v-cl-selo.2011t

# v-cl-selo.2011t - a P P(3-stooges has many elements b Let P...

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Sets and Logic MHF32022787 Class-V Prof. JLF King 2Nov2011 OYOP: For your 2 Essays: Write your grammatical English sentences on every third line, so that I can easily write between the lines. Start each essay on a new sheet of paper. V1: Cantor Diagonalization Thm : For each set B , there does not exist a surjection B ± P ( B ) . V2: For N Z + , suppose Γ = ( V,E ) is a complete digraph on N vertices. I.e, we have N towns, with each pair connected by a one-way road. From a Γ -good town w V we can legally bike to every town. A town w is Γ -great if we can get to each town with a path of length 6 2. GD Prove that each such Γ has a Γ-good town. GT Prove that each such Γ has a Γ -great town. [ Hint: Induction on N , works. In addition to text, your essay should have pictures illustrating your argument; they should be large . When refering to a good/great town, be careful to specify w.r.t. what network. ] V3: Short answer. Show no work. Please write DNE in a blank if the described object does not exist or if the indicated operation cannot be performed.
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Unformatted text preview: a P ( P (3-stooges) ) has .............. many elements. b Let P ∞ denote the collection of all inﬁnite subsets of N . Deﬁne a relation ≈ on P ∞ by: A ≈ B IFF A ∩ B is inﬁnite. Stmt “ Relation ≈ is transitive ” is: T F c An explicit bijection F : N , ± Z is If n is even , then F ( n ) := .......................... . If n is odd , then F ( n ) := ........................... . d To the interval J := ( π 2 , π 2 ) , deﬁne a bijection g : ( , 1 ) , ± J by g ( x ) := ............................. . In terms of this g and a trigonometric function, deﬁne a bijection h : ( , 1 ) , ± R by h ( x ) := .................... . e The map f ( k,n ) := 2 k · [1 + 2 n ] is a bijection from N × N → Z + . And f 1 (112) = ( ....... , ....... ). End of Class-V V1: 65pts V2: 85pts V3: 95pts Total: 245pts...
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