A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (infimum, meet) and a least upper bound (supremum, join). These are denoted by: A (meet) and A (join). Note that in the special case where A is the empty set the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete latt ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite en ...

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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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Real number - Construction from the rational numbers. The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers. Real number - Axiomatic approach. Let R denote the set of all real numbers. Then: The set R is a field, meaning that addition and multiplication are defined and have the usual properties. The field R is ordered, meaning th ...

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Real number, Real number - History, Real number - Definition, Real number - Construction from the rational numbers, Real number - Axiomatic approach, Real number - Properties, Real number - Completeness, Real number - The complete ordered field, Real number - Advanced properties, Real number - Generalizations and extensions

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Projective line - Real projective line. The projective line over the real numbers is called the real projective line. It is given by projecting points in R2 onto the unit circle and then identifying diametrically opposite points. In terms of group theory we can take the quotient by the subgroup {1,−1}. Topologically, it is again a circle. One may also think of gluing the two "ends" of the real line onto a new point ∞ resulting in a circle. Compare the extended real number line, which distinguishes ∞ and −∞. Projective line - Complex ...

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Projective line, Projective line - Homogeneous coordinates, Projective line - Examples, Projective line - Real projective line, Projective line - Complex projective line: the Riemann sphere, Projective line - For a finite field, Projective line - Symmetry group, Projective line - As algebraic curve

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In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In theory, as long as there is no external interaction, the loop will continue to run for all time. In practice however, some programming loops co ...

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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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Zero is the smallest element: 0 ≤ a for all a in A. The sum a + b is the least upper bound of a and b: we have a ≤ a + b and b ≤ a + b and if x is an element of A with a ≤ x and b ≤ x, then a + b ≤ x. Similarly, a1 + ... + an is the least upper bound of the elements aSee also:

Kleene algebra, Kleene algebra - Definition, Kleene algebra - Examples, Kleene algebra - Properties, Kleene algebra - History

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Limit of a function - Real-valued functions. Limit of a function - Functions on metric spaces. If z is a complex number with |z| < 1, then the sequence z, z2, z3, ... of complex numbers converges with limit 0. Geometrically, these numbers "spiral into" the origin, following a logarithmic spiral. In the metric space C[a,b] of all continuous functions defined on the interval [a,b], wit ...

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Limit of a function, Limit of a function - History, Limit of a function - Formal definition, Limit of a function - Functions on metric spaces, Limit of a function - Real-valued functions, Limit of a function - Complex-valued functions, Limit of a function - Limit of a function of more than one variable, Limit of a function - Examples, Limit of a function - Real-valued functions, Limit of a function - Functions on metric spaces, Limit of a function - Properties

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To say that the limit of a function f at p is L is equivalent to saying for every convergent sequence (xn) in M with limit equal to p, the sequence (f(xn)) converges with limit L. If the sets A, B, ... form a finite partition of the function domain, , ... and the relative limit for each of those sets exist and is the equal to, say, L, then t ...

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Limit of a function, Limit of a function - History, Limit of a function - Formal definition, Limit of a function - Functions on metric spaces, Limit of a function - Real-valued functions, Limit of a function - Complex-valued functions, Limit of a function - Limit of a function of more than one variable, Limit of a function - Examples, Limit of a function - Real-valued functions, Limit of a function - Functions on metric spaces, Limit of a function - Properties

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Let Σ be a finite set (an "alphabet") and let A be the set of all regular expressions over Σ. We consider two such regular expressions equal if they describe the same language. Then A forms a Kleene algebra. In fact, this is a free Kleene algebra in the sense that any equation among regular expressions follows from the Kleene algebra axioms and is therefore valid in every Kleene algebra. Again let Σ be an alphabet. Let A be the set of all regular languages over Σ (or the set of all context-free languages over ...

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Kleene algebra, Kleene algebra - Definition, Kleene algebra - Examples, Kleene algebra - Properties, Kleene algebra - History

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Consider This integral involves a function with a vertical asymptote at x = 0. One can evaluate this integral by evaluating from b to 1, and then take the limit as b approaches 0. One should note that the antiderivative of the above function is (3)(x1/3); which can be evaluated by direct substitution to give the value 3 × (1 − ...

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Improper integral, Improper integral - Infinite bounds of integration, Improper integral - Vertical asymptotes at bounds of integration, Improper integral - Cauchy principal values

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Consider the following function: f(x)=xn if n-1<x≤n. Then the limit of the sequence of xn is just the limit of f(x) at infinity. A function f, defined on a first-countable space, is continuous if and only if it is compatible with limits in that (f(xn)) converges to f(L) given that (xn) converges to < ...

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Limit of a sequence, Limit of a sequence - Formal definition, Limit of a sequence - Comments, Limit of a sequence - Examples, Limit of a sequence - Properties, Limit of a sequence - History

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The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involved limiting processes. Leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series. Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite seri ...

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Limit of a sequence, Limit of a sequence - Formal definition, Limit of a sequence - Comments, Limit of a sequence - Examples, Limit of a sequence - Properties, Limit of a sequence - History

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Quite generally, the group of Möbius transformations with coefficients in K acts on the projective line P1(K). This group action is transitive, so that P1(K) is a homogeneous space for the group, often written PGL2(K) to emphasise its definition as a projective linear group. Transitivity says that any point Q may be transformed to any other point R by a Möbius transformation. The point at infinity on P1(K) is therefore an artefact of choice of coordinates: homogeneous coordinates ...

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Projective line, Projective line - Homogeneous coordinates, Projective line - Examples, Projective line - Real projective line, Projective line - Complex projective line: the Riemann sphere, Projective line - For a finite field, Projective line - Symmetry group, Projective line - As algebraic curve

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If f is non-negative, then ∫f dμ is precisely the area under the curve as measured by the product measure μ × λ where λ is the Lebesgue measure for R. One can try to circumvent measure theory entirely. The Riemann integral exists for any continuous function f of compact support. Then we use functional analysis to obtain the integral for more general functions. Let Cc be the space of all real-valued compactly supported continuous functions of RSee also:

Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

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Given a normed vector space (X,||.||) we can define a metric on X by d(x,y):=||x-y||. The metric d is called induced by ||.||. Conversely if a metric d on a vector space X satisfies the properties d(x,y) = d(x+a,y+a) (translation invariance) d(αx,αy) = |α|d(x,y) (homogenity) then we can define a ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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The definition of infima easily generalizes to subsets of arbitrary partially ordered sets and as such plays a vital role in order theory. In this context, especially in lattice theory, greatest lower bounds are also called meets. Formally, the infimum of a subset S of a partially ordered set (P, ≤) is an element l of P such that l ≤ x for all x in S, and for any p in P such that p ≤ x for all x ...

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Infimum, Infimum - Infima of real numbers, Infimum - Infima in partially ordered sets, Infimum - Least upper bound property

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Least upper bounds are important concepts in order theory, where they are also called joins (especially in lattice theory). As in the special case treated above, a supremum of a given set is just the least element of the set of its upper bounds, provided that such an element exists. Formally, we have: For subsets S of arbitrary partially ordered sets (P, ≤), a supremum or least upper bound of S is an element u in P such that x ≤ u for all x i ...

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Supremum, Supremum - Supremum of a set of real numbers, Supremum - Approximation property, Supremum - Additive property, Supremum - Comparison property, Supremum - Suprema within partially ordered sets, Supremum - Comparison with other order theoretical notions, Supremum - Greatest elements, Supremum - Maximal elements, Supremum - Minimal upper bounds, Supremum - Least-upper-bound property

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For a given set X two metrics d1 and d2 are called topological equivalent (uniformly equivalent) if the identity mapping id: (X,d1) → (X,d2) is a homeomorphism (uniform isomorphism). ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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These conditions express intuitive notions about the concept of distance. For example, that the distance between distinct points is positive and the distance from x to y is the same as the distance from y to x. The triangle inequality means that the distance traversed directly between x and z, is not larger than the distance to traverse in going first from x to y, and then from y to z. Euclid in his work proved that the shortest distance between two points is a line; that w ...

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Metric mathematics, Metric mathematics - Definition, Metric mathematics - Notes, Metric mathematics - Examples, Metric mathematics - Equivalence of metrics, Metric mathematics - Relation of norms and metrics, Metric mathematics - Related concepts and alternative axiom systems

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The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are characterized as functions that preserve all joins and all meets. Explicitly, this means that a function f: L→M between two complete lattices L and M is a complete homomorphism if and , for all subsets A of L. Such functions are automatically monotonic, but the condition of being a complete homomorphism is in fact much more spe ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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There are various other mathematical concepts that can be used to represent complete lattices. One means of doing so is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is a complete lattice of sets which is isomorphic to the original one. Thus we immediately find that every complete lattice is isomorphic to a complete lattice of sets. Another representation is obtained by noting that the image of any closure operator on a complete lattice is again a comp ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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