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Math · Advanced Math
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Please help I did send the definitions as well
Exercises Exercise 1.7.1. [Used in Exercise 1.7.7.1 Let rE Q. (1) Prove that D -Dr, using only Definition 1.6.4 and Definition 1.7.3. (2) Prove that Dr-ー[Dr]-1, using only Definition 1.7.5 and Definition 1.7.3 Exercise 1.7.2. [Used in Theorem 1.7.6.] Let A, B E R. Suppose that A>Do a B> Do. For this exercise, you may use only reculte nui
Delinition 1.7.3. The binary operation + on R is defined by A+B reQIr-a+b for some a E A and beB) or all A, BE R. The unary operation - on R is defined by -A-rEQI-r<c for some cEQ-A r all A R.
Definition 1.6.4. Letr e Q. The rational cut at r, denoted D,, is the Ded rational n An irra tional cut is a Dedekind cut that is not a rational cut at any Before using Dedekind cuts to form the set of real numbers in Section 1.7,w take the remainder of the present section to prove some technically useful propertie Dedekind cuts, starting with the following simple lemma that will be used frequ Lemma 1.6.5. Let A C Q be a Dedekind cut. We 1. Q-A={x EQlx<afor all a EA}. 2. Let x EQ-A. Ify E Q and y S x, then y EQ-A oof (1) Lete
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