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w-cl-selo.2011t - or if the indicated operation cannot be...

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Sets and Logic MHF3202 2787 Class-W Prof. JLF King 21Nov2011 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. W1: Let J := [ 0 , 1 ] . You may use, without proof, the Schr¨ oder-Bernstein thm and the following. a 1 : R { 0 , 1 } N . a 2 : For each three sets Ω ,B,D : Ω B × D Ω B D . a 3 : The set S := Q J is countable. Prove that C ( J ) , the set of continuous functions J R , is bijective with R . Cite each ( a i ) where you use it. Specify what Ω , B, D are, when you apply ( a 2 ). [ Note: Does your proof split into easily-understood lemmas? ] W2: Between sets A := Z + and B := N , consider injections f : A , B and g : B , A , defined by f ( α ) := 3 α and g ( β ) := β + 5 . The S-B thm produces a set W g ( B ) A so that, letting U := A r W , function ϕ : A , B is a bijection , where ϕ U := f U and ϕ W := g 1 W . * : i Prove , for these particular injections, that there is only one set W which makes ( * ) a bijection. ii Compute ϕ (56)= . . . . . . . . and ϕ (83)= . . . . . . . . . W3: 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. a Repeating decimal 0
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Unformatted text preview: or if the indicated operation cannot be performed. a Repeating decimal 0 . 3 12 equals n d , where posints n ⊥ d are n = ............... and d = ............... . b An explicit bijection F : N × N , ± [ 4 .. ∞ ) is F ( n,k ) := ....................................... . c 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 d Each three sets Ω ,B,C engender a natural bijection, Θ : Ω B × C , ± [Ω B ] C . defined, for each f ∈ Ω B × C , by Θ( f ) := h c 7→ [ ................................... ] i . Its inverse-map Υ : [Ω B ] C , ± Ω B × C has, for g ∈ [Ω B ] C , Υ( g ) := h ( b,c ) 7→ [ ............................... ] i . End of Class-W W1: 75pts W2: 60pts W3: 110pts Total: 245pts...
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