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Description. The simplest and most common form of mathematical induction infers that a statement involving a natural number n (that is, an integer n ≥ 0 or 1) holds for all values of n. The proof consists of two steps: The base case (or initial case): prove that the statement holds for 0, or 1. The induction step (or inductive step, or step ...
All horses are the same color. All horses are the same color is a falsidical paradox that arises from a flawed use of mathematical induction to prove the statement All horses are the same color. [1] There is no actual contradiction, as these arguments have a crucial flaw that makes them incorrect. This example was originally raised by George ...
Well-founded induction is sometimes called Noetherian induction, [4] after Emmy Noether. On par with induction, well-founded relations also support construction of objects by transfinite recursion . Let ( X , R ) be a set-like well-founded relation and F a function that assigns an object F ( x , g ) to each pair of an element x ∈ X and a ...
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic [1] – do not vary smoothly in this way, but have distinct, separated values. [2]
Definition. Formally, let be any graph, and let be any subset of vertices of G. Then the induced subgraph is the graph whose vertex set is and whose edge set consists of all of the edges in that have both endpoints in . [1] That is, for any two vertices , and are adjacent in if and only if they are adjacent in .
Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets [ 4 ] (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics".
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 .
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly ...