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Unformatted text preview: CSE105: Automata and Computability Theory Winter 2011 Homework #1 Due: Tuesday, January 11th, 2011 Problem 1 Let A be the set { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 } and B be the set { 1 , 3 , 5 , 7 } . a. How many elements in the set A \ B ? b. How many elements in the set A × B 2 ? c. How many elements in the set 2 B (aka P ( B ) aka the powerset of B )? d. How many distinct functions exist mapping from A to B ? e. How many distinct functions exist mapping from B to A that are onto ? For the purpose of this question, assume that functions are defined on their entire domain. Problem 2 The Fibonacci sequence is defined as follows: F = 0, F 1 = 1, and, for n > 1, F n = F n 2 + F n 1 . The first few elements of the sequence are 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 ,... Let φ represent the “golden ratio”: φ = 1+ √ 5 2 ≈ 1 . 618033989. Prove, by an inductive argument, that for all n ≥ 1 we have F n ≤ φ n 1 . Be sure to label clearly each part of your proof....
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 Spring '99
 Paturi
 Fibonacci number, Golden ratio, DFAS, unary notation

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