klarfeld-sec-4-1-4-2 - Jennifer Klarfeld Week 10 Edition 8...

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Jennifer’Klarfeld’ –’ Week’ 10’ –’ Edition’ 8’ 1 Section’4.1 ’ 6.’ (b)’Let’A’be’the’set’{1,2,3},’and’let’R’be’the’relation’on’A’given’by’{ ’ (x,y)’:’3x+y’is’ prime}.’ Prove’ that’R’is’a’function’with’domain’A. If’x ’ =’ 1,’then’3x+y ’ =’ 3+y.’For’y’=’1,’2,’or’3,’3+y’is’prime’if’and’only’if’y’=’2. If’x ’ =’2,’then’3x+y’=’ 6+y.’For’y’=’1,’2,’or’3,’6+y’is’prime’if’and’only’if’y’=’1. If’x’=’3,’the n’3x+y’=’9+y.’For’y’=’1,’2,’or’3,’9+y’is’prime’if’and’only’if’y=2. For’every’x A,’there’is’a’unique’value’y A’ such’that’3x+y’is’prime. So,’R’i s’a’function’with’domain’A.
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Jennifer’Klarfeld’ –’ Week’ 10’ –’ Edition’ 8’ 2 6.’ ’ (d)’Let’R’=’{ ’ (x,y)’ ℕ×ℕ ’ :’ ࠵± − ² = ³ }.’Prove’that’R’is’a’function’with’ domain’ .’ Let’x .’Then’ ࠵± − ³ ’ is’a’natural’number,’so’(x,’ ࠵± − ³ = ² ) ’ R.’ Therefore,’Dom (R)’ =’ .’ Now’suppose’that’(x,y)’ ’ R’and’(x,z)’ ’ R.’ Then’ ࠵± − ³ =y’and’ ࠵± − ³ =z.’ So’y=z. ’ Therefore,’R’is’a’function’with’domain’ .’
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Jennifer’Klarfeld’ –’ Week’ 10’ –’ Edition’ 8’ 3 7.’Complete’the’proof’of’Theorem’4.1.1.’That’is,’prove’that’if’ (i)’Dom( )’=’Dom( ± )’and’ ’ (ii)’for’all’x Dom( ),’ (x)’=’ ± (x),’ then’ = ± .’ Suppose’that’ x Dom( ± ).’ Then’(x,y)’ ± ’ for’some’y’ ’ Rng( ± ).’ Thus,’ ± (x)=y.’ Since’Dom ( )’ =’ Dom( ± ),’ x Dom( ).’ For’all’ x Dom( ),’ (x)’=’ ± (x),’ (x)’=’ ± (x)=y.’ So’(x,y) ’ ’ ’ and’ (x,y)’ ’ ± .’ We’conclude’that’ ’ =’ ± .’
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Jennifer’Klarfeld’ –’ Week’ 10’ –’ Edition’ 8’ 4 9.’Let’the’universe’be’ ,’and’A’=’[1,3).’Find (a)’ ± ² ± ² ’ =’1,’since’1 A.’ (b)’ ± ³ ± ³ ’ =’0,’since’3’ ’ A.’ (c)’ ± ´ ± ´ ’ =’0.’ Since’ ´ ’ >’3 ,’ ´ ’ A.’ (d)’ ± µ ’ -’ ± ¶.µ ± µ ’ =’ 1,’since’2 A.’ ± ’ =’0,’since’0.2’ ’ A.’ Therefore,’ ± µ ’ -’ ± ’ =’1’ –’ 0’ =’1. ’
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Jennifer’Klarfeld’ –’ Week’ 10’ –’ Edition’ 8’ 5 10.’Let’U’be’the’un iverse.’Suppose’A’ ’ U’with’A’
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  • Spring '16
  • Math, Edition, Jennifer, Jennifer  Klarfeld, Klarfeld

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