# lect13 - 1 Mathematical Induction • Induction is the most...

This preview shows pages 1–7. Sign up to view the full content.

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

View Full Document

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

View Full Document

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.

Unformatted text preview: 1 Mathematical Induction. • Induction is the most important proof method in computer science. • Suppose you want to prove that the proposition P ( n ) is true for all n ∈ N , where N = {0, 1, 2, …} (an infinite set). ( i. e. every natural number n has some property P ) • You might be able to prove it for 0, 1, 2, … But you can’t check one-by-one that all natural numbers have property P. 2 • The key idea of mathematical induction is start with 0 and repeatedly add 1. • Suppose you can show that i) 0 has property P and ii) whenever you add 1 to a number that has property P the resulting number also has property P. This guarantees that as you go through the list of all natural numbers, every number you encounter must have property P . 3 • The proof method of mathematical induction says that you just need to do two things: i) prove it for n = 0 (basis case) ii) prove “if it’s true for n=k, then it’s true for n =k+ 1” Induction Hypothesis: • fix some k ≥ 0 • assume P ( k ) Induction Step • Using assumption P ( k ) prove that P ( k +1) In this way we prove that for any n [ P ( n ) → P ( n +1)] 4 Why does this prove it for all n ∈ N ? Actually it relies upon the property of integers N= {0, 1, 2, …} The pattern is: To prove 2200 n ≥ 0 ( P ( n )) it’s enough to show: 1) (Base) P (0) 2) 2200 n ≥ 0 ( P ( n ) → P ( n +1)) So, in the second step we say: “take some k ≥ 0 and assume that property P holds for n = k“ (Induction Hypothesis). Then, (based upon this assumption) we need to show that the property P can be implied for n = k +1. How can we prove 2200 n ≥ 0 ( P ( n ) → P ( n +1)) ? Pick arbitrary k ≥ 0 and prove it for n = k. 5 Suppose we want to prove the formula for the sum of ‘geometric progression’ q q q q q q n n-- = + + + + + + 1 1 ... 1 1 3 2 where 1 < < q , and n is any integer, n ≥ How can we prove that the formula is correct for any n ≥ ? • Show it for arbitrary fixed n ≥ • Use induction on n ≥ Notation: ∑ = = + + + + = n i i n n q q q q S 2 ... 1 q q S n n-- = + 1 1 1 6 • Show it for some n ≥ 1 2 1 2 1 ) ... 1 ( ......
View Full Document

## This document was uploaded on 07/14/2011.

### Page1 / 21

lect13 - 1 Mathematical Induction • Induction is the most...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online