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Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]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.
Pages in category "Mathematical induction" The following 8 pages are in this category, out of 8 total. This list may not reflect recent changes. ...
The induction, bounding and least number principles are commonly used in reverse mathematics and second-order arithmetic. For example, I Σ 1 {\displaystyle {\mathsf {I}}\Sigma _{1}} is part of the definition of the subsystem R C A 0 {\displaystyle {\mathsf {RCA}}_{0}} of second-order arithmetic.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms ...
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 .
In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction. [17] As Katz says . Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences.
Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for r ∈ { 0 , 1 } {\displaystyle r\in \{0,1\}} ,
Induction (play), an opening scene; Induction (teachers), support of novice teachers; Inductive reasoning aptitude; Collective Induction, in psychology; Hypnotic induction, causing hypnosis "Induction", a song by Broken Spindles from Fulfilled/complete "Induction" (short story), a short story by Greg Egan