CMPSC360hw3 - CMPSC 360 Discrete Mathematics for Computer...

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Homework 3 Instructions: 1. [A Formula for the Fibonacci Numbers] Recall the defnition oF the ±ibonacci numbers F ( n ) ±ibonacci number is given by F ( n )= φ n - (1 - φ ) n 5 , where φ = 1+ 5 2 is the Golden Ratio, and is a solution oF the equation φ 2 - φ - 1 = 0 . 2. [Strengthening the Induction Hypothesis] Prove that, For all n 1 , all entries oF the matrix ± 10 11 ² n are bounded above by n . [HINT: Look at the title oF this problem!] 3. [ Well-Ordering Principle] This problem concerns the Well-Ordering Principle. ±irst, we will see that the set oF all pairs oF natural numbers N × N = { ( a,b ): a N ,b N } is a well-ordered set, provided we defne the ordering correctly. Then we will use this Fact to do an inductive prooF over pairs oF natural numbers. (a) Suppose we defne the ordering 1 on N × N by ( ) 1 ( c,d ) iF a b < c d , or iF a b = c d and a < c this defnition, we assume that ( ) 1 ( c, 0) whenever b> 0 , and that ( a, 0) 1 ( c, 0) iF and only a < c .) Show that, with this ordering, N × N is not well-ordered.
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CMPSC360hw3 - CMPSC 360 Discrete Mathematics for Computer...

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