u-cl-selo.2011t

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

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Sets and Logic MHF3202 2787 Class-U Prof. JLF King 3Oct2011 OYOP: Write each essay on new sheets of paper, writing every third line, sothatIcan easily writebetweenthelines. In grammatical English sentences , prove the following: U1: Define a sequence ~ b = ( b 0 , b 1 , b 2 , . . . ) by b 0 := 0 and b 1 := 3 and b n +2 := 7 b n +1 - 10 b n , for n = 0 , 1 , . . . . Use induction to prove, for all k [ 0 .. ) , that b k = 5 k - 2 k . 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 ] . 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. 1 45 M = . . . . . . . . [ 0 .. M ) . [ Hint: ] So x = . . . . . . . . . . [ 0 .. M ) solves 4 - 45 x M 7. b Posints K = . . . . , N = . . . . , α = . . . . , β = . . . . , are st. α K β , yet N α = . . . . is not K to N β = . . . . . c Stmt C B has contrapositive . . . . . . . . . . . . . . . . and converse . . . . . . . . . . . . . . . Recall & , , ¬ mean And, Or, Not . Using only & , , ¬ , B , C , ] , [ , write C B as . . . . . . . . . . . . . . . . d Define h : [ 1 .. 12 ] where
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