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Tautological (other)
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In mathematics, tautological may refer to: Logic: * Tautological consequence Geometry, where it is used as an alternative to canonical: *Tautological bundle *Tautological one-form * Tautology (grammar), unnecessary repetition, or more words than necessary, to say the same thing. See also * Tautology (other) * List of tautological place names A toponymy, place name is tautology (grammar), tautological if two differently sounding parts of it are synonymous. This often occurs when a name from one language is imported into another and a standard descriptor is added on from the second lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological Consequence
In propositional logic, tautological consequence is a strict form of logical consequenceBarwise and Etchemendy 1999, p. 110 in which the tautologousness of a proposition is preserved from one line of a proof to the next. Not all logical consequences are tautological consequences. A proposition Q is said to be a tautological consequence of one or more other propositions (P_1, P_2, ..., P_n) in a proof with respect to some logical system if one is validly able to introduce the proposition onto a line of the proof within the rules of the system; and in all cases when each of (P_1, P_2, ..., P_n) are true, the proposition Q also is true. Another way to express this preservation of tautologousness is by using truth tables. A proposition Q is said to be a tautological consequence of one or more other propositions (P_1, P_2, ..., P_n) if and only if in every row of a joint truth table that assigns "T" to all propositions (P_1, P_2, ..., P_n) the truth table also assigns "T" to Q. Exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical (other)
The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical example'' is often used to mean 'archetype'. Science and technology * Canonical form, a natural unique representation of an object, or a preferred notation for some object Mathematics * * Canonical coordinates, sets of coordinates that can be used to describe a physical system at any given point in time * Canonical map, a morphism that is uniquely defined by its main property * Canonical polyhedron, a polyhedron whose edges are all tangent to a common sphere, whose center is the average of its vertices * Canonical ring, a graded ring associated to an algebraic variety * Canonical injection, in set theory * Canonical representative, in set theory a standard member of each element of a set partition Differential geometry * Canonical one-form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological Bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector subspace W \subseteq V, the fiber over W is the subspace W itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is :\mathcal_(-1), the dual of the hyperplane bundle or Serre's twisting sheaf \mathcal_(1). The hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological One-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics and Hamiltonian mechanics (on the manifold Q). The exterior derivative of this form defines a symplectic form giving T^Q the structure of a symplectic manifold. The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector field on the tangent bundle. Definition in coordinates To define the tautological one-form, select a coordinate chart U on T^*Q and a canonical coordinate system on U. Pick an arbitrary point m \in T^*Q. By defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (grammar)
In literary criticism and rhetoric, a tautology is a statement that repeats an idea using near-synonymous morphemes, words or phrases, effectively "saying the same thing twice". Tautology and pleonasm are not consistently differentiated in literature. Like pleonasm, tautology is often considered a fault of style when unintentional. Intentional repetition may emphasize a thought or help the listener or reader understand a point. Sometimes logical tautologies like "Boys will be boys" are conflated with language tautologies, but a language tautology is not inherently true, while a logical tautology always is. Etymology The word was coined in Koine Greek from ('the same') plus ('word' or 'idea'), and transmitted through 3rd-century Latin and French . It first appeared in English in the 16th century. The use of the term logical tautology was introduced in English by Wittgenstein in 1919, perhaps following Auguste Comte's usage in 1835. Examples * "Convicted felon", a common E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (other)
Tautology may refer to: *Tautology (language), a redundant statement in literature and rhetoric *Tautology (logic), in formal logic, a statement that is true in every possible interpretation *Tautology (rule of inference), a rule of replacement for logical expressions See also *Pleonasm *Redundancy (other) *Tautological (other) *Tautonym A tautonym is a scientific name of a species in which both parts of the name have the same spelling, such as '' Rattus rattus''. The first part of the name is the name of the genus and the second part is referred to as the ''specific epithet'' i ..., a scientific name of a species in which both parts of the name have the same spelling {{Disambiguation hu:Tautológia#Nyelvtudományi és irodalmi tautológia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
