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Unformatted text preview: Illinois Institute of Technology Department of Computer Science Solutions to First Examination CS 330 Discrete Structures Spring Semester, 2008 1. Mathematical Induction. Prove by induction that for n ≥ 1, ∑ n 1 k =1 k × k ! = n ! 1. Basis step: for n = 1 , P (1) = 0 = 0. Inductive step: assume P ( m ) is true, show P ( m +1) is true. Let P ( m ) be the statement ∑ m 1 k =1 k × k ! = m ! 1. Then P ( m +1) is the statement ∑ m k =1 k × k ! = ( m +1)! 1. Rewrite LHS of P ( m +1) in terms of P ( m ) by induction: ∑ m 1 k =1 k × k ! + m × m ! = m ! 1 + m × m !. But m ! 1 + m × m ! = ( m + 1)! 1 and P ( m + 1) follows. 2. Growth rates. (a) Is ( n k ) ∈ O (2 n ) for k ≤ 10? Prove your answer. ( n k ) = n k /k ! + O ( n k 1 ) is a polynomial in n of degree k ; in this case k ≤ 10 is constant, and 2 n grows faster than any polynomial in n . If, say, k = n/ 2, the growth rate would be n n/ 2 which grows faster than 2 n ....
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This test prep was uploaded on 04/15/2008 for the course CS 330 taught by Professor Reingold,edwardm. during the Spring '08 term at Illinois Tech.
 Spring '08
 Reingold,EdwardM.
 Computer Science

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