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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 ...
Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring. The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤": For any predicate φ, if φ(0) is true, and
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction.
In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. 1 In other words, the set of nonnegative integers is well-ordered by its "natural" or "magnitude" order in which precedes if and only if is either or the sum of and some nonnegative integer (other orderings include ...
Then P(n) is true for all natural numbers n. For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent " 2n − 1 is odd": (i) For n = 1, 2n − 1 = 2 (1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.
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 .
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 .
Hockey-stick identity. Recurrence relations of binomial coefficients in Pascal's triangle. Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n =6, r =2: 1+3+6+10+15=35. In combinatorial mathematics, the hockey-stick identity, [1] Christmas stocking identity, [2] boomerang identity, Fermat's identity or ...
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