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Many theorems provable using choice are of an elegant general character: the cardinalities of any two sets are comparable, every nontrivial ring with unity has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among many others. Frequently, the axiom of ...
The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be ...
Using the axiom of choice, one can show that for any family S of sets | ⋃S | ≤ | S | × sup { |s| : s ∈ S} (A). [5] Moreover, by Tarski's theorem on choice, another equivalent of the axiom of choice, | X | n = | X | for all finite n (B). Let X be an infinite set and let F denote the set of all finite subsets of X. There is a natural ...
Biology portal; Pages in category "Biological theorems" The following 6 pages are in this category, out of 6 total. This list may not reflect recent changes. B. Bet ...
Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2m < n + 1 every map f : M m → N n of an m-dimensional manifold to an n-dimensional manifold is homotopic to an immersion, and in fact to an embedding for 2m < n; these are the Whitney immersion theorem and Whitney embedding theorem.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic.
In Mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
The choice of the "shortest tree" relative to a not-so-short tree under any optimality criterion (smallest distance, fewest steps, or maximum likelihood) is always based on parsimony. [61] Francis Crick has commented on potential limitations of Occam's razor in biology. He advances the argument that because biological systems are the products ...