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303a4solns - ASSIGNMENT 4 SOLUTIONS MATH 303 FALL 2011 If...

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ASSIGNMENT 4 SOLUTIONS MATH 303, FALL 2011 If you find any errors please let me know. Manipulation (M1) No the integers with the usual are not well ordered. We can see this because the set of all integers has no least element, since for any n Z , n - 1 Z and n - 1 < n . So it is certainly not the case that all subsets of the set of all integers have a least element, and hence this is not a well order. (M2) A × B = { (1 , 4) , (1 , 5) , (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) } , which has size 6. Notice that 6 = #( A )#( B ) . (M3) B A = {{ (8 , 6) , (96 , 6) } , { (8 , 6) , (96 , 7) } , { (8 , 6) , (96 , 8) } , { (8 , 7) , (96 , 6) } , { (8 , 7) , (96 , 7) } , { (8 , 7) , (96 , 8) } , { (8 , 8) , (96 , 6) } , { (8 , 8) , (96 , 7) } , { (8 , 8) , (96 , 8) }} which has size 9. Notice that 9 = (#( B )) #( A ) . (M4) P ( A ) = {∅ , { 5 } , { 7 } , { 83 } , { 251 } { 5 , 7 } , { 5 , 83 } , { 5 , 251 } , { 7 , 83 } , { 7 , 251 } , { 83 , 251 } , { 5 , 7 , 83 } , { 5 , 7 , 251 } , { 5 , 83 , 251 } , { 7 , 83 , 251 } , { 5 , 7 , 83 , 251 }} which has size 16. Notice that 16 = 2 #( A ) . Pure Math (P1) Define f : ( Z Y ) X Z X × Y as follows. For every g ( Z Y ) X , f ( g ) is the function in Z X × Y which takes ( x, y ) X × Y to ( g ( x ))( y ). Now we just need to check f is one-to-one and onto. one-to-one: Suppose g 1 , g 2 ( Z Y ) X and f ( g 1 ) = f ( g 2 ) then for every ( x, y ) X × Y , ( g 1 ( x ))( y ) = f ( g 1 )( x, y ) = f ( g 2 )( x, y ) = ( g 2 ( x ))( y )
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