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Tree (set Theory)
In set theory, a tree is a partially ordered set (''T'', <) such that for each ''t'' ∈ ''T'', the set is by the relation <. Frequently trees are assumed to have only one root (i.e. ), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees. Definition A tree is a (poset) (''T'', <) such that for each ''t'' ∈ ''T'', the set is by the ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kőnig's Lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory. This theorem also has important roles in constructive mathematics and proof theory. Statement of the lemma Let G be a connected, locally finite, infinite graph. This means that every two vertices can be connected by a finite path, the graph has infinitely many vertices, and each vertex is adjacent to only finitely many other vertices. Then G contains a ray: a simple path (a path with no repeated vertices) that starts at one vertex and continues from it through infinitely many vertices. A useful special case of the lemma is that every infinite tree contains either a vertex of infinite degree or an in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix Order
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous progress and continuous branching. Natural prefix orders often occur when considering dynamical systems as a set of functions from ''time'' (a totally-ordered set) to some phase space. In this case, the elements of the set are usually referred to as ''executions'' of the system. The name ''prefix order'' stems from the prefix order on words, which is a special kind of substring relation and, because of its discrete character, a tree. Formal definition A prefix order is a binary relation "≤" over a set ''P'' which is antisymmetric, transitive, reflexive, and downward total, i.e., for all ''a'', ''b'', and ''c'' in ''P'', we have that: *''a ≤ a'' (reflexivity); *if ''a ≤ b'' and ''b ≤ a'' then ''a'' = ''b'' (antisymmetry); *if ''a ≤ b'' and ''b ≤ c'' then ''a ≤ c'' (transitivity); *if ''a ≤ c'' and ''b ≤ c'' t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Graph
GraphOn GO-Global is a multi-user remote access application for Windows. Overview GO-Global allows multiple users to concurrently run Microsoft Windows applications installed on a Windows server or server farm from network-connected locations and devices. GO-Global redirects the user interface of Windows applications running on the Windows server to the display or browser on the user's device. Applications look and feel like they are running on the user's device. Supported end-user devices include Windows, Mac and Linux personal computers, iOS and Android mobile devices, and Chromebooks. GO-Global is used by Independent Software Vendors (ISVs), Hosted Service Providers (HSPs), and Managed Service Providers (MSPs) to publish Windows applications without modification of existing code for the use of local and remote users. Architecture GO-Global enables multi-user remote access to Windows applications without the use of Microsoft Remote Desktop Services (RDS) or th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (descriptive Set Theory)
In descriptive set theory, a tree on a set X is a collection of finite sequences of elements of X such that every prefix of a sequence in the collection also belongs to the collection. Definitions Trees The collection of all finite sequences of elements of a set X is denoted X^. With this notation, a tree is a nonempty subset T of X^, such that if \langle x_0,x_1,\ldots,x_\rangle is a sequence of length n in T, and if 0\le m and called the ''body'' of the tree T. A tree that has no branches is called ''wellfounded''; a tree with at least one branch is ''illfounded''. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded. Terminal nodes A finite sequence that belongs to a tree T is called a terminal node if it is not a prefix of a longer sequence in T. Equivalently, \langle x_0,x_1,\ldots,x_\rangle \in T is terminal if there is no element x of X such that that \langle x_0,x_1,\ldots,x_,x\rangle \in T. A tree that does not h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laver Tree
In mathematics, forcing is a method of constructing new models ''M'' 'G''of set theory by adding a generic subset ''G'' of a poset ''P'' to a model ''M''. The poset ''P'' used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable ''P''. This article lists some of the posets ''P'' that have been used in this construction. Notation *''P'' is a poset with order < *''V'' is the universe of all sets *''M'' is a countable transitive model of set theory *''G'' is a generic subset of ''P'' over ''M''. Definitions *''P'' satisfies the if every antichain in ''P'' is at most countable. This implies that ''V'' and ''V'' 'G''have the same car ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurepa Tree
In set theory, a Kurepa tree is a tree (''T'', <) of height ω1, each of whose levels is at most countable, and has at least ℵ2 many branches. This concept was introduced by . The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's |
Cantor Tree
In mathematical set theory, the Cantor tree is either the full binary tree of height ω + 1, or a topological space related to this by joining its points with intervals, that was introduced by Robert Lee Moore in the late 1920s as an example of a non-metrizable Moore space . References * * *{{Citation , last1=Steen , first1=Lynn Arthur , author1-link=Lynn Arthur Steen , last2=Seebach , first2=J. Arthur Jr. , author2-link=J. Arthur Seebach, Jr. , title=Counterexamples in Topology , origyear=1978 , publisher=Springer-Verlag , location=Berlin, New York , edition=Dover Dover () is a town and major ferry port in Kent, South East England. It faces France across the Strait of Dover, the narrowest part of the English Channel at from Cap Gris Nez in France. It lies south-east of Canterbury and east of Maidstone ... reprint of 1978 , isbn=978-0-486-68735-3 , mr=507446 , year=1995 Trees (set theory) Topological spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (totally ordered subsets) into which the set can be partitioned. Dually, the height of the partially ordered set (the length of its longest chain) equals by Mirsky's theorem the minimum number of antichains into which the set can be partitioned. The family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Uncountable Ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of \omega_1 are the countable ordinals (including finite ordinals), of which there are uncountably many. Like any ordinal number (in von Neumann's approach), \omega_1 is a well-ordered set, with set membership serving as the order relation. \omega_1 is a limit ordinal, i.e. there is no ordinal \alpha such that \omega_1 = \alpha+1. The cardinality of the set \omega_1 is the first uncountable cardinal number, \aleph_1 (aleph-one). The ordinal \omega_1 is thus the initial ordinal of \aleph_1. Under the continuum hypothesis, the cardinality of \omega_1 is \beth_1, the same as that of \mathbb—the set of real numbers. In most constructions, \omega_1 and \aleph_1 are considered equal as sets. To generalize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Ordered Set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Cardinal
Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, see List of animal names * Singular matrix, a matrix that is not invertible * Singular measure, a measure or probability distribution whose support has zero Lebesgue (or other) measure * Singular cardinal, an infinite cardinal number that is not a regular cardinal * The property of a ''singularity'' or ''singular point'' in various meanings; see Singularity (other) * Singular (band), a Thai jazz pop duo *'' Singular: Act I'', a 2018 studio album by Sabrina Carpenter *'' Singular: Act II'', a 2019 studio album by Sabrina Carpenter See also * Singulair, Merck trademark for the drug Montelukast * Cingular Wireless AT&T Mobility LLC, also known as AT&T Wireless and marketed as simply AT&T, is an American telecommunications company ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
