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Unformatted text preview: Rob Bayer Math 55 Worksheet July 14, 2009 Structural Induction 1. Whats wrong with the following proof that every Fibonacci number is even? F = 0 is even. If F n and F n- 1 are even, then F n +1 = F n + F n- 1 is the sum of two even numbers and thus is even. Thus, by induction, all fibonacci numbers are even. 2. Consider the set S defined recursively as follows: 0 S, 1 S, S and whenever w S , both 1 w 1 and w S . What is simple definition for this set? 3. Prove that in any bit string, 01 occurs at most one more time that 10. 4. Consider the set S defined by: 3 S and whenever x,y S,x + y S and x- y S . Show that every element of S is divisible by 3. 5. Prove that in any well-formed formula, the parentheses are balanced in the following sense: If we start from 0 and scan the formula from left to right, adding 1 every time we see a ( and subtracting 1 everytime we see a ), then our running total will always be non-negative at every step.a ), then our running total will always be non-negative at every step....
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