# A First Course in Abstract Algebra 3rd Edition Rotman Test Bank

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# A First Course in Abstract Algebra 3^{rd} Edition Rotman Test Bank

ISBN-13: 978-0131862678

ISBN-10: 0131862677

# A First Course in Abstract Algebra 3^{rd} Edition Rotman Test Bank

ISBN-13: 978-0131862678

ISBN-10: 0131862677

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1

Solution Manual for

A First Course in Abstract Algebra, with Applications

Third Edition

by Joseph J. Rotman

Exercises for Chapter 1

1.1 True or false with reasons.

(i) There is a largest integer in every nonempty set of negative integers.

Solution.

True. If C is a nonempty set of negative integers, then

−

C

={n

:

n

∈

C

}

is a nonempty set of positive integers. If

a

is the smallest element

of

C,

which exists by the Least Integer Axiom, then

a

≤c

for all c

∈

C, so that a

≥

c for all c

∈

C.

(ii) There is a sequence of 13 consecutive natural numbers containing

exactly 2 primes.

Solution. True. The integers 48 through 60 form such a sequence;

only 53 and 59 are primes.

(iii) There are at least two primes in any sequence of 7 consecutive

natural numbers.

Solution. False. The integers 48 through 54 are 7 consecutive

natural numbers, and only 53 is prime.

(iv) Of all the sequences of consecutive natural numbers not containing

2 primes, there is a sequence of shortest length.

Solution. True. The set C consisting of the lengths of such (ﬁnite)

sequences is a nonempty subset of the natural numbers.

(v) 79 is a prime.

Solution. True. √79 < √81

=

9, and 79 is not divisible by 2, 3,

5, or 7.

(vi) There exists a sequence of statements S(1), S(2),… with S(2n)

true for all n

≥

1 and with S(2n

1)

false for every n

≥

1.

Solution. True. Deﬁne S(2n

1)

to be the statement n

=

n, and

deﬁne S(2n) to be the statement n

=

n.

(vii) For all n

≥

0, we have n

≤

Fn , where Fn is the nth Fibonacci

number.