 

 

1

 

2

 

 

 

 

 

 

 

Lattice

3

5

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

Lattice

2.59. The primes pi .

 

 

 

2.67. d = a b c .

 

2.65. Yes.

 

 

 

 

 

 

CHAPTER 3

 

 

 

 

 

 

 

3.1.

 

 

 

 

 

 

 

 

 

·

e

g

g2

g3

g4

 

 

 

e

e

g

g2

g3

g4

 

 

 

 

g

g

g2

g3

g4

e

 

 

 

g2

g2

g3

g4

e

g

 

 

 

g3

g3

g4

e

g

g2

 

 

 

g4

g4

e

g

g2

g3

 

 

 

3.3. See Table 8.3.

 

 

 

 

 

 

3.5. Abelian group.

 

 

 

 

3.7. Abelian group.

 

3.9. Not a group; the operation is not closed.

3.11. Abelian group. 3.13. Abelian group.

3.25. No.

3.29. D6.

3.35. Z, generated by a glide reflection.

ANSWERS TO THE ODD-NUMBERED EXERCISES

311

3.37.C6

e, g3

e, g2, g4

 

e

3.39. No.

3.41. For any c Q, f : Z → Q defined by f (n) = cn for all n Z.

3.43. No; Q has an element of order 2, whereas Z does not.

3.45.The identity has order 1; (12) Ž (34), (13) Ž (24), (14) Ž (23) have order 2, and all the other elements have order 3.

3.47.

 

 

 

 

 

 

 

 

 

·

1

−1

i

−i

j

−j

k

−k

 

 

1

1 −1

i

−i

j

−j

k

−k

 

−1

−1

1 −i

i

−j

j

−k

k

 

i

i

−i

−1

1

k

−k

−j

j

 

−i

−i

i

1 −1 −k

k

j

−j

 

j

j

−j

−k

k

−1

1

i

−i

 

−j

−j

j

k

−k

1 −1 −i

i

 

k

k

−k

j

−j

−i

i

−1

1

 

−k

−k

k −j

j

i

−i

1

−1

The identity, 1, has order 1; −1 has order 2; all the other elements have

order 4.

3.55.

 

1

3

4

2

.

3.53. {(1), (123), (132)}.

 

 

 

1

2

3

4

 

3.57. (12435).

3.59. (165432) is of order 6 and is odd.

3.61.(1526) Ž (34) is of order 4 and is even.

3.63.

 

1

2

3

4

5 .

3.65. (132).

3.67.

 

2

3

4

5

1

Ž (34), (13) Ž (24), (14) Ž (23), (1324), (1423) .

{

(1), (12), (34), (12)

3.69.

(1), (13), (24), (13)

}

{

Ž (24), (12) Ž (34), (14) Ž (23), (1234), (1432) .

 

 

 

 

 

 

}

3.73. φ(n), the number of positive integers less than n that are relatively prime

 

to n.

3.77. {e}.

3.75. 52; 8.

3.83.

(1) Achievable; (3) achievable.

3.87. S2.

3.85.

S3.

3.89.

F is not abelian; y−1x−1y−1xx.

 

CHAPTER 4

4.1. Equivalence relation whose equivalence classes are the integers.

4.3. Not an equivalence relation.

312

 

 

 

 

ANSWERS TO THE ODD-NUMBERED EXERCISES

4.5.

 

 

 

 

Left Cosets

 

 

 

 

Right Cosets

 

 

 

=

 

{

 

}

 

 

=

{

(1), (12), (34), (12)

}

 

H

=

 

 

(1), (12), (34), (12) Ž (34)

 

H

 

 

Ž (34)

(13)H

 

{(13), (123), (134), (1234)}

 

H (13) = {(13), (132), (143), (1432)}

(14)H

=

 

{(14), (124), (143), (1243)}

 

H (14) = {(14), (142), (134), (1342)}

(23)H

=

 

{(23), (132), (234), (1342)}

 

H (23) = {(23), (123), (243), (1243)}

(24)H

=

 

{(24), (142), (243), (1432)}

 

H (24) = {(24), (124), (234), (1234)}

(1324)H

=

 

{(1324), (14) Ž (23),

 

H (1324) =

{(1324), (13) Ž (24),

 

 

 

 

 

 

 

}

 

 

 

 

}

 

 

 

 

 

(13) Ž (24), (1423)

 

 

 

(14) Ž (23), (1423)

 

 

4.7. Not a morphism.

= 4Z, and Imf

= {(0, 0), (1, 1), (0, 2), (1, 3)}.

 

4.9. A morphism; Kerf

 

4.11. Not a morphism.

 

4.19. No.

 

 

 

 

 

4.21. f : C3 → C4

defined by f (gr ) = e.

 

 

 

 

 

 

 

4.23. fk : C6 → C6

 

defined by fk (gr ) = gkr for k = 0, 1, 2, 3, 4, 5.

C10 × C6

4.25. Not isomorphic; C60 contains elements of order 4, whereas

does not.

 

 

 

 

 

 

 

 

 

 

 

 

4.27. Not isomorphic; Cn × C2 √is commutative, whereas Dn is not.

4.29. Not isomorphic; (1 + i)/

2

has order 8, whereas Z4 × Z2 contains no

element of order 8.

 

4.39. (R+, ·).

4.33. C10 and D5.

 

4.49. G2 S3.

 

 

=

 

 

4.59. 5 is a generator of Z6, and 3 is a generator of Z17.

CHAPTER 5

 

 

 

 

 

 

 

 

 

 

 

 

 

5.1. C2

and C2.

 

 

 

 

 

 

 

5.3. C2

and C2 × C2.

5.5. C3

and D3.

 

 

 

 

 

 

 

5.7. C9

and C9.

5.9. D4.

 

 

 

 

 

 

 

 

5.11. S4.

 

 

5.13. S4.

 

 

 

 

 

 

 

 

 

5.15. S4.

 

 

5.17. A5.

 

 

 

 

 

 

 

 

 

5.19. A5.

 

 

5.21. A5.

 

1

0

and

 

1

0

0

5.25. S4.

 

 

5.27. −0

.

 

 

 

 

 

 

 

1

0

0

 

 

 

0

0

1

 

 

 

 

 

 

 

 

0

−0 1

 

0

1 0

 

 

 

 

 

5.29. D6.

 

 

 

 

 

 

 

 

5.31. C2.

 

 

 

 

 

 

 

 

 

1

0

 

0

and

 

0

1

0

.

5.33. D4

generated by

−0

1

 

0

1

−0

0

 

 

 

 

 

 

 

0 0

 

−1

 

 

0

0 1

 

CHAPTER 6

 

 

 

 

 

 

 

 

 

 

 

 

 

6.1.

3.

 

 

 

 

 

 

 

 

 

6.3.

38.

 

 

6.5.

78.

 

 

 

 

 

 

 

 

 

6.7.

35.

 

 

ANSWERS TO THE ODD-NUMBERED EXERCISES

313

6.9.

333.

6.11.

(n6 + 3n4 + 8n2)/12.

6.13.

1.

6.15.

96.

6.17.

30.

6.19.

396.

6.21.

126.

6.23.

96.

CHAPTER 7

 

 

 

 

 

 

7.1. Monoid with identity 0.

 

7.3. Semigroup.

7.5. Neither.

 

 

 

 

 

7.7. Semigroup.

7.9. Semigroup.

 

 

 

 

7.11. Monoid with identity 1.

7.13. Neither.

 

 

 

 

 

 

7.15.

 

 

 

 

 

 

 

 

 

gcd

1

2

3

4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

1

1

1

1

 

 

 

 

2

 

1

2

1

2

 

 

 

 

3

 

1

1

3

1

 

 

 

 

4

 

1

2

1

4

 

 

 

7.17.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

·

 

e

c

c2

 

c3

c4

 

 

e

 

e

c

c2

 

c3

c4

 

c

 

c

c2

c3

 

c4

c2

 

c2

 

c2

c3

c4

 

c2

c3

 

c3

 

c3

c4

c2

 

c3

c4

 

c4

 

c4

c2

c3

 

c4

c2

 

7.19. No; 01 = 1 in the free semigroup.

 

 

 

 

 

7.29.

 

 

 

 

 

 

 

0

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

 

1

 

 

0

 

1

 

 

0

 

Sends an

 

 

 

s1

 

 

s2

 

 

 

s3

 

 

s4

 

s5

output signal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

1

 

0

 

7.31. A congruence relation with quotient semigroup = {2N, 2N + 1}.

7.33. Not a congruence relation.

 

 

 

 

 

 

 

7.35.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[0]

[1]

[00]

[10]

[01]

[010]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[0]

 

[00]

[01]

[0]

[010]

[1]

[10]

 

 

 

 

 

[1]

 

[10]

[1]

[1]

[10]

[01]

[010]

 

 

 

 

 

[00]

 

[0]

[1]

[00]

[10]

[01]

[010]

 

 

 

 

 

[10]

 

[1]

[01]

[10]

[010]

[1]

[10]

 

 

 

 

 

[01]

 

[010]

[01]

[01]

[010]

[1]

[10]

 

 

 

[010]

 

[01]

[1]

[010]

[10]

[01]

[010]

 

 

7.37.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

24.

 

 

 

 

 

 

 

 

 

 

 

 

 
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