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by: Cydney Conroy


Cydney Conroy

GPA 3.65

Agnes Szendrei

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About this Document

Agnes Szendrei
Class Notes
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This 1 page Class Notes was uploaded by Cydney Conroy on Thursday October 29, 2015. The Class Notes belongs to MATH 3000 at University of Colorado at Boulder taught by Agnes Szendrei in Fall. Since its upload, it has received 10 views. For similar materials see /class/231823/math-3000-university-of-colorado-at-boulder in Mathematics (M) at University of Colorado at Boulder.




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Date Created: 10/29/15
Introduction to Abstract Mathematics MATH 3000401 HANDOUT 1 September 19 2007 Review for Exam 1 Practice Problems 1 Which of the following statements are true Justify your answer a For arbitrary integers in and n m n is even only ifm and n are both even b For arbitrary integers in and n m n is even ifm and n are both even c If is a rational number then 2 is both even and odd d For arbitray real number x z gt 0 if and only if p2 gt 0 2 Let A and B be arbitrary statements Are the following pairs of statements logically equivalent a AAB EAR b AAB EVE c EVE AaF d AaB B A e Ast BsxA f AeB AA VZB 3 For each statement below 0 nd a formula involving quanti ers which represents the logical structure of the statement and 0 use the formula to negate the statement and state the negation in English in such a way that the statement does not use phrases equivalent to there is no7 or not all7 a All teenagers know all the rock stars b Some rock stars know some teenagers c There is a rock star who knows some teenagers but does not know all of them 4 Prove or disprove a For all positive real numbers z and y y 1 gt z 1 b There is no largest natural number c For every positive real number x there is a positive real number y with the property that ify lt p then for all positive real numbers 2 yz 2 z 5 Prove each statement below by the methods indicated it n 1 a n 1 lt n 2 Direct proof Proof by contradiction b For all integers n if n is odd then 2n2 3n 4 is also odd Direct proof Prove the contrapositive c The sum of three consecutive integers is divisible by 3 Direct proof holds for all positive integers n 1


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