sets

Consider the following statement: For any natural number n, there is a natural number s such that s = n × n. This is clearly true; it just asserts that every number has a square. The meaning of the assertion in which the quantifiers are turned around is quite different: There is a natural number s such that for any natural number n, s = n × n. This is clearly false; it asserts that there is a single natural number s t ...

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Quantification, Quantification - Quantification in natural language, Quantification - Need for quantifiers in mathematical assertions, Quantification - Nesting of quantifiers, Quantification - Range of quantification, Quantification - Notation for quantifiers, Quantification - Formal semantics, Quantification - Paucal multal and other degree quantifiers, Quantification - History of formalisation

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We will begin by discussing quantification in informal mathematical discourse. Consider the following statement 1·2 = 1 + 1, and 2·2 = 2 + 2, and 3 · 2 = 3 + 3, ...., and n · 2 = n + n, etc. This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since we expect syntax rules to generate finite objects. Putting aside this objection, also note that in this example we were lucky in that there is a proc ...

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Quantification, Quantification - Quantification in natural language, Quantification - Need for quantifiers in mathematical assertions, Quantification - Nesting of quantifiers, Quantification - Range of quantification, Quantification - Notation for quantifiers, Quantification - Formal semantics, Quantification - Paucal multal and other degree quantifiers, Quantification - History of formalisation

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So far we have only considered universal, existential and uniqueness quantification as used in mathematics. None of this applies to a quantification such as There were many dancers out on the dance floor this evening. Though we will not consider semantics of natural language in this article, we will attempt to provide a semantics for assertions in a formal language of the type There are many integ ...

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Quantification, Quantification - Quantification in natural language, Quantification - Need for quantifiers in mathematical assertions, Quantification - Nesting of quantifiers, Quantification - Range of quantification, Quantification - Notation for quantifiers, Quantification - Formal semantics, Quantification - Paucal multal and other degree quantifiers, Quantification - History of formalisation

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A common geeky joke (for example [1]) is the following "definition" of recursion. Recursion See "Recursion". This is a parody on references in dictionaries, which in some careless cases may lead to circular definitions. Every joke has an element of wisdom, and also an element of misunderstanding. This one is also the second-shortest possible example of an erroneous recursive definition of an object, the error being the absence of the termination condition (or lack of the initial state, i ...

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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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A typical sequent might be: This claims that either α or β can be derived from φ and ψ. ...

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Sequent, Sequent - Explanation, Sequent - Intuitive meaning, Sequent - Example, Sequent - Property, Sequent - Rules, Sequent - Variations, Sequent - History

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A sequent has the form where both Γ and Σ are sequences of logical formulae (i.e., both the number and the order of the occurring formulae matter). The symbol is usually referred to as turnstile or tee and is often read, suggestively, as "yields" or "proves". It is not a symbol in the language, rather it is a symbol in the metalanguage used to discuss proofs. In a sequent, Γ is called the antecedent and Σ is s ...

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Sequent, Sequent - Explanation, Sequent - Intuitive meaning, Sequent - Example, Sequent - Property, Sequent - Rules, Sequent - Variations, Sequent - History

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Self-balancing binary search trees can be used in a natural way to construct associative arrays; key-value pairs are simply inserted with an ordering based on the key alone. In this capacity, self-balancing BSTs have a number of advantages and disadvantages over their main competitor, hash tables. Lookup is somewhat complicated in the case where the same key can be used multiple times. Many algorithms can exploit self-balancing BSTs to achieve good worst-case bounds with very little effort. For example, if binary tree sort is done wit ...

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Self-balancing binary search tree, Self-balancing binary search tree - Overview, Self-balancing binary search tree - Implementations, Self-balancing binary search tree - Applications

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According to Beyer and Zardecki (2004), the earliest known paper about the ham sandwich theorem, specifically the d = 3 case of bisecting three solids with a plane, is by Steinhaus and others (1938). Beyer and Zardecki's paper includes a translation of the 1938 paper. It attributes the posing of the problem to Hugo Steinhaus, and attributes Stefan Banach as the first to solve the problem, by a reduction to the Borsuk-Ulam theorem. The paper poses the problem in two ways: first, formally, as "Is it always possible to bisect t ...

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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 ham sandwich theorem takes its name from the case when n = 3 and the three objects of any shape are a chunk of ham and two chunks of bread — notionally, a sandwich — which can then each be bisected with a single cut (i.e., a plane). In two dimensions, the theorem is known as the pancake theorem of having to cut two infinitesimally thin pancakes on a plate each in ...

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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 category C is called cartesian closed iff it satisfies the following three properties: it has a terminal object any two objects X and Y of C have a product X×Y in C any two objects Y and Z of C have an exponential ZY in C For the first two conditions above, it is the same to require that any finite (possibly empty) family of objects of C admit a product in C, because of t ...

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

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An earlier proof by Cantor relied, in effect, on the axiom of choice by inferring the result as a corollary of the well-ordering theorem. The argument given above shows that the result can be proved without using the axiom of choice. The theorem is also known as the Schroeder-Bernstein theorem, but the trend has been to add Cantor's name, thus crediting him for the original version. It is also called the Cantor-Bernstein theorem. ...

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

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A bracket may be an inverted "L" shape, such as is usually used to hold up a shelf, or a rafter extension and its diagonal brace supporting an overhanging roof over a gable. Decorative brackets used in furniture and mantlepieces are called corbels. ...

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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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In addition to the use of parentheses to specify the order of operations, both parentheses and square brackets can also be used to denote an interval. The notation [a, c) is used to indicate a sequence from a to c that is inclusive of a but exclusive of c. That is, [5, 12) would be the set of all real numbers between 5 and 12, including 5 but except 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In Europe, ...

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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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In cartesian closed categories, a "function of two variables" (a morphism f:X×Y → Z) can always be represented as a "function of one variable" (the morphism λf:X → ZY). In computer science applications, this is known as currying; it has led to the realization that simply-typed lambda calculus can be interpreted in any cartesian closed category. Certain cartesian closed categories, the topoi, have been proposed as a general s ...

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

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Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Stanislaw Ulam, that comparable ideas were current in the late 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy N ...

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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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A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel. This may also be denoted DiffKer(f,g), Ker(f,g), or Ker(f - g). The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f - g. Conversely, the kernel of a single function f can be reconstructed as the difference kernel Eq(f,0), w ...

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Equaliser, Equaliser - Definitions, Equaliser - Difference kernels, Equaliser - In category theory

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Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically: The equaliser may be denoted Eq(f,g) or a variation on that theme (such as with lowercase letters "eq"). In informal contexts, the ...

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Equaliser, Equaliser - Definitions, Equaliser - Difference kernels, Equaliser - In category theory

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An interesting complete Ring cycle was begun in 2004. The production is notable for its use of contemporary minimalist sets and costumes. Many of the scenes look like rooms from Ikea and indeed the production is sponsored by the MFI furniture company. Some critics have described Phyllida Lloyd's Cycle as superior to that at the Royal Opera House in almost every way, although many others thought it was muddled and that its "relentlessly trivialising" approach served only to belittle W ...

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English National Opera, English National Opera - History, English National Opera - Repertoire, English National Opera - Home, English National Opera - Education, English National Opera - Ring Cycle, English National Opera - Gilbert and Sullivan Opera

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The Coliseum Theatre, near Trafalgar Square, is one of London's largest and best equipped theatres, opening in 1904. Built by Frank Matcham, a famous theatrical architect who designed two famous London theatres: the London Palladium and the London Coliseum. It underwent extensive renovations between 2000 and 2004 and has the widest proscenium arch in London as well as being one of the earliest to have electric lighting. It was built with a revolving stage although this was rarely used. It was the creation of the most powerful theatre ...

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English National Opera, English National Opera - History, English National Opera - Repertoire, English National Opera - Home, English National Opera - Education, English National Opera - Ring Cycle, English National Opera - Gilbert and Sullivan Opera

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Also, in many computer languages: "(" and ")" are used to contain the arguments to functions: substring($val,10,1). Parentheses are so ubiquitous in the Lisp programming language that the name is said to be an acronym for "Lots of Irritating Superfluous Parentheses". They may also be used to indicate the start and end of lists. "[" and "]" are used to define the number of elements in an array, or reference one of those elements: $queue[3]. In MediaWiki's syntax, a double square-bracket set is u ...

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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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Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression: Unary relation or property: L(u) Binary relation: L(u, v) or u L v Ternary relation: L(u, v, w) Quaternary relation: L(u, v, w, x) Relations with more than four terms are usually referred to a ...

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Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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Informally, a restriction of a function f is the result of trimming its graph to a smaller domain. More precisely, if f is a function from a X to Y, and S is any subset of X, the restriction of f to S is the function f|S from S to Y such that f|S(s) = f(s) for all s in S. The restriction f|S can also be expressed as the composition f incS,X, where incSSee also:

Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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