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c-cl-selo.Prac - g x:= Essay questions For each question...

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Sets and Logic MHF3202 4628 Practice-C Prof. JLF King 07Feb2008 C0: Show no work; no partial credit. Write DNE ( for Does Not Exist ) in a blank if the indicated operation cannot be performed, or if the described object does not exist. Use V () for the Vince invariant of a cell in a TicTacToe board. Use TTT to abbreviate “TicTacToe”. a LBolt: Gcd(70 , 42)= . . . . . . . · 70 + . . . . . . . · 42. So ( LBolt again ) G := Gcd(70 , 42 , 30)= . . . . . . . . . . and . . . . . . . . · 70 + . . . . . . . . · 42 + . . . . . . . . · 30 = G . b Algebraic numbers α, β C have resp. degrees K and N . The max possible degree of α + β is . . . . . . . . . . . . . . . . c In Qubic , V ( Center cell )= . . . . . . . . And V ( Edge cell )= . . . . . . . . There are . . . . . . . many cells whose Vince-inv. equals V ( Edge cell ). d The 9 × 9 TTT board has . . . . many TicTacToes. And 3 × 3 × 3 has . . . . . . . . . . . . . . . . . many TicTacToes. e An explicit bijection
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Unformatted text preview: g ( x ) := ........................................... . Essay questions: For each question, carefully write a triple– spaced, grammatical, essay solving the problem. C1: Given an explicit bijection f : C → H between intervals C := [ , 4 ] and H := ( , 4 ) . C2: α The powerset P ( S ) of set S , is ... . β Give a complete proof that there is no surjection h : S ± P ( S ). C3: i A number β ∈ C is algebraic IFF ··· . ii The degree of β is ... . C4: A subset M of the plane is convex IFF ··· . C5: Let L ( n ) := n [ n + 1][2 n + 1]. And let R ( n ) := 6 · ∑ n r =1 r 2 . By induction on n , prove that ± ² ³ ´ ∀ n ∈ N : L ( n ) = R ( n ) ....
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