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Choice Function
Let ''X'' be a set of sets none of which are empty. Then a choice function (selector, selection) on ''X'' is a mathematical function ''f'' that is defined on ''X'' such that ''f'' is a mapping that assigns each element of ''X'' to one of its elements. An example Let ''X'' = . Then the function ''f'' defined by ''f''() = 7, ''f''() = 9 and ''f''() = 2 is a choice function on ''X''. History and importance Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem, which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, some sets can still be shown to have a choice function. *If X is a finite set of nonempty sets, then one can construct a choice function for X by picking on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Of Sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I, known as the index set, to F, in which case the sets of the family are indexed by members of I. In some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class. A finite family of subsets of a finite set S is also called a ''hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the power set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Inclusion
In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form :\frac(t)\in F(t,x(t)), where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point in \R^d. Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, Moreau's sweeping process, linear and nonlinear complementarity dynamical systems, discontinuous ordinary differential equations, switching dynamical systems, and fuzzy set arithmetic. For example, the basic rule for Coulomb friction is that the friction force has magnitude ''μN'' in the direction opposite to the direction of slip, where ''N'' is the normal force and ''μ'' is a constant (the friction coefficient). However, if the slip is zero, the friction force can be ''any'' force in the correct plane with magnitude smaller than or equal to ''μN''. Thus, writing the friction force as a function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemicontinuity
In mathematics, upper hemicontinuity and lower hemicontinuity are extensions of the notions of semicontinuity, upper and lower semicontinuity of single-valued function (mathematics), functions to Set-valued function, set-valued functions. A set-valued function that is both upper and lower hemicontinuous is said to be continuous in an analogy to the property of the same name for single-valued functions. To explain both notions, consider a sequence a of points in a domain, and a sequence b of points in the range. We say that ''b corresponds to a'' if each point in b is contained in the image of the corresponding point in a. * Upper hemicontinuity requires that, for any convergent sequence a in a domain, and for any convergent sequence b that corresponds to a, the image of the limit of a contains the limit of b. * Lower hemicontinuity requires that, for any convergent sequence a in a domain, and for any point ''x'' in the image of the limit of a, there exists a sequence b that corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Paradox
The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (the surface of a 3-dimensional ball in ). It states that if a certain countable subset is removed from , then the remainder can be divided into three disjoint subsets and such that and are all congruent. In particular, it follows that on S^2 there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of is simultaneously 1/3, 1/2, and 2/3 of the non-zero measure of the whole sphere). The paradox was published in ''Mathematische Annalen'' in 1914 and also in Hausdorff's book, '' Grundzüge der Mengenlehre'', the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas. The proof of this paradox relies on the axiom of choice. This paradox shows that there is no finitely additive measure on a sphere defined on ''all'' subsets which is equal on congr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Dependent Choice
In 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 ar ..., the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article in reverse mathematics that explores which set-theoretic axioms are needed to develop analysis."The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame." The axiom of dependent choice is stated on p. 86. Formal statement A homogeneous relation R on X is called a total relation if for every a \in X, there exists some b \in X such that a\,R~b is true. The axiom of dependent choice can be stated as follows: For every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Countable Choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain \mathbb (where \mathbb denotes the set of natural numbers) such that A(n) is a non-empty set for every n\in\mathbb, there exists a function f with domain \mathbb such that f(n)\in A(n) for every n\in\mathbb. Applications ACω is particularly useful for the development of mathematical analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point x of a set S\subseteq\mathbb is the limit of some sequence of elements of S\setminus\, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω. The ability to perform analysis using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Global Choice
In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set. Statement The axiom of global choice states that there is a global choice function τ, meaning a function such that for every non-empty set ''z'', τ(''z'') is an element of ''z''. The axiom of global choice cannot be stated directly in the language of Zermelo–Fraenkel set theory (ZF) with the axiom of choice (AC), known as ZFC, as the choice function τ is a proper class and in ZFC one cannot quantify over classes. It can be stated by adding a new function symbol τ to the language of ZFC, with the property that τ is a global choice function. This is a conservative extension of ZFC: every provable statement of this extended theory that can be stated in the language of ZFC is already provable in ZFC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon Calculus
In logic, Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...'s epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language. The ''epsilon operator'' and ''epsilon substitution method'' are typically applied to a first-order predicate calculus, followed by a demonstration of consistency. The epsilon-extended calculus is further extended and generalized to cover those mathematical objects, classes, and categories for which there is a desire to show consistency, building on previously-shown consistency at earlier levels. Epsilon operator Hilbert notation For any formal language ''L'', extend ''L'' by addin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in mathematical analysis, analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the ''Éléments de mathématique'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups, and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Cartan co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selection Theorem
In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics. Preliminaries Given two sets ''X'' and ''Y'', let ''F'' be a set-valued function from ''X'' and ''Y''. Equivalently, F:X\rightarrow\mathcal(Y) is a function from ''X'' to the power set of ''Y''. A function f: X \rightarrow Y is said to be a selection of ''F'' if : \forall x \in X: \,\,\, f(x) \in F(x) \,. In other words, given an input ''x'' for which the original function ''F'' returns multiple values, the new function ''f'' returns a single value. This is a special case of a choice function. The axiom of choice implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Economics
Mathematical economics is the application of Mathematics, mathematical methods to represent theories and analyze problems in economics. Often, these Applied mathematics#Economics, applied methods are beyond simple geometry, and may include differential and integral calculus, Recurrence relation, difference and differential equations, Matrix (mathematics), matrix algebra, mathematical programming, or other Computational economics, computational methods.TOC. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity. Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
