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Field Of Sets
In mathematics, a field of sets is a mathematical structure consisting of a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X called an algebra over X that contains the empty set as an element, and is closed under the operations of taking complements in X, finite unions, and finite intersections. Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over X" is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory. Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets. Definitions A field of sets is a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X, called an algebra over X, that has the following properties: : X \setminus F \in \mathcal \text F \in \mathcal. as an element: \varnothing ... [...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] |
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion of ''size''. In formal terms, a σ-algebra (also σ-field, where the σ comes from the German "Summe", meaning "sum") on a set ''X'' is a nonempty collection Σ of subsets of ''X'' closed under complement, countable unions, and countable intersections. The ordered pair (X, \Sigma) is called a measurable space. The set ''X'' is understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die ), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to cap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, partition of the rational numbers into two Set (mathematics) , sets ''A'' and ''B'', such that each element of ''A'' is less than every element of ''B'', and ''A'' contains no greatest element. The set ''B'' may or may not have a smallest element among the rationals. If ''B'' has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between ''A'' and ''B''. In other words, ''A'' contains every rational number less than the cut, and ''B'' contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filter (set Theory)
In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A\subset B\subset X and A\in \mathcal, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book '' Topologie Générale'' as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (order Theory)
In mathematics, mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. Definitions A subset of a partially ordered set (P, \leq) is an ideal, if the following conditions hold: # is non-empty, # for every ''x'' in and ''y'' in ''P'', implies that ''y'' is in ( is a lower set), # for every ''x'', ''y'' in , there is some element ''z'' in , such that and ( is a directed set). While this is the most general way to define an ideal for arbitrary posets, it was originally defined for Lattice (order), lattices only. In this case, the following equivalent definition can be given: a subset of a lattice (P, \leq) is an ideal if and only if it is a lower set that is closed under finite join and me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general (although one usually is also interested in the actual difference of two numbers, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Representation Theorem For Boolean Algebras
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. Stone spaces Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a basis consisting of all sets of the form \, where ''b'' is an element of ''B''. These sets are also closed and so are clopen (both closed and open). This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter (set Theory)
In the mathematical field of set theory, an ultrafilter on a set (mathematics), set X is a ''maximal filter'' on the set X. In other words, it is a collection of subsets of X that satisfies the definition of a filter (set theory), filter on X and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of X that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set X can also be characterized as a filter on X with the property that for every subset A of X either A or its complement X\setminus A belongs to the ultrafilter. Ultrafilters on sets are an important special instance of Ultrafilter, ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\subseteq. This article deals specifically with ultrafilters on a set and does not cover the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic (order Theory)
Atomic may refer to: * Of or relating to the atom, the smallest particle of a chemical element that retains its chemical properties * Atomic physics, the study of the atom * Atomic Age, also known as the "Atomic Era" * Atomic scale, distances comparable to the dimensions of an atom * Atom (order theory), in mathematics * Atomic (coffee machine), a 1950s stovetop coffee machine * Atomic (cocktail), a champagne cocktail * Atomic (magazine), ''Atomic'' (magazine), an Australian computing and technology magazine * Atomic Skis, an Austrian ski producer Music * Atomic (band), a Norwegian jazz quintet * Atomic (Lit album), ''Atomic'' (Lit album), 2001 * Atomic (Mogwai album), ''Atomic'' (Mogwai album), 2016 * ''Atomic'', an album by Rockets (band), Rockets, 1982 * Atomic (EP), ''Atomic'' (EP), by , 2013 * Atomic (song), "Atomic" (song), by Blondie, 1979 * "Atomic", a song by Tiger Army from ''Tiger Army III: Ghost Tigers Rise'' See also * * * Atom (other) * Atomicity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom (order Theory)
In the mathematical field of order theory, an element ''a'' of a partially ordered set with least element 0 is an atom if 0 < ''a'' and there is no ''x'' such that 0 < ''x'' < ''a''. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0. Atomic orderings Let <: denote the covering relation in a partially ordered set. A partially ordered set with a least element 0 is atomic if every element ''b'' > 0 has an atom ''a'' below it, that is, there is some ''a'' such that ''b'' ≥ ''a'' :> ''0''. Every finite partially ordered set with 0 i ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
