lecture04_induction

# lecture04_induction - COMPSCI 102 Discrete Mathematics for...

This preview shows pages 1–17. Sign up to view the full content.

COMPSCI 102 Discrete Mathematics for Computer Science

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

View Full Document
Lecture 4 Inductive Reasoning
Dominoes Domino Principle: Line up any number of dominos in a row; knock the first one over and they will all fall

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

View Full Document
Dominoes Numbered 1 to n F k = “The k th domino falls” If we set them up in a row then each one is set up to knock over the next: For all 1 k < n: F k F k+1 F 1 F 2 F 3 F 1 All Dominoes Fall
Standard Notation “for all” is written “ 2200 Example: For all k>0, P(k) 2200 k>0, P(k) =

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

View Full Document
Dominoes Numbered 0 to n-1 F k = “The k th domino falls” 2200 k, 0 k < n-1: F k F k+1 F 0 F 1 F 2 F 0 All Dominoes Fall
The Natural Numbers One domino for each natural number: 0 1 2 3 = { 0, 1, 2, 3, . . . }

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

View Full Document
Plato (428-328BC): The Domino Principle works for an infinite row of dominoes Aristotle (384-322BC): Never seen an infinite number of anything, much less dominoes.
Mathematical Induction statements proved instead of dominoes fallen Infinite sequence of dominoes Infinite sequence of statements: S 0 , S 1 , … F k = “domino k fell” F k = “S k proved” Conclude that F k is true for all k Establish: 1. F 0 2. For all k, F k F k+1

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

View Full Document
Inductive Proofs To Prove 2200 k , S k Establish “Base Case”: S 0 Establish that 2200 k, S k S k+1 2200 k, S k S k+1 Assume hypothetically that S k for any particular k; Conclude that S k+1
Theorem ? The sum of the first n odd numbers is n 2 Check on small values: 1 = 1 1+3 = 4 1+3+5 = 9 1+3+5+7 = 16

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

View Full Document
Theorem ? The sum of the first n odd numbers is n 2 The k th odd number is (2k – 1), when k > 0 S n is the statement that: “1+3+5+(2k-1)+. ..+(2n-1) = n 2
S n = “1 + 3 + 5 + (2k-1) + . . +(2n-1) = n 2 Establishing that 2200 n 1 Base Case: S 1 Assume “Induction Hypothesis”: S k That means: 1+3+5+…+ (2k-1) = k 2 1+3+5+…+ (2k-1) +(2k+1) = k 2 +(2k+1) Sum of first k+1 odd numbers = (k+1) 2 Domino Property:

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

View Full Document
Theorem The sum of the first n odd numbers is n 2
Primes: Note: 1 is not considered prime A natural number n > 1 is a prime if it has no divisors besides 1 and itself

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

View Full Document
Theorem ? Every natural number > 1 can be
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 55

lecture04_induction - COMPSCI 102 Discrete Mathematics for...

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online