1. Recall that we discussed defining the real numbers as an axiomatic system with addition axioms A1-A4, multiplication axioms M1-M4, a distributive law (DL), and order axioms 01-05.
(a) Which of the properties Al-A4, M1-M4, DL, 01-05 fail for N? In each of those cases, demonstrate that the property fails through an example. (b) Repeat part (a) for Z instead of N. 2. (a) For a, b E R, prove that Ibl
3. (a) Use the triangle inequality to prove that la + b + ci
dal+ a2 + • • + an.1 5_ lad ± lath + • • lard for all al, a2, , E H, where n is an integer with n > 2. 4. Let a and ,3 be Dedekind cuts. (a) Show that the set a + ,3 defined as a + = {r+slrect,sE 3}
is a Dedekind cut. (b) Explain why the following definition for "multiplication" of Dedekind sets is a poor definition: a • 13 = {r•sir a,s 3}