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Assignment 4 Solutions

# Assignment 4 Solutions - Assignment#4 Due Tuesday 2/10...

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Assignment #4 Due Tuesday, 2/10 Complete the following exercises from chapter 1.7: �   12, 18, 32 Complete the following exercises from chapter 2.1: 5   4, 8, 10, 16, 20, 27, 28 p   (For 20 in particular, justify your answer) Chapter 1.7 Exercise 12: Prove or disprove that if a and b are rational numbers, then a b is also rational. Disprove by counterexample: Let a = 2 and b = ½. Then 2 ½ = 2 , which is irrational from Example 10 in p. 80. Chapter 1.7 Exercise 18: Prove that given a real number x there exist unique numbers n and ε such that x = n + ε , n is an integer, and 0 ε < 1. Given x , let n be the greatest integer less than or equal to x , and let ε = x – n. Clearly 0 ε 1, and ε is unique for this n . Any other choice of n would cause the required ε to be less than 0 or greater than or equal to 1, so n is unique as well. Chapter 1.7 Exercise 32: Prove that 3 2 is irrational. This is similar to Example 10 in Section 1.6 in p. 80. Let us assume that 3 2 is rational.

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