Concrete Maths Notes 1
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Transcript of Concrete Maths Notes 1
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7/23/2019 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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7/23/2019 Concrete Maths Notes 1
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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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