lecture04_induction

lecture04_induction - COMPSCI 102 Discrete Mathematics for...

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

View Full Document Right Arrow Icon
COMPSCI 102 Discrete Mathematics for Computer Science
Background image of page 1

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

View Full DocumentRight Arrow Icon
Lecture 4 Inductive Reasoning
Background image of page 2
Dominoes Domino Principle: Line up any number of dominos in a row; knock the first one over and they will all fall
Background image of page 3

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

View Full DocumentRight Arrow Icon
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
Background image of page 4
Standard Notation “for all” is written “ 2200 Example: For all k>0, P(k) 2200 k>0, P(k) =
Background image of page 5

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

View Full DocumentRight Arrow Icon
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
Background image of page 6
The Natural Numbers One domino for each natural number: 0 1 2 3 = { 0, 1, 2, 3, . . . }
Background image of page 7

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

View Full DocumentRight Arrow Icon
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.
Background image of page 8
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
Background image of page 9

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

View Full DocumentRight Arrow Icon
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
Background image of page 10
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
Background image of page 11

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

View Full DocumentRight Arrow Icon
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
Background image of page 12
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:
Background image of page 13

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

View Full DocumentRight Arrow Icon
Theorem The sum of the first n odd numbers is n 2
Background image of page 14
Primes: Note: 1 is not considered prime A natural number n > 1 is a prime if it has no divisors besides 1 and itself
Background image of page 15

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

View Full DocumentRight Arrow Icon
Theorem ? Every natural number > 1 can be
Background image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

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 Right Arrow Icon
Ask a homework question - tutors are online