sets

There are three axioms of IST to add to the established ZFC set theoretic axioms (note that use of the ZFC axiom schemas is restricted: the axiom schemas of separation and replacement can only be used with classical formulas, just as in ZFC proper) - conveniently one for each letter in the name: Idealisation, Standardisation, and Transfer. All the principles described above can b ...

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Internal set theory, Internal set theory - Intuitive justification, Internal set theory - Principles of the standard predicate, Internal set theory - Formal axioms for IST, Internal set theory - I : Idealisation, Internal set theory - S : Standardisation, Internal set theory - T : Transfer, Internal set theory - Formal justification for the axioms

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Java programming language - Early history. The Java platform and language began as an internal project at Sun Microsystems in December of 1990. Engineer Patrick Naughton had become increasingly frustrated with the state of Sun's C++ and C APIs (application programming interfaces) and tools. While considering moving to NeXT, Naughton was offered a chance to work on new techno ...

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Java programming language, Java programming language - History, Java programming language - Early history, Java programming language - Java meets the Internet, Java programming language - Recent history, Java programming language - Version history, Java programming language - Philosophy, Java programming language - Object orientation, Java programming language - Platform independence, Java programming language - Automatic garbage collection, Java programming language - Criticism, Java programming language - Language, Java programming language - Library, Java programming language - Performance, Java programming language - Syntax, Java programming language - Hello world, Java programming language - Resources, Java programming language - Java Runtime Environment, Java programming language - APIs, Java programming language - Extensions and related architectures, Java programming language - Lists, Java programming language - Notes

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Inverse limit - Algebraic objects. We start with the definition of an inverse system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (Ai)i∈I be a family of groups and suppose we have a family of homomorphisms fij : Aj → Ai for all i ≤ j (note the order) with the following properties:< ...

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Inverse limit, Inverse limit - Formal definition, Inverse limit - Algebraic objects, Inverse limit - General definition, Inverse limit - Examples, Inverse limit - Related concepts and generalizations

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A completely analogous definition holds for the case of linear representations of G. Specifically, if X and Y are two linear representations of G then a linear map f : X → Y is called an intertwiner of the representations if it commutes with the action of G. Alternatively, an intertwiner for representations of G over a field K is the same thing as a module homomorphism of K[G]-modules, where K< ...

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Equivariant, Equivariant - Intertwiners, Equivariant - Categorical description

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The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider the following example. The class Grp of groups consists of all objects having a "group structure". More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms t ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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Recursion in computer programming defines a function in terms of itself. One example application of recursion is in parsers for programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program. One basic form of recursive computer program is to define one or a few base cases, and then define rules to break down other cases into the base case. This is analytic, and is the mo ...

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Recursion, Recursion - Recursion in mathematics, Recursion - Functional Recursion, Recursion - Recursive Proofs, Recursion - Recursion in computing, Recursion - Recursion in language, Recursion - Recurrence relations or algorithms, Recursion - Recursively defined sets, Recursion - Example: the natural numbers, Recursion - Example: The set of true reachable propositions, Recursion - Recursively defined functions, Recursion - Recursive algorithms, Recursion - The recursion theorem, Recursion - Proof of uniqueness, Recursion - Proof of existence, Recursion - Recursion in plain English, Recursion - Recursive humour

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Direct limit - Algebraic objects. In this section we will understand objects to be sets with a given algebraic structure such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. We will also understand homomorphisms in the corresponding setting (group homomorphisms, etc.). We start with the definition of a direct system of objects and homomorphisms. Let (I, ≤) be a directed poset. Let {Ai | i ∈ I} be a fam ...

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Direct limit, Direct limit - Formal definition, Direct limit - Algebraic objects, Direct limit - General definition, Direct limit - Examples, Direct limit - Related constructions and generalizations

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Definitions in graph theory vary in the literature. Here are the conventions used in this encyclopedia. Graph mathematics - Undirected graph. An undirected graph or graph G is an ordered pair G:=(V, E) with V, a set of vertices or nodes, E, a set of unordered pairs of distinct vertices, called edges or lines. The vertices belonging to an edge are called the ends, endpoint ...

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Graph mathematics, Graph mathematics - Definitions, Graph mathematics - Undirected graph, Graph mathematics - Directed graph, Graph mathematics - Mixed graph, Graph mathematics - Variations in the definitions, Graph mathematics - Further definitions, Graph mathematics - Examples, Graph mathematics - Important graphs, Graph mathematics - Operations on graphs, Graph mathematics - Unary operations, Graph mathematics - Binary operations, Graph mathematics - Generalizations

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Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (that is, depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as op ...

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Sheaf mathematics, Sheaf mathematics - Introduction, Sheaf mathematics - The formal definition, Sheaf mathematics - Definition of a presheaf, Sheaf mathematics - The gluing axiom, Sheaf mathematics - Examples, Sheaf mathematics - Morphisms of sheaves, Sheaf mathematics - Stalks of a sheaf at a point and germs of functions, Sheaf mathematics - The étale space of a sheaf, Sheaf mathematics - Generalizations, Sheaf mathematics - History

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The ham sandwich theorem can be proved as follows using the Borsuk-Ulam theorem. This proof follows the one described by Steinhaus and others (1938), attributed there to Stefan Banach, for the n = 3 case. Let A1, A2, ..., An denote the n objects that we wish to simultaneously bisect. Let S be the unit (n − 1)-sphere in , centered at the origin. For each point p on the surface of the sphere S, we can define a con ...

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Ham sandwich theorem, Ham sandwich theorem - Naming, Ham sandwich theorem - History, Ham sandwich theorem - Reduction to the Borsuk-Ulam theorem, Ham sandwich theorem - Measure theoretic statement, Ham sandwich theorem - Discrete and computational geometry versions

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The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois ...

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Areas of mathematics, Areas of mathematics - Foundations / general, Areas of mathematics - Algebra, Areas of mathematics - Analysis, Areas of mathematics - Geometry, Areas of mathematics - Applied mathematics, Areas of mathematics - Probability and statistics, Areas of mathematics - Computational sciences, Areas of mathematics - Physical sciences, Areas of mathematics - Non-physical sciences

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Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as homomorphisms. For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). Note that f(x) = 3x is a homomorphism, since f(a + b< ...

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Homomorphism, Homomorphism - Informal discussion, Homomorphism - Formal definition, Homomorphism - Types of homomorphisms, Homomorphism - Kernel of a homomorphism

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Let C be a locally small category (i.e. a category for which Hom-classes are actually sets and not proper classes). For all objects A in C we define a functor Hom(A,–) : C → Set to the category of sets as follows: Hom(A,–) maps each object X in C to the set of morphisms, Hom(A, X) Hom(A,–) maps each morphism f : X → Y to the function Hom(A, f) : Hom(A, XSee also:

Hom functor, Hom functor - Formal definition, Hom functor - Yoneda's lemma

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Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a musician of German descent, Maria Anna Böhm. In 1856, the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867. In 1890, he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and ...

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Georg Cantor, Georg Cantor - Biography

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In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Introducing new function symbols from old function symbols is easy; given function symbols F and G, there is a new function symbol F o G, the composition of F and G, satisfying (F o G)(X) = F(G(X)), for all X. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of F matches the codomain ...

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Functional predicate, Functional predicate - Introducing new function symbols, Functional predicate - Doing without functional predicates

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The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler, in the 18th century. But the intuitive idea of a function as any rule or procedure that assigns an output to each given input proved to be naive. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. The concept of a function was not put on a rigorous basis u ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

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Philosophy of language - Composition and parts. A major question in the field - perhaps the single most important question for formalist and structuralist thinkers - is, "how does the meaning of a sentence emerge out of its parts?" Much about composition of sentences is addressed in the work of linguistics of syntax. More logic-oriented semantics tend to look towards the principle of compositionality in order to explain the relationship between meaningful parts and whole sentences. The princ ...

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Philosophy of language, Philosophy of language - Overview, Philosophy of language - History, Philosophy of language - Major problems and sub-fields, Philosophy of language - Composition and parts, Philosophy of language - The nature of meaning, Philosophy of language - Language and the world, Philosophy of language - Mind and language, Philosophy of language - Social interaction and language, Philosophy of language - Miscellaneous, Philosophy of language - Important theorists, Philosophy of language - Important topics and terms, Philosophy of language - References

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Brackets are punctuation marks, used in pairs to set apart or interject text within other text. Types of brackets include parentheses ( ) (the singular is parenthesis), box brackets or square brackets [ ], curly brackets or braces { }, and angle brackets 〈 〉. All these forms may be used according to typographical conventions that may vary from publication to publication and may vary even more from language to language. Some typical uses in English texts follow. Brack ...

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Bracket, Bracket - In writing, Bracket - Types of brackets, Bracket - In computing, Bracket - Layout rules, Bracket - In mathematics, Bracket - In sports, Bracket - In mechanics and structures, Bracket - Reference

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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The definition of h can be visualized with the following diagram. Displayed are parts of the (disjoint) sets A and B together with parts of the mappings f and g. If the set A ∪ B, together with the two maps, is interpreted as a directed graph, then this bipartite graph has several connected components. These can be divided into four types: paths extending infinitely to both directions, finite cycles of even length, infinite paths starting in the set A, and infini ...

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Cantor–Bernstein–Schroeder theorem, Cantor–Bernstein–Schroeder theorem - Visualization, Cantor–Bernstein–Schroeder theorem - Original proof

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Examples of cartesian closed categories include: The category Set of all sets, with functions as morphisms, is cartesian closed. The product X×Y is the cartesian product of X and Y, and ZY is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X×Y → Z is naturally identified with the function g : X → ZY defined by gSee also:

Cartesian closed category, Cartesian closed category - Definition, Cartesian closed category - Examples, Cartesian closed category - Applications, Cartesian closed category - Equational theory

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Tennis - The court. Main article: Tennis court Tennis is played on a rectangular flat surface, usually of grass, clay, or concrete (hard court). The court is 78 feet (23.77 m) long, and its width is 27 feet (8.23 m) for singles matches and 36 feet (10.97 m) for doubles matches. Additional clear space around the court is required in order for players to reach overrun balls. A net is stretched across the full width of the court, parallel with the baselines, dividing it into two equal ends. The net is 3 feet 6 inches (1.07 m) high at the posts ...

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Tennis, Tennis - Manner of play, Tennis - The court, Tennis - Play of a single point, Tennis - Scoring, Tennis - Officials, Tennis - Miscellaneous, Tennis - Shots, Tennis - Serve, Tennis - Forehand, Tennis - Backhand, Tennis - Other shots, Tennis - Tournaments, Tennis - History, Tennis - Great players

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