# problems05-sol - MATH S-104 Lecture 5 In-class Problem...

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Unformatted text preview: MATH S-104 Lecture 5 In-class Problem Solutions July 7, 2009 Problem 1 † : Find the first 5 terms of the following recursively defined function. G ( n ) = 1 if n is 1 1 + G ( n 2 ) if n is even G ( 3 n- 1 ) if n is odd and n > 1 G ( 1 ) = 1 G ( 2 ) = 1 + G ( 1 ) = 2 G ( 3 ) = G ( 8 ) = 1 + G ( 4 ) = 1 + 1 + G ( 2 ) = 4 G ( 4 ) = 1 + G ( 2 ) = 3 G ( 5 ) = G ( 14 ) = 1 + G ( 7 ) = 1 + G ( 20 ) = 1 + 1 + G ( 10 ) = 1 + 1 + 1 + G ( 5 ) G ( 5 ) uncovers a problem with the function. We get G ( 5 ) = 3 + G ( 5 ) , which means 0 = 3, a contradiction. So G is not a well-defined function. Problem 2 † : Consider the following recursively defined set S : Base step : λ ∈ S ( λ is the empty string) Recursive step : If s ∈ S , then a. bs ∈ S b. sb ∈ S c. saa ∈ S d. aas ∈ S e. asa ∈ S Form a conjecture about the strings in S , and prove it using structural induction. The strings in S all contain an even number of a ’s....
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## This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.

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problems05-sol - MATH S-104 Lecture 5 In-class Problem...

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