Principles of Analysis

Homework Assignment #1

Due Wednesday, January 23, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to Bilodeau and Thie's book, the required text for the course. You should write up solutions neatly to all problems, paying particular attention to your arguments and proofs. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work.

Section 1.1
Problems:   2c, 4, 6 (all parts), 10

Prime Numbers
(1) Using Euclid's proof, but in your own words, prove that the set of all prime numbers is an infinite set.

(2) The odd positive integers can be broken into two disjoint sets: A = {4k + 1 where k = 0,1,2,3,...} and B = {4k + 3 where k = 0,1,2,3,...}. Thus A = {1,5,9,13,...} and B = {3,7,11,15,...}.

a. Show that A is closed under multiplication. In other words, show that the product of any two elements in A is also in A. (Note that this is not true for the set B.)

b. Following Euclid's proof, show that the set B contains an infinite number of primes.
Hint: By contradiction, suppose that there were a finite number of primes in the set B labeled {3, p1, p2, ... pn} and form the number P = 4(p1 p2 ... pn) + 3. (Note that 3 is not a factor of P.) Now use part a. and find a contradiction.