Mathematical induction
LECTURE NOTES ON MATHEMATICAL INDUCTION
PETE L. CLARK
Contents
1. Introduction
2. The (Pedagogically) First Induction Proof
3. The (Historically) First(?) Induction Proof
4. Closed Form Identities
5. More on Power Sums
6. Inequalities
7. Extending binary properties to n-ary properties
8. Miscellany
9. The Principle of Strong/Complete Induction
10. Solving Homogeneous Linear Recurrences
11. The Well-Ordering Principle
12. Upward-Downward Induction
13. The Fundamental Theorem of Arithmetic
13.1. Euclid’s Lemma and the Fundamental Theorem of Arithmetic
13.2. Rogers’ Inductive Proof of Euclid’s Lemma
13.3. The Lindemann-Zermelo Inductive Proof of FTA
References
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1. Introduction
Principle of Mathematical Induction for sets
Let S be a subset of the positive integers. Suppose that:
(i) 1 ∈ S, and
(ii) ∀ n ∈ Z+ , n ∈ S =⇒ n + 1 ∈ S.
Then S = Z+ .
The intuitive justification is as follows: by (i), we know that 1 ∈ S. Now apply (ii) with n = 1: since 1 ∈ S, we deduce 1 + 1 = 2 ∈ S. Now apply (ii) with n = 2: since 2 ∈ S, we deduce 2 + 1 = 3 ∈ S. Now apply (ii) with n = 3: since
3 ∈ S, we deduce 3 + 1 = 4 ∈ S. And so forth.
This is not a proof. (No good proof uses “and so forth” to gloss over a key point!)
But the idea is as follows: we can keep iterating the above argument as many times as we want, deducing at each stage that since S contains the natural number which is one greater than the last natural number we showed that it contained. Now it is a fundamental part of the structure of the positive integers that every positive
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PETE L. CLARK
integer can be reached in this way, i.e., starting from 1 and adding 1 sufficiently many times. In other words, any rigorous definition of the natural numbers (for instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical
PETE L. CLARK
Contents
1. Introduction
2. The (Pedagogically) First Induction Proof
3. The (Historically) First(?) Induction Proof
4. Closed Form Identities
5. More on Power Sums
6. Inequalities
7. Extending binary properties to n-ary properties
8. Miscellany
9. The Principle of Strong/Complete Induction
10. Solving Homogeneous Linear Recurrences
11. The Well-Ordering Principle
12. Upward-Downward Induction
13. The Fundamental Theorem of Arithmetic
13.1. Euclid’s Lemma and the Fundamental Theorem of Arithmetic
13.2. Rogers’ Inductive Proof of Euclid’s Lemma
13.3. The Lindemann-Zermelo Inductive Proof of FTA
References
1
4
5
6
7
10
11
13
14
16
20
21
23
23
24
25
25
1. Introduction
Principle of Mathematical Induction for sets
Let S be a subset of the positive integers. Suppose that:
(i) 1 ∈ S, and
(ii) ∀ n ∈ Z+ , n ∈ S =⇒ n + 1 ∈ S.
Then S = Z+ .
The intuitive justification is as follows: by (i), we know that 1 ∈ S. Now apply (ii) with n = 1: since 1 ∈ S, we deduce 1 + 1 = 2 ∈ S. Now apply (ii) with n = 2: since 2 ∈ S, we deduce 2 + 1 = 3 ∈ S. Now apply (ii) with n = 3: since
3 ∈ S, we deduce 3 + 1 = 4 ∈ S. And so forth.
This is not a proof. (No good proof uses “and so forth” to gloss over a key point!)
But the idea is as follows: we can keep iterating the above argument as many times as we want, deducing at each stage that since S contains the natural number which is one greater than the last natural number we showed that it contained. Now it is a fundamental part of the structure of the positive integers that every positive
1
2
PETE L. CLARK
integer can be reached in this way, i.e., starting from 1 and adding 1 sufficiently many times. In other words, any rigorous definition of the natural numbers (for instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical
References: [Ac00] F. Acerbi, Plato: Parmenides 149a7-c3. A Proof by Complete Induction? Archive for History of the Exact Sciences 55 (2000), 57–76. [Li33] F.A Lindemann, The Unique Factorization of a Positive Integer. Quart. J. Math. 4, 319– 320, 1933. [Mu63] A.A. Mullin, Recursive function theory (A modern look at a Euclidean idea). Bulletin of the American Mathematical Society 69 (1963), 737. [Ro63] K. Rogers, Classroom Notes: Unique Factorization. Amer. Math. Monthly 70 (1963), no. [Ze34] E. Zermelo, Elementare Betrachtungen zur Theorie der Primzahlen. Nachr. Gesellsch. Wissensch. G¨ ottingen 1, 43–46, 1934.