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 ofsuch (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.