Concrete Maths Notes 1

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The Proof Technique CheatSheet

Ravi Mohan

January 5, 2012

1 Prelims: Proof By Induction

(1) Axiom: Mathematical Induction Over N . Form 1

(∀n :  N |: (∀i | 0 ≤ i < n :  P i) ⇒ P n) ⇒ (∀n :  N   :  P n)used to prove Universal Quantification By Induction

(2) Theorem: Mathematical Induction Over N . Form 2

(∀n :  N |: (∀i | 0 ≤ i < n :  P i) ⇒ P n) ≡ (∀n :  N   :  P n)

used to prove properties of Induction

(3) Theorem: Mathematical Induction Over N . Form 3

P 0 ∧ (∀n :  N |: (∀i | 0 ≤ i ≤ n :  P i) ⇒ P n+1) ⇒ (∀n :  N   :  P n)

restatement of (1) used for inductive proofs

(4) Parts of Mathematical Induction axiom

P 0  is the base case

(∀n  :  N |: (∀i |  0  ≤   i ≤   n  :   P i) ⇒   P n+1) ⇒  (∀n  :  N   :   P n) is the inductivecase

(∀n :  N |: (∀i | 0 ≤ i ≤ n :  P i) ⇒ P n+1) is the inductive hypothesis

(5) Normal Proof Method

(1) Prove Base Case.   P 0

(2) Assume arbitrary n  ≥  0

(3) Assume Inductive Case (∀i | 0 ≤ i ≤ n :  P i)

(4) Prove   P n+1

(6) Induction starting at other integers

Let a sequence of integers start at   n0  Then Inductive Theorem is

P n0 ∧ (∀n :  n0 ≤ n |: (∀i | n0  ≤ i ≤ n :  P i) ⇒ P n+1) ⇒ (∀n :  n0 ≤ n :  P n)

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2 Chapter One

3 Chapter Two - Summation

4 Chapter Five - Binomials

5 Chapter Seven - Generating Functions

6 Chapter Eight - Probability

7 Chapter Nine - Asymptotics

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