Search results
Results from the WOW.Com Content Network
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.
In first-order set theories, the common framework, the set induction principle is an axiom schema, granting an axiom for any predicate (i.e. class). In contrast, the axiom of regularity is a single axiom, formulated with a universal quantifier only over elements of the domain of discourse, i.e. over sets.
Bar induction is a reasoning principle used in intuitionistic mathematics, introduced by L. E. J. Brouwer. Bar induction's main use is the intuitionistic derivation of the fan theorem, a key result used in the derivation of the uniform continuity theorem. It is also useful in giving constructive alternatives to other classical results.
A simple common example is the Universe à la Tarski type former. It creates some inductive type : and some inductive predicate :.For every type in the type theory (except itself!), there will be some element of which may be seen as some code for this corresponding type; The predicate inductively encodes each possible type to the corresponding element of ; and constructing new codes in will ...
As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is the following: Transfinite Recursion Theorem (version 2). Given a set g 1, and class functions G 2, G 3, there exists a unique function F: Ord → V such that F(0) = g 1, F(α + 1) = G 2 (F(α)), for all α ∈ Ord,
Since each of the latter is a substructure of the whole tree, it can be assumed to satisfy the property to be proven (a.k.a. the induction hypothesis). That is, p ≤ 2 g − 1 and q ≤ 2 h − 1 can be assumed, where g and h denotes the number of generations the father's and the mother's subtree extends over, respectively, and p and q denote ...
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. [1]
Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field. Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction .