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Sigma-ring
In mathematics, a nonempty collection of sets is called a -ring (pronounced ''sigma-ring'') if it is closed under countable union and relative complementation. Formal definition Let \mathcal be a nonempty collection of sets. Then \mathcal is a -ring if: # Closed under countable unions: \bigcup_^ A_ \in \mathcal if A_ \in \mathcal for all n \in \N # Closed under relative complementation: A \setminus B \in \mathcal if A, B \in \mathcal Properties These two properties imply: \bigcap_^ A_n \in \mathcal whenever A_1, A_2, \ldots are elements of \mathcal. This is because \bigcap_^\infty A_n = A_1 \setminus \bigcup_^\left(A_1 \setminus A_n\right). Every -ring is a δ-ring but there exist δ-rings that are not -rings. Similar concepts If the first property is weakened to closure under finite union (that is, A \cup B \in \mathcal whenever A, B \in \mathcal) but not countable union, then \mathcal is a ring but not a -ring. Uses -rings can be used instead of -fields ... [...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] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set (mathematics), set is closed under an Operation (mathematics), operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set (mathematics), set equipped with one or several methods for producing elements of from other elements of .Operation (mathematics), Operations and (partial function, partial) multivar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of Set (mathematics), sets is the set of all element (set theory), elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of Zero, zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the List of mathematical symbols, table of mathematical symbols. Binary union The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, : A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by A^c (or ), is the set of Element (mathematics), elements not in . When all elements in the Universe (set theory), universe, i.e. all elements under consideration, are considered to be Element (mathematics), members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family 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 pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta-ring
In mathematics, a non-empty collection of sets \mathcal is called a -ring (pronounced "") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durchschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a -ring which is closed under countable unions. Definition A family of sets \mathcal is called a -ring if it has all of the following properties: #Closed under finite unions: A \cup B \in \mathcal for all A, B \in \mathcal, #Closed under relative complementation: A - B \in \mathcal for all A, B \in \mathcal, and #Closed under countable intersections: \bigcap_^ A_n \in \mathcal if A_n \in \mathcal for all n \in \N. If only the first two properties are satisfied, then \mathcal is a ring of sets but not a -ring. Every -ring is a -ring, but not every -ring is a -ring. -rings can be used instead of σ-algebras in the developmen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Sets
(The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a Japanese horror media franchise based on the novel series by Koji Suzuki ** ''Ring'' (film), or ''The Ring'', a 1998 Japanese horror film by Hideo Nakata *** ''The Ring'' (2002 film), an American horror film, remake of the 1998 Japanese film ** ''Ring'' (1995 film), a TV film ** ''Rings'' (2005 film), a short film by Jonathan Liebesman ** ''Rings'' (2017 film), an American horror film * "Ring", a season 3 episode of ''Servant'' (TV series) Gaming * ''Ring'' (video game), 1998 * Rings (''Sonic the Hedgehog''), a collectible in ''Sonic the Hedgehog'' games Literature * ''Ring'' (Baxter novel), a 1994 science fiction novel * ''Ring'' (Alexis novel), a 2021 Canadian novel by André Alexis * ''Ring'' (novel series), a Japanese nov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universe (mathematics)
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often class (set theory), classes that contain (as element (set theory), elements) all sets for which one hopes to Mathematical proof, prove a particular theorem. These classes can serve as Inner model, inner models for various axiomatic systems such as Zermelo–Fraenkel set theory, ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the List of mathematical jargon#canonical, canonical motivating example of a category is Category of sets, Set, the category of all sets, which cannot be formalized in a set theory without some notion of a universe. In type theory, a universe is a type whose elements are types. In a specific cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
