Nicholson - Introduction to Abstract Algebra, Solutions Manual

Copyright 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Nicholson, W. Keith.

Introduction to abstract algebra / W. Keith Nicholson. 4th ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-28815-3 (cloth)

1. Algebra, Abstract. I. Title.

QA162.N53 2012

512'.02dc23 2011031416

Chapter 0

Preliminaries

a.

If n = 2 k , k an integer, then n 2 = (2 k )2 = 4 k 2 is a multiple of 4.

The converse is true: If n 2 is a multiple of 4 then n must be even because n 2 is odd when n is odd (Example 1).

c.

Verify: 23 6 22 + 11 2 6 = 0 and 33 6 32 + 11 3 6 = 0.

The converse is false: x = 1 is a counterexample. because

a. Either n = 2 k or n = 2 k + 1, for some integer k . In the first case n 2 = 4 k 2; in the second n 2 = 4( k 2 + k ) + 1.

c. If n = 3 k , then n 3 n = 3(9 k 3 k ); if n = 3 k + 1, then

if n = 3 k + 2, then n 3 n = 3(9 k 3 + 18 k 2 + 11 k + 2).

a.

If n is not odd, then n = 2 k , k an integer, k 1, so n is not a prime.

The converse is false: n = 9 is a counterexample; it is odd but is not a prime.

c.

If then , that is a > b , contrary to the assumption.

The converse is true: If then , that is a b .

a. If x > 0 and y > 0 assume . Squaring gives , whence . This means xy = 0 so x = 0 or y = 0, contradicting our assumption.

c. Assume all have birthdays in different months. Then there can be at most 12 people, one for each month, contrary to hypothesis.

a. n = 11 is a counterexample because then n 2 + n + 11 = 11 13 is not prime. Note that n 2 + n + 11 is prime if 1 n 9 as is readily verified, but n = 10 is also a counterexample as 102 + 10 + 11 = 112.

c. n = 6 is a counterexample because there are then 31 regions. Note that the result holds if 2 n 5.

0.2 Sets

a. A = { x x = 5 k , , k 1}

a. {1, 3, 5, 7,... }

c. { 1, 1, 3}

e. { } = is the empty set by Example 3.

a. Not equal: 1 A but 1 B .

c. Equal to { a , l , o , y }.

e. Not equal: 0 A but 0 B .

g. Equal to { 1, 0, 1}.

a. , {2}

c. {1}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}

a. True. B C means each element of B (in particular A ) is an element of C .

c. False. For example, A = {1}, B = C = {{1}, 2}.

a. Clearly A B A and A B B ; If X A and X B , then x X implies x A and x B , that is x A B . Thus X A B .

If x A ( B 1 B 2... B n ), then x A or x B i for all i . Thus x A B i for all i , that is x ( A B 1) ( A B 2)... ( A B n ). Thus