A First Course in Abstract Algebra 3rd Edition Rotman Test Bank
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Solution Manual for
A First Course in Abstract Algebra, with Applications
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.
True. If C is a nonempty set of negative integers, then
is a nonempty set of positive integers. If
is the smallest element
which exists by the Least Integer Axiom, then
for all c
CRemember, that a
c for all 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
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
false for every n
Solution. True. Deﬁne S(2n
to be the statement n
deﬁne S(2n) to be the statement n
(vii) For all n
0, we have n
Fn , where Fn is the nth Fibonacci