Discrete Structures - CM0246 Mathematical Induction
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1 Discrete Structures - CM0246 Mathematical Induction Andrés Sicard-Ramírez EAFIT University Semester
2 Motivation Example Conjecture a formula for the sum of the first n positive odd integers. Discrete Structures - CM0246. Mathematical Induction 2/39
3 Motivation Example Conjecture a formula for the sum of the first n positive odd integers. Problem Let P (n) be a propositional function. How can we proof that P (n) is true for all n Z +? Discrete Structures - CM0246. Mathematical Induction 3/39
4 Definition (Principle of mathematical induction) Let P (n) be a propositional function. To prove that P (n) is true for all n Z +, we must make two proofs: Basis step: Prove P (1) Discrete Structures - CM0246. Mathematical Induction 4/39
5 Definition (Principle of mathematical induction) Let P (n) be a propositional function. To prove that P (n) is true for all n Z +, we must make two proofs: Basis step: Prove P (1) Inductive step: Prove P (k) P (k + 1) for all k Z + P (k) is called the inductive hypothesis. Discrete Structures - CM0246. Mathematical Induction 5/39
6 How mathematical / Induction induction and Recursion works FIGURE 2 Illustrating How Mathematical Induction Works Using Dom WAYS TO REMEMBER HOW MATHEMATICAL INDUCTION WORK Fig. 2 of [Rosen 2012, the 5.1]. infinite ladder and the rules for reaching steps can help you remember ho Discrete Structures - CM0246. Mathematical induction Inductionworks. Note that statements (1) and (2) for the infinite ladder6/39 are e
7 Definition (Principle of mathematical induction) Inference rule version: P (1) k(p (k) P (k + 1)) (PMI) np (n) or equivalently [P (1) k(p (k) P (k + 1)] np (n) (PMI) Discrete Structures - CM0246. Mathematical Induction 7/39
8 Methodology for using the principle of mathematical induction Discrete Structures - CM0246. Mathematical Induction 8/39
9 Methodology for using the principle of mathematical induction 1. State the propositional function P (n). Discrete Structures - CM0246. Mathematical Induction 9/39
10 Methodology for using the principle of mathematical induction 1. State the propositional function P (n). 2. Prove the basis step, i.e. P (1). Discrete Structures - CM0246. Mathematical Induction 10/39
11 Methodology for using the principle of mathematical induction 1. State the propositional function P (n). 2. Prove the basis step, i.e. P (1). 3. Prove the induction step, i.e. k(p (k) P (k + 1)). Remark: In this proof you need to use the inductive hypothesis P (n). Discrete Structures - CM0246. Mathematical Induction 11/39
12 Methodology for using the principle of mathematical induction 1. State the propositional function P (n). 2. Prove the basis step, i.e. P (1). 3. Prove the induction step, i.e. k(p (k) P (k + 1)). Remark: In this proof you need to use the inductive hypothesis P (n). 4. Conclude np (n) by the principle of mathematical induction. Discrete Structures - CM0246. Mathematical Induction 12/39
13 Example Prove that the sum of the first n odd positive integers is n 2. Whiteboard. Historical remark. From 1575, it could be the first property proved using the PMI [Gunderson 2011, 1.8]. Discrete Structures - CM0246. Mathematical Induction 13/39
14 Exercise Prove that if n Z +, then n = n(n + 1)/2. Discrete Structures - CM0246. Mathematical Induction 14/39
15 Exercise (cont.) Proof. 1. P (n): n = n(n + 1)/2. Discrete Structures - CM0246. Mathematical Induction 15/39
16 Exercise (cont.) Proof. 1. P (n): n = n(n + 1)/2. 2. Basis step P (1): 1 = 1(1 + 1)/2. Discrete Structures - CM0246. Mathematical Induction 16/39
17 Exercise (cont.) Proof. 1. P (n): n = n(n + 1)/2. 2. Basis step P (1): 1 = 1(1 + 1)/2. 3. Inductive step: Inductive hypothesis P (k): k = k(k + 1)/2. Let s prove P (k + 1): k + (k + 1) = k(k + 1)/2 + (k + 1) (by IH) = (k + 1)(k/2 + 1) (by arithmetic) = (k + 1)(k + 2)/2 (by arithmetic) Discrete Structures - CM0246. Mathematical Induction 17/39
18 Exercise (cont.) Proof. 1. P (n): n = n(n + 1)/2. 2. Basis step P (1): 1 = 1(1 + 1)/2. 3. Inductive step: Inductive hypothesis P (k): k = k(k + 1)/2. Let s prove P (k + 1): k + (k + 1) = k(k + 1)/2 + (k + 1) (by IH) = (k + 1)(k/2 + 1) (by arithmetic) = (k + 1)(k + 2)/2 (by arithmetic) 4. np (n) by the principle of induction mathematical. Discrete Structures - CM0246. Mathematical Induction 18/39
19 Exercise Prove that if n N, then n = 2 n+1 1 Discrete Structures - CM0246. Mathematical Induction 19/39
20 Exercise (cont.) Proof. 1. P (n): n = 2 n+1 1 Discrete Structures - CM0246. Mathematical Induction 20/39
21 Exercise (cont.) Proof. 1. P (n): n = 2 n Basis step P (0): 2 0 = 1 = Discrete Structures - CM0246. Mathematical Induction 21/39
22 Exercise (cont.) Proof. 1. P (n): n = 2 n Basis step P (0): 2 0 = 1 = Inductive step: Inductive hypothesis P (k): k = 2 k+1 1 Let s prove P (k + 1): k + 2 k+1 = 2 k k+1 (by IH) = 2(2 k+1 ) 1 (by arithmetic) = 2 k+2 1 (by arithmetic) Discrete Structures - CM0246. Mathematical Induction 22/39
23 Exercise (cont.) Proof. 1. P (n): n = 2 n Basis step P (0): 2 0 = 1 = Inductive step: Inductive hypothesis P (k): k = 2 k+1 1 Let s prove P (k + 1): k + 2 k+1 = 2 k k+1 (by IH) = 2(2 k+1 ) 1 (by arithmetic) = 2 k+2 1 (by arithmetic) 4. np (n) by the principle of induction mathematical. Discrete Structures - CM0246. Mathematical Induction 23/39
24 Strong Induction Definition (Strong (or course-of-values) induction) Let P (n) be a propositional function. To prove that P (n) is true for all n Z +, we must make two proofs: Basis step: Prove P (1) Discrete Structures - CM0246. Mathematical Induction 24/39
25 Strong Induction Definition (Strong (or course-of-values) induction) Let P (n) be a propositional function. To prove that P (n) is true for all n Z +, we must make two proofs: Basis step: Prove P (1) Inductive step: Prove [P (1) P (2) P (k)] P (k + 1) for all k Z + Discrete Structures - CM0246. Mathematical Induction 25/39
26 Strong Induction Definition (Strong (or course-of-values) induction) Let P (n) be a propositional function. To prove that P (n) is true for all n Z +, we must make two proofs: Basis step: Prove P (1) Inductive step: Prove [P (1) P (2) P (k)] P (k + 1) for all k Z + The (strong) inductive hypothesis is given by P (j) is true for j = 1, 2,, k. Discrete Structures - CM0246. Mathematical Induction 26/39
27 Strong Induction Definition (Strong induction) Inference rule version: P (1) k[(p (1) P (2) P (k)) P (k + 1)] np (n) (strong induction) Discrete Structures - CM0246. Mathematical Induction 27/39
28 Strong Induction Example (A part of the fundamental theorem of arithmetic) Prove that if n is an integer greater than 1, either is prime itself or is the product of prime numbers. Discrete Structures - CM0246. Mathematical Induction 28/39
29 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. Discrete Structures - CM0246. Mathematical Induction 29/39
30 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. 2. Basis step P (2): 2 is a prime number. Discrete Structures - CM0246. Mathematical Induction 30/39
31 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. 2. Basis step P (2): 2 is a prime number. 3. Inductive step: Inductive hypothesis: P (j) is true for j = 1, 2,, k. Discrete Structures - CM0246. Mathematical Induction 31/39
32 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. 2. Basis step P (2): 2 is a prime number. 3. Inductive step: Inductive hypothesis: P (j) is true for j = 1, 2,, k. Let s prove that k + 1 satisfies the property: Discrete Structures - CM0246. Mathematical Induction 32/39
33 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. 2. Basis step P (2): 2 is a prime number. 3. Inductive step: Inductive hypothesis: P (j) is true for j = 1, 2,, k. Let s prove that k + 1 satisfies the property: 3.1 If k + 1 is a prime number then it satisfies the property. Discrete Structures - CM0246. Mathematical Induction 33/39
34 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. 2. Basis step P (2): 2 is a prime number. 3. Inductive step: Inductive hypothesis: P (j) is true for j = 1, 2,, k. Let s prove that k + 1 satisfies the property: 3.1 If k + 1 is a prime number then it satisfies the property. 3.2 If k + 1 is a composite number: k + 1 = ab where 2 a b < k + 1. Since P (a) and P (b) by the inductive hypothesis, then P (k + 1). Discrete Structures - CM0246. Mathematical Induction 34/39
35 Strong Induction Example (cont.) Proof. 1. P (n): n is prime itself or it is the product of prime numbers. 2. Basis step P (2): 2 is a prime number. 3. Inductive step: Inductive hypothesis: P (j) is true for j = 1, 2,, k. Let s prove that k + 1 satisfies the property: 3.1 If k + 1 is a prime number then it satisfies the property. 3.2 If k + 1 is a composite number: k + 1 = ab where 2 a b < k + 1. Since P (a) and P (b) by the inductive hypothesis, then P (k + 1). 4. P (n) is true for all integer n greater than 1 by strong induction. Discrete Structures - CM0246. Mathematical Induction 35/39
36 First-Order Peano Arithmetic Axioms of first-order Peano arithmetic n. 0 n m n. m = n m = n n. 0 + n = n m n. m + n = (m + n) n. 0 n = 0 m n. m n = n + (m n) Giuseppe Peano ( ) See, for example, [Hájek and Pudlák 1998]. Discrete Structures - CM0246. Mathematical Induction 36/39
37 First-Order Peano Arithmetic Axioms of first-order Peano arithmetic n. 0 n m n. m = n m = n n. 0 + n = n m n. m + n = (m + n) n. 0 n = 0 m n. m n = n + (m n) For all formulae A, Giuseppe Peano ( ) [A(0) ( n. A(n) A(n ))] na(n) See, for example, [Hájek and Pudlák 1998]. Discrete Structures - CM0246. Mathematical Induction 37/39
38 First-Order Peano Arithmetic Theorem The principle of mathematical induction and strong induction are equivalent. See, for example, [Gunderson 2011]. Discrete Structures - CM0246. Mathematical Induction 38/39
39 References Gunderson, D. S. (2011). Handbook of Mathematical Induction. Chapman & Hall. Hájek, P. and Pudlák, P. (1998). Metamathematics of First-Order Arithmetic. 2nd printing. Springer. Rosen, K. H. (2012). Discrete Mathematics and Its Applications. 7th ed. McGraw-Hill. Discrete Structures - CM0246. Mathematical Induction 39/39
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