u-cl-selo.2011t - Sets and Logic MHF3202 2787 Prof. JLF...

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Unformatted text preview: Sets and Logic MHF3202 2787 Prof. JLF King 3Oct2011 U1: 60pts U2: 30pts U3: 150pts Total: Class-U 240pts OYOP: Write each essay on new sheets of paper, writing every third line, so that I can easily write between the lines. In grammatical English sentences, prove the following: U1: Define a sequence b = ( b0 , b1 , b2 , . . . ) by b0 := 0 and b1 := 3 and bn+2 := 7bn+1 − 10bn , An extra question, for your posting pleasure: for n = 0, 1, . . . . Use induction to prove, for all k ∈ [0 .. ∞), that bk = 5k − 2k . f U2: For a posint K , let ≡ mean ≡K . Defn: Expression “x ≡ y ” means. . . . Please prove: Thm: For all b, β, g, γ ∈ Z, if b ≡ β and g ≡ γ then [b · g ] ≡ [β · γ ]. Write the free vars in each of these expressions. E3 ∃n ∈ N: f (n) < r+7 =r−4 x∈Z E1 E2 U3: 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 Mod M :=229, the recipr. [Hint: ] So x= b .......... Posints K = .... are st. α ≡K β , yet N α = c 1 45 M = ∈[0 .. M ) , N= .... .... .............. ........ ∈[0 .. M ). solves 4 − 45x ≡M 7. , α= .... , β= is not ≡K to N β = Stmt C ⇒B has contrapositive converse .... .... ................ . Recall &, ∨, ¬ mean Using only &, ∨, ¬, B, C, ], [, write C ⇒B as , . and . And, Or, Not ................ d Define h:[1 .. 12] where h(n) is the number of letters in the nth Gregorian month. So h(2) = 8, since the 2nd month is “February”; 8 letters. The only fixed-point of h is . The set of posints k where h◦k (12) = h◦k (7) ..... is . ..................................................... e · n ≡5 x2 TF T F T F ∀x,z ∈ Z with x < z , ∃y ∈ Z st.: x < y < z . ∀x,z ∈ Q with x = z , ∃y ∈ R st.: x < y < z . For all sets Ω, there exists a fnc f : R→Ω. End of Class-U E 1: ........... . E 2: ........... . E 3: ........... . ...
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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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