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 indexpair 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.
 Fall '09
 LARSON

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