CMPSC360hw3

# CMPSC360hw3 - CMPSC 360 Discrete Mathematics for Computer...

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/08/2008 for the course CMPSC 360 taught by Professor Haullgren during the Fall '08 term at Penn State.

### Page1 / 2

CMPSC360hw3 - CMPSC 360 Discrete Mathematics for Computer...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online