1 = SOLUTION MANUAL FOR A FIRST COURSE IN ABSTRACT ALGEBRA, WITH APPLICATIONS 8TH EDITION BY JOSEPH J. ROTMAN LATEST 2024 Exercises for Chapter 1 1.1 True or false with reasons. (i) There is a largest integer in every nonempty set of negative inte- gers. 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