Mathematical Induction Jason Filippou CMSC250 @ UMCP 06-27-2016 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 1 / 48
Outline 1 Sequences and series Sequences Series and partial sums 2 Weak Induction Intro to Induction Practice 3 Strong Induction 4 Errors in proofs by mathematical induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 2 / 48
Sequences and series Sequences and series Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 3 / 48
Sequences and series Sequences Sequences Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 4 / 48
Sequences and series Sequences Definitions Definition (Sequence) A function a : N �→ R is called a sequence . Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 5 / 48
Sequences and series Sequences Definitions Definition (Sequence) A function a : N �→ R is called a sequence . Examples: 2 , 4 , 6 , . . . 10 , 20 , 30 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , . . . Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 5 / 48
Sequences and series Sequences Definitions Definition (Sequence) A function a : N �→ R is called a sequence . Examples: 2 , 4 , 6 , . . . 10 , 20 , 30 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , . . . So, sequences can be either finite or infinite . We will mostly care about infinite sequences. Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 5 / 48
Sequences and series Sequences Denoting sequences A sequence can be enumerated ... a : a 1 , a 2 , . . . (or just a 1 , a 2 , . . . ) c 0 , c 1 , c 2 , . . . (notice the indices) Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 6 / 48
Sequences and series Sequences Denoting sequences A sequence can be enumerated ... a : a 1 , a 2 , . . . (or just a 1 , a 2 , . . . ) c 0 , c 1 , c 2 , . . . (notice the indices) Described through an explicit formula ... b k = 2 k r n = ( n + 1)! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 6 / 48
Sequences and series Sequences Denoting sequences A sequence can be enumerated ... a : a 1 , a 2 , . . . (or just a 1 , a 2 , . . . ) c 0 , c 1 , c 2 , . . . (notice the indices) Described through an explicit formula ... b k = 2 k r n = ( n + 1)! Or a recursive formula... F n +1 = F n + F n − 1 ∀ n ≥ 1 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 6 / 48
Sequences and series Sequences The arithmetic sequence Definition Let a : a 0 , a 1 , . . . be a sequence and ω ∈ R . If a j = a j − 1 + ω ∀ j ∈ N ∗ , a is an arithmetic sequence (or progression ). Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 7 / 48
Sequences and series Sequences The arithmetic sequence Definition Let a : a 0 , a 1 , . . . be a sequence and ω ∈ R . If a j = a j − 1 + ω ∀ j ∈ N ∗ , a is an arithmetic sequence (or progression ). a 0 and ω fully define the sequence. So, how can I write a r ? a 1 + r ∗ ω r ∗ a 0 a 0 + r ∗ ω r ∗ a 0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 7 / 48
Sequences and series Sequences The geometric sequence Definition Let a : a 0 , a 1 , . . . be a sequence and k ∈ R ∗ . If a j = c ∗ a j − 1 ∀ j ∈ N ∗ , a is a geometric sequence (or progression ). Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 8 / 48
Sequences and series Sequences The geometric sequence Definition Let a : a 0 , a 1 , . . . be a sequence and k ∈ R ∗ . If a j = c ∗ a j − 1 ∀ j ∈ N ∗ , a is a geometric sequence (or progression ). a 0 and c fully define the sequence. a r . How can I write it? c r ∗ a 0 a r a rc a 0 + c r 0 0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 8 / 48
Sequences and series Series and partial sums Series and partial sums Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 9 / 48
Sequences and series Series and partial sums Definitions Definition (Series) + ∞ � Let a 0 , a 1 , . . . be any sequence. Then, the sum a i is called a series . i =0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 10 / 48
Sequences and series Series and partial sums Definitions Definition (Series) + ∞ � Let a 0 , a 1 , . . . be any sequence. Then, the sum a i is called a series . i =0 Definition (Partial sum) + ∞ � Let n ∈ N . Then, the n -th partial sum of the series a i , denoted i =0 n � S n , is the sum a i . i =0 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 10 / 48
Sequences and series Series and partial sums Definitions Definition (Series) + ∞ � Let a 0 , a 1 , . . . be any sequence. Then, the sum a i is called a series . i =0 Definition (Partial sum) + ∞ � Let n ∈ N . Then, the n -th partial sum of the series a i , denoted i =0 n � S n , is the sum a i . i =0 The partial sums themselves also form a sequence! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 10 / 48
Sequences and series Series and partial sums Statements to prove! To kickstart the discussion on induction, here are two theorems concerning partial sums: Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 11 / 48
Sequences and series Series and partial sums Statements to prove! To kickstart the discussion on induction, here are two theorems concerning partial sums: Theorem (Closed form of the arithmetic progression partial sum) If a is an arithmetic progression, S n = n ( a 1 + a n ) . 2 Theorem (Closed form of the geometric progression partial sum) If a is a geometric progression and c � = 1 , S n = a 1 ( c n − 1) . c − 1 Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 11 / 48
Sequences and series Series and partial sums Statements to prove! To kickstart the discussion on induction, here are two theorems concerning partial sums: Theorem (Closed form of the arithmetic progression partial sum) If a is an arithmetic progression, S n = n ( a 1 + a n ) . 2 Theorem (Closed form of the geometric progression partial sum) If a is a geometric progression and c � = 1 , S n = a 1 ( c n − 1) . c − 1 Both of those theorems can be proven via (weak) mathematical induction! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 11 / 48
Weak Induction Weak Induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 12 / 48
Weak Induction Intro to Induction Intro to Induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 13 / 48
Weak Induction Intro to Induction Proof methods: The story so far... S Existential Stmt. ? - + + - Universal Existential Proof Proof Non- Direct Indirect Constructive constructive Contradiction ? Generic ? ? Particular Contraposition Universal Statement Division into cases Exhaustion Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 14 / 48
Weak Induction Intro to Induction Where induction fits S Existential Stmt. Universal Stmt - + + - Universal Existential Proof Proof Non- Indirect Constructive Direct constructive Contradiction Contraposition Generic Exhaustion Cases Particular Induction Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 15 / 48
Weak Induction Intro to Induction The penny proposition: Statement Suppose I have at least 4 ¢ in my wallet. Then, it turns out that all my money can be stacked as 2 ¢ and 5 ¢ coins! Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 16 / 48
Weak Induction Intro to Induction The penny proposition: Direct (non-inductive) proof The penny proposition Every dollar amount greater than 3 ¢ s can be paid with only 2 ¢ and 5 ¢ coins. Proof (Direct, by cases). Suppose we have a total amount of C cents in our wallet. If C is an even number, then the statement is trivial by the definition of even numbers: We can just stack k 2 ¢ coins for some positive integer k . If C is an odd number greater than 3, then it is 5 or greater. If it is 5, the problem is trivial: We only need one 5 ¢ coin. But every odd dollar amount after 5 ¢ can be retrieved by adding any number of 2 ¢ coins, because of the definition of parity. We are therefore done in both cases. Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 17 / 48
Weak Induction Intro to Induction The penny proposition: Direct (non-inductive) proof The penny proposition Every dollar amount greater than 3 ¢ s can be paid with only 2 ¢ and 5 ¢ coins. Proof (Direct, by cases). Suppose we have a total amount of C cents in our wallet. If C is an even number, then the statement is trivial by the definition of even numbers: We can just stack k 2 ¢ coins for some positive integer k . If C is an odd number greater than 3, then it is 5 or greater. If it is 5, the problem is trivial: We only need one 5 ¢ coin. But every odd dollar amount after 5 ¢ can be retrieved by adding any number of 2 ¢ coins, because of the definition of parity. We are therefore done in both cases. What do you think of this proof? Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 17 / 48
Weak Induction Intro to Induction The principle Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 18 / 48
Weak Induction Intro to Induction The principle Principle of Weak Mathematical Induction Assume P ( n ) is a predicate applied on any natural number n , and a ∈ N . If: P ( a ) is true P ( k + 1) is true when P ( k ) is true ∀ k ≥ a then, ∀ n ≥ a, P ( n ) is true. Jason Filippou (CMSC250 @ UMCP) Induction 06-27-2016 19 / 48
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