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Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold. This is done by first proving a simple case, then also showing that if we assume the claim is true for a given case, then the next case is also true.
The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n): ( + ) = + . For n > 0, we proceed by mathematical induction.
Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further generalized to arbitrary Noetherian induction .
Pages in category "Mathematical induction" The following 8 pages are in this category, out of 8 total. ... This page was last edited on 2 June 2023, ...
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion.
For example, for the condition F is "Why do apples fall?". The answer is a theory T that implies that apples fall; = Inductive inference is of the form, All observed objects in a class C have a property P. Therefore there is a probability that all objects in a class C have a property P.
Backward induction was first utilized in 1875 by Arthur Cayley, who discovered the method while attempting to solve the secretary problem. [2] In dynamic programming, a method of mathematical optimization, backward induction is used for solving the Bellman equation.