Complete induction | Wikipedia audio article
Channel: wikipedia tts
Date: 23/01/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Mathematical_induction


00:02:40 1 History
00:05:07 2 Description
00:05:41 3 Example
00:06:57 4 Variants
00:07:26 4.1 Induction basis other than 0 or 1
00:12:24 4.1.1 Example: forming dollar amounts by coins
00:12:44 4.2 Induction on more than one counter
00:14:00 4.3 Infinite descent
00:26:03 4.4 Prefix induction
00:26:49 4.5 Complete (strong) induction
00:28:13 4.5.1 Example: Fibonacci numbers
00:32:47 4.5.2 Example: prime factorization
00:36:10 4.5.3 Example: dollar amounts revisited
00:37:11 4.6 Forward-backward induction
00:37:38 5 Example of error in the inductive step
00:38:04 6 Formalization
00:38:08 7 Transfinite induction
00:38:18 8 Equivalence with the well-ordering principle
00:39:45 9 See also
00:43:43 10 Notes
00:44:51 11 References
00:46:14 11.1 Introduction
00:46:38 11.2 History
00:46:49 Formalization
00:51:17 Transfinite induction
00:52:21 Equivalence with the well-ordering principle
00:55:31 See also



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SUMMARY
=======
Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on.  Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).

The method of induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0.  The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1.  These two steps establish the property P(n) for every natural number n = 0, 1, 2, 3, ... The base step need not begin with zero.  Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.  Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy (also see Problem of induction). Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning.