{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter_04-4up

# Chapter_04-4up - Sequences Sequences Induction Sums...

This preview shows pages 1–4. Sign up to view the full content.

Comp Sci 360 - Fall 2009 Induction Definitions and Proofs Sequences Sums Products John Hannan - Comp Sci 360 - Fall 2009 Sequences John Hannan - Comp Sci 360 - Fall 2009 Sets & Properties Given: Set S of elements & some property (predicate) P Goal: Prove that P holds for all elements of S: ! x " S.P(x) Techniques? John Hannan - Comp Sci 360 - Fall 2009 Observing a Set 0 7 8 167 168

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

View Full Document
John Hannan - Comp Sci 360 - Fall 2009 Observations? 2 3 4 6 9 8 10 12 14 15 16 18 20 21 John Hannan - Comp Sci 360 - Fall 2009 Observations? 4 2 6 8 10 12 14 16 3 6 9 12 15 18 Sets of Numbers Natural Numbers Real Numbers? John Hannan - Comp Sci 360 - Fall 2009 Inductive Definitions John Hannan - Comp Sci 360 - Fall 2009 Mathematical Induction To prove ! x " N.P(x) holds Inductive Definition of N: 1. 0 " N 2. if n " N then n+1 " N 1. Show P(0) holds 2. Show that if P(n) holds then P(n+1) must also hold
John Hannan - Comp Sci 360 - Fall 2009 Examples ! i = n(n+1)/2 i=0 n 3 2n -1 is divisible by 8 A set with n elements has 2 n distinct subsets Sum of squares of the natural numbers up to n is n(n+1)(2n+1)/5 John Hannan - Comp Sci 360 - Fall 2009 Steps to Remember

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.

{[ snackBarMessage ]}

### Page1 / 6

Chapter_04-4up - Sequences Sequences Induction Sums...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online