Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Discrete Mathematics with Applications Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) March 21, 2019 Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Strong induction is similar to weak induction, but more general. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Strong induction is similar to weak induction, but more general. The difference is that the strong form of induction assumes several cases for the basis step. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Strong induction is similar to weak induction, but more general. The difference is that the strong form of induction assumes several cases for the basis step. Rather than assume that P ( k ) is true to prove that P ( k + 1) is true, we assume that P ( i ) is true for every integer i such that a ≤ i < k , where i is the base case, to help prove that P ( k ) is true. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Strong induction is similar to weak induction, but more general. The difference is that the strong form of induction assumes several cases for the basis step. Rather than assume that P ( k ) is true to prove that P ( k + 1) is true, we assume that P ( i ) is true for every integer i such that a ≤ i < k , where i is the base case, to help prove that P ( k ) is true. We use strong induction instead of weak induction whenever the inductive hypothesis of P ( k ) being true by itself isn’t enough for a complete proof. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Process: Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Process: 1 “Basis Step” or “Base Case”: Prove that P ( a ) is true. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Process: 1 “Basis Step” or “Base Case”: Prove that P ( a ) is true. 2 “Inductive Step”: Prove “ ∀ k ≥ a , if P ( i ) holds for all integers i such that a ≤ i < k , then P ( k ) also holds” via the following: Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Process: 1 “Basis Step” or “Base Case”: Prove that P ( a ) is true. 2 “Inductive Step”: Prove “ ∀ k ≥ a , if P ( i ) holds for all integers i such that a ≤ i < k , then P ( k ) also holds” via the following: “Inductive Hypothesis”: Assume P ( i ) is true for every integer 1 i between a (inclusive) and k (exclusive). Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Process: 1 “Basis Step” or “Base Case”: Prove that P ( a ) is true. 2 “Inductive Step”: Prove “ ∀ k ≥ a , if P ( i ) holds for all integers i such that a ≤ i < k , then P ( k ) also holds” via the following: “Inductive Hypothesis”: Assume P ( i ) is true for every integer 1 i between a (inclusive) and k (exclusive). 2 Deduce that P ( k ) must also be trues. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem The Principle of Strong Mathematical Induction Let P ( n ) be a predicate where the domain of n is Z , and let a be a fixed integer. Goal: Prove a theorem of the form “ ∀ n ≥ a , P ( n ) . ” Process: 1 “Basis Step” or “Base Case”: Prove that P ( a ) is true. 2 “Inductive Step”: Prove “ ∀ k ≥ a , if P ( i ) holds for all integers i such that a ≤ i < k , then P ( k ) also holds” via the following: “Inductive Hypothesis”: Assume P ( i ) is true for every integer 1 i between a (inclusive) and k (exclusive). 2 Deduce that P ( k ) must also be trues. Note: The base case is often vacuously true. In such situations, proving a base case is not necessary. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Theorem If { b n } ∞ n =0 is the sequence defined recursively by b 0 = 1 , b 1 = 2 , b 3 = 3 , and b n = b n − 3 + b n − 2 + b n − 1 for n ≥ 3 , then b n ≤ 3 n for all nonnegative integers n . Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Theorem If { b n } ∞ n =0 is the sequence defined recursively by b 0 = 1 , b 1 = 2 , b 3 = 3 , and b n = b n − 3 + b n − 2 + b n − 1 for n ≥ 3 , then b n ≤ 3 n for all nonnegative integers n . This is an example of when strong induction is needed instead of weak induction, because the recurrence relation is defined using the three prior terms of the sequence instead of just the immediate previous one. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Proof. We proceed by strong induction on n . Base Case 1: n = 0 . Then b 0 = 1 ≤ 3 0 . Base Case 2: n = 1 . Then b 1 = 2 ≤ 3 1 . Base Case 3: n = 2 . Then b 2 = 3 ≤ 3 2 . Inductive Step: Let k be an arbitrary integer that is at least 3, and suppose that b i ≤ 3 i for every nonnegative integer i < k . Then b k = b k − 1 + b k − 2 + b k − 3 ≤ 3 k − 1 + 3 k − 2 + 3 k − 3 (inductive hypothesis) ≤ 3 k − 1 + 3 k − 1 + 3 k − 1 = 3 ∗ 3 k − 1 = 3 k Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Theorem (The Fundamental Theorem of Arithmetic) Every integer greater than 1 is either a prime number or can be uniquely written as the product of prime factors. Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
Strong Induction The Well-Ordering Principle The Quotient-Remainder Theorem Theorem (The Fundamental Theorem of Arithmetic) Every integer greater than 1 is either a prime number or can be uniquely written as the product of prime factors. This is possibly the most famous example of a theorem requiring a strong inductive proof! (Note: a base case is unnecessary here. Why?) Chapter 5: Sequences, Mathematical Induction, and Recursion (Part 2) Discrete Mathematics with Applications
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