mt3 - , 1) and [0 , 1]. Prove that the function you found...

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Homework 3, Math 245, Fall 2010 Due Friday, October 1 in class. The solution of each exercise should be at most one page long. If you can, try to write your solutions in LaTex. Each question is worth 2 points. 1. The Fibonacci numbers ( F n ) n 1 are defined as follows: F 1 = F 2 = 1 and F n = F n - 1 + F n - 2 for every n 2. Prove that F n = ± 1+ 5 2 ² n - ± 1 - 5 2 ² n 5 for every n 1. 2. Let f : A B and g : B C be two functions. Show that if g f is injective, then f is injective. Show that if g f is surjective, then g is surjective. 3. Find a bijection between the intervals (0 , 1) and ( a,b ) where a < b are two given real numbers. Prove that the function you found is a bijection. 4. Find a bijection between the intervals [0
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Unformatted text preview: , 1) and [0 , 1]. Prove that the function you found is a bijection. 5. Prove that the function f : (0 , 1) R ,f ( x ) = 2 x-1 2 x (1-x ) is a bijection between (0 , 1) and R . 6. Bonus Question! Prove that every natural number n 2 can be written as a sum of dierent Fibonacci numbers such that the sum does not include any two consecutive Fibonacci numbers. (For example, 2 = F 3 , 3 = F 4 , 4 = F 4 + F 2 , 5 = F 5 , 6 = F 5 + F 2 , 7 = F 5 + F 3 , 8 = F 6 , 9 = F 6 + F 2 , 10 = F 6 + F 3 , 11 = F 6 + F 4 , 12 = F 6 + F 4 + F 2 , 13 = F 7 etc.)...
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This note was uploaded on 12/07/2011 for the course MATH 245 taught by Professor Cioaba during the Fall '10 term at University of Delaware.

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