sets

The most general notion is the union of an arbitrary collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of M if and only if for at least one element A of M, x is an element of A. In symbols: That this union of M is a set no matter how large a set M itself might be, is the co ...

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Union set theory, Union set theory - Basic definition, Union set theory - Finite unions, Union set theory - Algebraic properties, Union set theory - Infinite unions

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Basic notions in the relational model are relation names and attribute names. We will represent these as strings such as "Person" and "name" and we will usually use the variables r, s, t, ... and a, b, c to range over them. Another basic notion is the set of atomic values that contains values such as numbers and strings. Our first definition concerns the notion of tuple, which formalizes the notion of row or record in a table: Def. A tuple ...

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Relational model, Relational model - The model, Relational model - Competition, Relational model - History, Relational model - Misimplementation, Relational model - Implementation, Relational model - Controversies, Relational model - Design, Relational model - Example database, Relational model - Set Theory Formulation, Relational model - Key constraints and functional dependencies

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However, once you consider subsets of a given set X (in Cantor's case, X = R), you may become interested in sets of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X. Of course, it doesn't stop there; you might next be interested in sets of sets of subsets of X, and so on. In another direction, you may become interest ...

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Universe mathematics, Universe mathematics - In a specific context, Universe mathematics - In ordinary mathematics, Universe mathematics - In set theory, Universe mathematics - In category theory

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Left-associative operations include the following. Subtraction and division of real numbers: Right-associative operations include the following. Exponentiation of real numbers: The reason exponentiation is right-associative is that a repeated left-associative exponentiation operation would be less useful. Multiple appearances could (and would) be rewritten with multiplication: ...

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Associativity, Associativity - Definition, Associativity - Examples, Associativity - Non-associativity, Associativity - More examples

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Minimum Intent: The following definition of the term 'definition' is presented as a reference, (a comparator, a norm) that must not be violated when defining scientific terms. Axioms: 1) ‘Something’ is a term that has a most general meaning, it can mean anything (but it does not automatically include ‘everything’). 2) 'Ambient' is anything in the vicinity of, and, to a certain degree, within something. 3) ‘Event’ is something that can be distinguished from its ambient. 4) ‘Relation’ is something that has, at l ...

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Definition, Definition - Kinds of definition, Definition - Determining meaning: extension intension ambiguity and vagueness, Definition - A definition of 'definition', Definition - A contribution to defining the term 'definition', Definition - Quotation

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The Kleene star is often generalized for any monoid (M, .), that is, a set M and binary operation '.' on M such that (closure) for all a and b in M, a . b in M (associativity) for all a, b and c in M, (a . b) . c = a . (b . c) (identity) there is an ε in M such that for all a, a . Π...

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Kleene star, Kleene star - Examples, Kleene star - Generalization

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A distance between two points P and Q in a metric space is d(P,Q), where d is the distance function that defines the given metric space. We can also define the distance between two sets A and B in a metric space as being the minimum (or infimum) of distances between any two points P in A and Q in B. Alternatively, the distance between sets may indicate "how different they are", by taking the supremum over one set of the distance from a point in that set to the other set, and conversely, and taking the ...

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Distance, Distance - Distance covered, Distance - Formal definition, Distance - The distance formula, Distance - Generalized distance in arbitrary dimensions: Norms, Distance - Distances in other spaces

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The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of GSee also:

Symmetry, Symmetry - Mathematical model for symmetry, Symmetry - Non-isometric symmetry, Symmetry - Reflection symmetry, Symmetry - Rotational symmetry, Symmetry - Translational symmetry, Symmetry - Glide reflection symmetry, Symmetry - Rotoreflection symmetry, Symmetry - Screw axis symmetry, Symmetry - Symmetry combinations, Symmetry - Color, Symmetry - Similarity vs. sameness, Symmetry - More on symmetry in geometry, Symmetry - Symmetry in mathematics, Symmetry - Symmetry in logic, Symmetry - Generalization of symmetry, Symmetry - Symmetry in physics, Symmetry - Symmetry in biology, Symmetry - Symmetry in chemistry, Symmetry - Symmetry in the arts and crafts, Symmetry - Architecture, Symmetry - Pottery, Symmetry - Quilts, Symmetry - Carpets rugs, Symmetry - Music, Symmetry - Other arts and crafts, Symmetry - Aesthetics, Symmetry - Symmetry in games and puzzles, Symmetry - Symmetry in literature, Symmetry - Symmetry in telecommunications, Symmetry - Moral symmetry, Symmetry - Related topics

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Theater structure - Ancient Greece. Greek theatre buildings were called a theatron ('seeing place'). The theatres were large, open-air structures constructed on the slopes of hills. They consisted of three principal elements: the orchestra, the skene, and the audience. The centrepiece of the theatre was the orchestra, or "dancing place", a large circular or rectangular area. The orchestra was the site the choral performances, the religious rites, and, possibly, the acting. An altar was located in the mid ...

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Theater structure, Theater structure - Basic elements of a theatre structure, Theater structure - History of theater construction, Theater structure - Ancient Greece, Theater structure - Ancient Rome, Theater structure - Elizabethan England, Theater structure - Contemporary theatres

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Most operations on a binary search tree take time directly proportional to the tree's height, which makes keeping the height small important. Ordinary binary search trees have the primary disadvantage that they can attain very large heights in rather ordinary situations, such as when the keys are inserted in order. The result is a data structure similar to a linked list, making all operations on the tree expensive. If we know all the data ahead of time, we can keep the height small on average by adding values in a random order, but we don' ...

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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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PROPOSITION 1: The empty set is a subset of every set. Proof: Given any set A, we wish to prove that ø is a subset of A. This involves showing that all elements of ø are elements of A. But there are no elements of ø. For the experienced mathematician, the inference " ø has no elements, so all elements of ø are elements of A" is immediate, but it may be more troublesome for the beginner. Since ø has no members at all, how can "they" be members of anything else? It may help to think of it ...

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Subset, Subset - Examples, Subset - Properties, Subset - Other properties of inclusion

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The algebra of sets is the development of the fundamental properties of set operations and set relations. These properties provide insight into the fundamental nature of sets. They also have practical considerations. Just like expressions and calculations in ordinary arithmetic, expressions and calculations involving sets can be quite complex. It is helpful to have systematic procedures available for manipulating and evaluating such expressions and performing such computations. In the case of arithmetic, it is elementary algebra that develops the fundamental pr ...

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Algebra of sets, Algebra of sets - Introduction, Algebra of sets - The fundamental laws of set algebra, Algebra of sets - The principle of duality, Algebra of sets - Some additional laws for unions and intersections, Algebra of sets - Some additional laws for complements, Algebra of sets - The algebra of inclusion, Algebra of sets - The algebra of relative complements

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It is a basic fact that every category has a skeleton. (This requires the axiom of choice.) Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories, so up to isomorphism of categories, the skeleton of a category is unique. The importance of skeletons comes from the fact that they are (up to isomorphism of categories), canonical representatives of the equivalence classes of categories under the equivalence relation of equivalence of categories. This follows from the fact that any skeleton of a category C is equivalent to C, and that two cat ...

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Skeleton category theory, Skeleton category theory - Definition, Skeleton category theory - Existence and uniqueness, Skeleton category theory - Examples, Skeleton category theory - Reference

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Lua is intended for use as an extension or scripting language, and is compact enough to fit on a variety of host platforms. It supports only a small number of atomic data structures such as boolean values, numbers (double-precision floating point by default), and strings. Typical data structures such as arrays, sets, hash tables, lists, and records can be represented using Lua's single native data structure, the table, which is essentially a heterogeneous map. Namespaces and objects can also be created using tables. By including only a minimum of data types, Lua ...

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Lua programming language, Lua programming language - Philosophy, Lua programming language - History, Lua programming language - Features, Lua programming language - Example code, Lua programming language - Tables, Lua programming language - Object-oriented programing, Lua programming language - Internals, Lua programming language - Applications, Lua programming language - Books

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Wallace and Gromit - Wallace. Wallace lives at 62 West Wallaby Street, Wigan, Lancs [1]. He can usually be found wearing a white shirt, brown wool trousers, green knitted vest and red tie. He loves cheese - preferably Wensleydale. The thought of Lancashire hotpot keeps him going in a crisis. He enjoys a nice cup of tea or a drop of Bordeaux red for those special occasions. He reads the Morning Post, the Afternoon Post, and the ...

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Wallace and Gromit, Wallace and Gromit - Characters, Wallace and Gromit - Wallace, Wallace and Gromit - Gromit, Wallace and Gromit - Studio Fire Incident, Wallace and Gromit - Films, Wallace and Gromit - Original 30-minute Shorts, Wallace and Gromit - Feature Film, Wallace and Gromit - New Shorts, Wallace and Gromit - Video Games, Wallace and Gromit - Stop-motion Technique

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The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". Correspondingly, quantified expressions are constructed as follows, where "P" denotes a formula. Many variant notations are used, such as All of these variations apply to univers ...

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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, Quantification - Links

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Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as Hom(F(X), Y< ...

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Adjoint functors, Adjoint functors - Motivation, Adjoint functors - Ubiquity of adjoint functors, Adjoint functors - Deep problems formulated with adjoint functors, Adjoint functors - Adjoint functors as solving optimization problems, Adjoint functors - The case of partial orders, Adjoint functors - Formal definitions, Adjoint functors - Examples, Adjoint functors - Properties, Adjoint functors - Uniqueness of adjoints, Adjoint functors - Relation to universal constructions, Adjoint functors - Characterization via unit and co-unit, Adjoint functors - Adjoints preserve certain limits, Adjoint functors - Additivity, Adjoint functors - Composition, Adjoint functors - Adjoint pairs extend equivalences, Adjoint functors - General existence theorem

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To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection . Intuitively, this means that it is possible to "pair off" elements of S with elements of T in such a fashion that every element of S is paired off with exactly one element of T and vice versa. Hence, the set {banana, appl ...

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Continuum hypothesis, Continuum hypothesis - The size of a set, Continuum hypothesis - Impossibility of proof and disproof, Continuum hypothesis - Arguments pro and con, Continuum hypothesis - The generalized continuum hypothesis

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Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, there may be other ways to construct it, but they are all equivalent. 1. Suppose that X and Y as above are sets. Then if we write Z for their intersection, there are morphisms f : Z → X and g : Z → Y given by inclusion. The pushout of f and g is the union of X and Y t ...

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Pushout category theory, Pushout category theory - Universal property, Pushout category theory - Examples of pushouts, Pushout category theory - Construction via coproducts and coequalizers, Pushout category theory - Application: The Seifert-van Kampen theorem

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To define the set of well typed lambda terms of a given type, we introduce typing contexts which are sequences of typing assumptions of the form x:σ where x is a variable. We introduce the judgment which means that t is a term of type σ in context Γ which is given by the following typing rules: Examples of closed terms are: (I), (K), and (S). These are the typed lambda calculus represen ...

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Simply typed lambda calculus, Simply typed lambda calculus - Types, Simply typed lambda calculus - Terms, Simply typed lambda calculus - Important results

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The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". Correspondingly, quantified expressions are constructed as follows, where "P" denotes a formula. Many variant notations are used, such as All of these variations apply to univers ...

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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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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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