d-cl-PRAC-selo - n k =1 1 k k 1 By induction on n prove...

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Sets and Logic MHF3202 4628 Prac-D Prof. JLF King Friday, 04Apr2008 D1: Show no work. Write DNE ( for “ Does Not Exist ) in a blank if the indicated operation cannot be per- formed, or if the described object does not exist. a The repeating decimal 7 . 45 123 equals n/d , where posints n d are n = ........... and d = ........... . b P ( P ( { 3 , 4 , 5 , 6 } ) ) has .......... many elements. c To the interval J := ( π 2 , π 2 ) , define a bijection g : ( 5 , 6 ) , ± J by g ( x ) := ............................ . In terms of this g and a trigonometric function, define a bijection h : ( 5 , 6 ) , ± R by h ( x ) := ................ . d Any three sets Λ ,B,C yield a natural bijection, Θ:Λ B × C B ] C . Define it, for each f Λ B × C , by Θ( f ) := h c 7→ [ .................................. ] i . e Mod M :=50, the reciprocal ± 1 21 ² M = .......... . [ Hint: ] So x = ...... [ 0 ..M ) solves 4 + 21 x M 1. f LBolt: Gcd(45 , 63)= ....... · 45 + ....... · 63. So (LBolt again) G := Gcd(45 , 63 , 105)= ......... and ........ · 45 + ........ · 63 + ........ · 105 = G . Essay questions: For each question, carefully write a triple– spaced, grammatical, essay solving the problem. D2: Let L ( n ) := n n +1 . And let R ( n ) :=
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Unformatted text preview: n k =1 1 k · [ k +1] . By induction on n , prove that ± ² ³ ´ ∀ n ∈ N : L ( n ) = R ( n ) . Explicitly prove the base case. Explicitly state the induction implication, then prove that it holds for each n ∈ N . D3: Prove that the map f : N × N → Z + , with f ( k,n ) := 2 k · [1+2 n ], is injective. Now prove that it is surjective. D4: Let P ∞ denote the collection of all infinite subsets of N . Define a relation ˙ ≡ on P ∞ by: A ˙ ≡ B IFF A ∩ B is infinite. Either prove that ˙ ≡ is transitive, or else produce three explicit sets A,B,C ∈ P ∞ showing that ˙ ≡ is not transitive. D5: Suppose I is a set, and { B j } j ∈ I is a collection with each B j ⊂ Z + . Each distinct index-pair y,z ∈ I has B y ∩ B z = ∅ . Construct, with proof , an injection f : I , → N , to conclude that I is only countable....
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This note was uploaded on 01/26/2012 for the course MHF 3202 taught by Professor Larson during the Fall '09 term at University of Florida.

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