Unformatted text preview: 9. Give an example of a function g : Z → Z that is injective but not surjective. 10. Let A = { 1 , 2 , 3 , 4 } . Give an example of an element X ∈ P ( A × A ) with the property that for any Y and Z in P ( A ) we have X 6 = Y × Z . Alternatively, assert that there is no such element. [Reminder: P ( B ) is the set of all subsets of B .] Proofs. [10 points each, do exactly 4 of the 5] I. Let a,b,c be positive integers. Suppose that (i) gcd( a,b ) = 1 and (ii) a  bc . Prove that a  c . II. Let n be a positive integer and suppose that 3  n 2 . Prove that 3  n . III. Let a,b,k be positive integers. Prove that gcd( ka,kb ) = k gcd( a,b ). IV. Prove that 7  (2 n +2 + 3 2 n +1 ) for all positive integers n . V. Let a,b,c be positive integers. Prove that gcd(gcd( a,b ) ,c ) ∈ D ( a, gcd( b,c )). [Reminder: D ( n ) = { m ∈ Z : m  n } , and D ( m,n ) = D ( m ) ∩ D ( n ).]...
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 Fall '07
 COURTNEY
 Math, Natural number, positive integers, Identity element

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