Unformatted text preview: Rob Bayer Math 55 Worksheet July 14, 2009 Structural Induction 1. What’s 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,λ ∈ 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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This note was uploaded on 09/10/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at Berkeley.
- Summer '08