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All theorems and corollaries are proven by exploring the implications of the axiomata and other theorems that have previously been developed. New terms are defined using the primitive terms and other derived definitions based on those primitive terms. In a deductive system, one can correctly use the term "proof", as applying to a theorem.
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of ...
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 ...
With the development of these three techniques, the field of structural biology expanded and also became a branch of molecular biology, biochemistry, and biophysics concerned with the molecular structure of biological macromolecules (especially proteins, made up of amino acids, RNA or DNA, made up of nucleotides, and membranes, made up of ...
In 1830, Étienne Geoffroy Saint-Hilaire argued a structuralist case against the functionalist (teleological) position of Georges Cuvier.Geoffroy believed that homologies of structure between animals indicated that they shared an ideal pattern; these did not imply evolution but a unity of plan, a law of nature.
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 ...
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 ...
The theorem implies the existence of a continuous approximate selection. Another set of sufficient conditions for the existence of a continuous approximate selection is given by the Deutsch–Kenderov theorem, [4] whose conditions are more general than those of Michael's theorem (and thus the selection is only approximate): X is a paracompact ...