As Ritter counts down the clock to morning — to the sun, which rises in the cardinality of his obsession — he turns to recounting times past, excavating the precipitous collapse of his profession in the last days "before Google" and ISIS, when he and his colleagues still traveled through the region, and lived it up on fieldwork grants in Tehran (until the Islamic Revolution), and Damascus, Aleppo and Palmyra (until the civil war).
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Barbieri 2008, pp. 337–341 One isomorphic note-layout, the Wicki, when mapped to a hexagonal array of buttons, is particularly well-suited to the control of enharmonic scales. The orientation of its hexagonal columns of octaves and tempered perfect fifths place all the notes of every well-formed scale – pentatonic (cardinality 5), diatonic (cardinality 7), chromatic (cardinality 12), and enharmonic (cardinality 19) – in a tight, contiguous cluster. The notes of each progressively-higher cardinality are appended to the outer edges of the lower- cardinality scale, such that each well-formed scale's note-controlling buttons are embedded, unchanged, within the set of those controlling the higher- cardinality scales.
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If is a finite set of cardinality , this number is the cardinality of the -fold Cartesian power . Tuples are elements of this product set.
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Then there is a bijection between the factors of for which and the subsets of the set of all prime factors of . The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets. This last fact can be shown easily by induction on the cardinality of a non-empty finite set . First, if , there is exactly one odd-cardinality subset of , namely itself, and exactly one even-cardinality subset, namely .
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If the axiom of choice holds and the cardinal κ is regular, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.
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In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim–Skolem theorem. If L is a first-order language with cardinality \kappa and T is a complete theory over L, then this theorem guarantees a model for T of cardinality \max(\kappa,\aleph_0). Therefore no prime model of T can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality.
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Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial.
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There are several styles for representing data structure diagrams, with the notable difference in the manner of defining cardinality. The choices are between arrow heads, inverted arrow heads (crow's feet), or numerical representation of the cardinality.
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The pigeonhole principle can be extended to infinite sets by phrasing it in terms of cardinal numbers: if the cardinality of set is greater than the cardinality of set , then there is no injection from to . However, in this form the principle is tautological, since the meaning of the statement that the cardinality of set is greater than the cardinality of set is exactly that there is no injective map from to . However, adding at least one element to a finite set is sufficient to ensure that the cardinality increases. Another way to phrase the pigeonhole principle for finite sets is similar to the principle that finite sets are Dedekind finite: Let and be finite sets.
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So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set Vω has the same cardinality as ω. The set Vω+1 has the same cardinality as the set of real numbers.
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In mathematics, the continuum function is \kappa\mapsto 2^\kappa, i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality.
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In set theory, there are exponential operations for cardinal and ordinal numbers. If κ and λ are cardinal numbers, the expression κλ represents the cardinality of the set of functions from any set of cardinality λ to any set of cardinality κ.Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5. If κ and λ are finite, then this agrees with the ordinary arithmetic exponential operation.
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The reals are uncountable; that is: there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e. the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly greater than the cardinality of N. Since the set of algebraic numbers is countable, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
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Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. and ), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal.
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Call this cover H_1. Third, find all covers of cardinality k that do not violate the budget. Using these covers of cardinality k as starting points, apply the modified greedy algorithm, maintaining the best cover found so far. Call this cover H_2.
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Before the 1930s, the measurability of utility functions was erroneously labeled as cardinality by economists. A different meaning of cardinality was used by economists who followed the formulation of Hicks-Allen. Under this usage, the cardinality of a utility function is simply the mathematical property of uniqueness up to a linear transformation. Around the end of the 1940s, some economists even rushed to argue that von Neumann-Morgenstern axiomatization of expected utility had resurrected measurability.
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The set of subsets of of cardinality less than or equal to is sometimes denoted by or , and the set of subsets with cardinality strictly less than is sometimes denoted or . Similarly, the set of non-empty subsets of might be denoted by or .
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To take the algorithmic interpretation above would seem at odds with classical notions of cardinality. By enumerating algorithms, we can show classically that the computable numbers are countable. And yet Cantor's diagonal argument shows that real numbers have higher cardinality. Furthermore, the diagonal argument seems perfectly constructive.
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The number of elements in a particular set is a property known as cardinality; informally, this is the size of a set. In the above examples, the cardinality of the set A is 4, while the cardinality of set B and set C are both 3\. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets.
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For a finite set of cardinality n, there are nn transformations and (n+1)n partial transformations.
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The density of a topological space X is the least cardinality of a dense subset of X.
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Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents.
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The operators and represent cycles of even and odd length, and sets of even and odd cardinality.
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For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types, see Section #Natural numbers for a simple example. For a countably infinite set, the set of possible order types is even uncountable.
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It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers. The α-th infinite initial ordinal is written \omega_\alpha, it is always a limit ordinal. Its cardinality is written \aleph_\alpha. For example, the cardinality of ω0 = ω is \aleph_0, which is also the cardinality of ω2 or ε0 (all are countable ordinals).
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Let be a vector space, be a linearly independent set of elements of , and be a generating set. One has to prove that the cardinality of is not larger than that of . If is finite, this results from the Steinitz exchange lemma. (Indeed, the Steinitz exchange lemma implies every finite subset of has cardinality not larger than that of , hence is finite with cardinality not larger than that of .) If is finite, a proof based on matrix theory is also possible.
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Cardinality equals variety and structure implies multiplicity are true of all collections with Myhill's property or maximal evenness.
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132 (1968) 217–230.B. L. Osofsky, Homological dimension and cardinality, Trans. Amer. Math. Soc. 151 (1970) 641–649.
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A theory is -categorical (or categorical in ) if it has exactly one model of cardinality up to isomorphism. Morley's categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. extended Morley's theorem to uncountable languages: if the language has cardinality and a theory is categorical in some uncountable cardinal greater than or equal to then it is categorical in all cardinalities greater than .
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Every Hilbert space admits an orthonormal basis, and any two such bases for a particular space have the same cardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.
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Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.
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In abstract algebra, a matrix field is a field with matrices as elements. In field theory we come across two types of fields: finite fields and infinite fields. There are several examples of matrix fields of different characteristic and cardinality. There is a finite matrix field of cardinality p for each positive prime p.
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He generalized the Cantor–Bernstein theorem, which said the collection of countable order types has the cardinality of the continuum and showed that the collection of all graded types of an idempotent cardinality has a cardinality of 2. For the summer semester 1910 Hausdorff was appointed as professor to the University of Bonn. In Bonn, he began a lecture on set theory, which he repeated in the summer semester 1912, substantially revised and expanded. In the summer of 1912 he also began work on his magnum opus, the book Basics of set theory.
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The problem has several variants. 1\. In maximum-cardinality RM matching, the goal is to find, among all different RM matchings, the one with the maximum number of matchings. 2\. In fair matching, the goal is to find a maximum-cardinality matching such that the minimum number of edges of rank r are used, given that - the minimum number of edges of rank r−1 are used, and so on. Both maximum-cardinality RM matching and fair matching can be found by reduction to maximum-weight matching. 3\.
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Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex.
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When agents have cardinality constraints (i.e., for each category of items, there is an upper bound on the number of items each agent an get from this category), the envy- graph algorithm might fail. However, combining it with the round-robin protocol gives an algorithm that finds allocations that are both EF1 and satisfy the cardinality constraints.
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It has a cardinality of one-to-many relationship. :b. Is Candidate For – candidate indicates the possible filling of the role by a resource. :c. Has Member – this is a kind of relations between organisation and person by denoting that a certain person has membership in the organisation. Has a cardinality of many-to-many relation. :d.
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As to cardinality, almost all elements of the Cantor set are not endpoints of intervals, and the whole Cantor set is not countable.
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As it is mentioned in the Application part of this article, the maximum cardinality bipartite matching is an application of maximum flow problem.
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Then (in the von Neumann formulation of cardinality) C is a set and therefore has a power set 2C which, by Cantor's theorem, has cardinality strictly larger than C. Demonstrating a cardinality (namely that of 2C) larger than C, which was assumed to be the greatest cardinal number, falsifies the definition of C. This contradiction establishes that such a cardinal cannot exist. Another consequence of Cantor's theorem is that the cardinal numbers constitute a proper class. That is, they cannot all be collected together as elements of a single set. Here is a somewhat more general result.
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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph null, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol \aleph (aleph) followed by a subscript, describe the sizes of infinite sets.
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It will be shown that , and hence a one-to-one correspondence between and the group exists. For , let be the subset of consisting of all subsets of cardinality exactly . Then is the disjoint union of the . The number of subsets of of cardinality is at most because every subset with elements is an element of the -fold cartesian product of .
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In particular, every uncountable Polish space has the cardinality of the continuum. Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces.
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This means that, for every set S of cardinality \kappa, and every partition of the ordered pairs of elements of S into two subsets P_1 and P_1, there exists either a subset S_1\subset S of cardinality \kappa or a subset S_2\subset S of cardinality \alef_0, such that all pairs of elements of S_i belong to P_i. Here, P_1 can be interpreted as the edges of a graph having S as its vertex set, in which S_1 (if it exists) is a clique of cardinality \kappa, and S_2 (if it exists) is a countably infinite independent set. If S is taken to be the cardinal number \kappa itself, the theorem can be formulated in terms of ordinal numbers with the notation \kappa\rightarrow(\kappa,\omega)^2, meaning that S_2 (when it exists) has order type \omega.
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Languages in the OWL family support various operations on classes such as union, intersection and complement. They also allow class enumeration, cardinality, disjointness, and equivalence.
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There is a subtle cardinal ≤ κ if and only if every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. An infinite ordinal κ is subtle if and only if for every λ < κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.
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The ' r(H) of a hypergraph H is the maximum cardinality of any of the edges in the hypergraph. If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph. A graph is just a 2-uniform hypergraph. The degree d(v) of a vertex v is the number of edges that contain it.
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Finding the best BST execution for the input sequence x_1, x_2, . . ., x_m is equivalent to finding the minimum cardinality superset of points (that contains the input in geometric representation) that is arborally satisfied. The more general problem of finding the minimum cardinality arborally satisfied superset of a general set of input points (not limited to one input point per coordinate), is known to be NP-complete.
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Although Jevons predated the debate about ordinality or cardinality of utility, his mathematics required the use of cardinal utility functions. For example, in "The Theory of Political Economy", Chapter II, the subsection on "Theory of Dimensions of Economic Quantities", Jevons makes the statement that "In the first place, pleasure and pain must be regarded as measured upon the same scale, and as having, therefore, the same dimensions, being quantities of the same kind, which can be added and subtracted...." Speaking of measurement, addition and subtraction requires cardinality, as does Jevons's heavy use of integral calculus. Note that cardinality does not imply direct measurability, in which Jevons did not believe.
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Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤B F, if and only if there is a Borel function : Θ : X → Y such that for all x,x' ∈ X, one has :x E x' ⇔ Θ(x) F Θ(x'). Conceptually, if E is Borel reducible to F, then E is "not more complicated" than F, and the quotient space X/E has a lesser or equal "Borel cardinality" than Y/F, where "Borel cardinality" is like cardinality except for a definability restriction on the witnessing mapping.
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Having an infinite cardinality (that of the continuum), this is far larger than the symmetry group of any possible embedding of the Möbius band in R3.
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Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any true infinity. This is the paradoxical nature of Cantor's "paradox".
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Shout (protocol) builds a spanning tree on a generic graph and elects its root as leader. The algorithm has a total cost linear in the edges cardinality.
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A semi-infinite integral is an improper integral over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.Cator, Pimentel, A shape theorem and semi-infinite geodesics for the Hammersley model with random weights, 2010. Most forms of semi-infiniteness are boundedness properties, not cardinality or measure properties: semi-infinite sets are typically infinite in cardinality and measure.
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In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers. Cantor's theorem implies that there are sets having cardinality greater than the infinite cardinality of the set of natural numbers. Cantor's argument for this theorem is presented with one small change.
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The set S1 of all first player strategies in an ω-game G has the same cardinality as the continuum. The same is true of the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in G is also the continuum. Let A be the subset of SG of all sequences that make the first player win.
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Every distributive (∨,0)-semilattice of cardinality at most ℵ1 satisfies Schmidt's Condition. Thus it is representable. By using different methods, Dobbertin got the following result. Theorem (Dobbertin 1986).
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In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces states that :Given a vector space , any two bases have the same cardinality. As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful: :In a vector space , if is a generating set, and is a linearly independent set, then the cardinality of is not larger than the cardinality of .
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Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is \aleph_0. Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.
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The 2007 HyperLogLog algorithm splits the multiset into subsets and estimates their cardinalities, then it uses the harmonic mean to combine them into an estimate for the original cardinality.
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Kneser-Haken finiteness says that for each 3-manifold, there is a constant C such that any collection of surfaces of cardinality greater than C must contain parallel elements.
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Edward Nelson's internal set theory enriches the Zermelo–Fraenkel set theory syntactically by introducing a unary predicate "standard". In this approach, infinitesimals are (non-"standard") elements of the set of the real numbers (rather than being elements of an extension thereof, as in Robinson's theory). The continuum hypothesis posits that the cardinality of the set of the real numbers is \aleph_1; i.e. the smallest infinite cardinal number after \aleph_0, the cardinality of the integers.
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There are at least two independent arguments in favor of a small set theory like PST. #One can get the impression from mathematical practice outside set theory that there are “only two infinite cardinals which demonstrably ‘occur in nature’ (the cardinality of the natural numbers and the cardinality of the continuum),”Pocket Set Theory, p.8. therefore “set theory produces far more superstructure than is needed to support classical mathematics.”Alternative Set Theories, p.35.
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A bitmap index is a special kind of database index that uses bitmaps. Bitmap indexes have traditionally been considered to work well for low-cardinality columns, which have a modest number of distinct values, either absolutely, or relative to the number of records that contain the data. The extreme case of low cardinality is Boolean data (e.g., does a resident in a city have internet access?), which has two values, True and False.
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In particular, no theory extending ZF can prove either the completeness or compactness theorems over arbitrary (possibly uncountable) languages without also proving the ultrafilter lemma on a set of same cardinality.
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The principle behind snowflaking is normalization of the dimension tables by removing low cardinality attributes and forming separate tables.Paulraj Ponniah. Data Warehousing Fundamentals for IT Professionals. Wiley, 2010, pp. 29–32. .
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The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.
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What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set. However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers. A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers (i.e.
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"I see it but I don't believe," Cantor wrote to Richard Dedekind after proving that the set of points of a square has the same cardinality as that of the points on just an edge of the square: the cardinality of the continuum. This demonstrates that the "size" of sets as defined by cardinality alone is not the only useful way of comparing sets. Measure theory provides a more nuanced theory of size that conforms to our intuition that length and area are incompatible measures of size. The evidence strongly suggests that Cantor was quite confident in the result itself and that his comment to Dedekind refers instead to his then-still-lingering concerns about the validity of his proof of it.
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Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph G, and the goal is to find a matching containing as many edges as possible, that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when G is a bipartite graph, whose vertices V are partitioned between left vertices in X and right vertices in Y, and edges in E always connect a left vertex to a right vertex.
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A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum. For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.
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In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by , states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is (\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega). The axiom of constructibility implies that Chang's conjecture fails.
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Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum). The theory was first suggested by Rudy Rucker in his Infinity and the Mind.Rucker, Rudy, Infinity of the Mind, Princeton UP, 1995, p.253. The details set out in this entry are due to the American mathematician Randall M. Holmes.
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In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined.
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In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph. The theorem was first published by , in both the form stated above and an equivalent complementary form: every infinite graph contains either a countably infinite clique or an independent set with equal cardinality to the whole graph. In their paper, they credited Paul Erdős with assistance in its proof. They applied these results to the comparability graphs of partially ordered sets to show that each partial order contains either a countably infinite antichain or a chain of cardinality equal to the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order. The same theorem can also be stated as a result in set theory, using the arrow notation of , as \kappa\rightarrow(\kappa,\alef_0)^2.
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As this yields a notional cardinality of the continuum, Hartman advises that when setting out to describe a person, a continuum of properties would be most fitting and appropriate to bear in mind. This is the cardinality of intrinsic value in Hartman's system. Although they play no role in ordinary mathematics, Hartman deploys the notion of aleph number reciprocals, as a sort of infinitesimal proportion. This, he contends goes to zero in the limit as the uncountable cardinals become larger.
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These notions can also be defined with respect to other logics. For each σ-structure A, there are several associated theories in a larger signature σ' that extends σ by adding one new constant symbol for each element of the domain of A. (If the new constant symbols are identified with the elements of A which they represent, σ' can be taken to be σ \cup A.) The cardinality of σ' is thus the larger of the cardinality of σ and the cardinality of A. The diagram of A consists of all atomic or negated atomic σ'-sentences that are satisfied by A and is denoted by diagA. The positive diagram of A is the set of all atomic σ'-sentences which A satisfies. It is denoted by diag+A.
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Cardinality equals variety in the diatonic collection and the pentatonic scale, and, more generally, what Carey and Clampitt (1989) call "nondegenerate well-formed scales." "Nondegenerate well-formed scales" are those that possess Myhill's property.
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If is finite, there is nothing to prove. Thus, we may assume that is also infinite. Let us suppose that the cardinality of is larger than that of .This uses the axiom of choice.
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A transformation semigroup is synchronizing if it contains an element of rank 1, that is, an element whose image is of cardinality 1.. A DFA corresponds to a transformation semigroup with a distinguished generator set.
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Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities (Cantor 1874). Over the next twenty years, Cantor developed a theory of transfinite numbers in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset.
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The following are equivalent for any uncountable cardinal κ: # κ is weakly compact. # for every λ<κ, natural number n ≥ 2, and function f: [κ]n → λ, there is a set of cardinality κ that is homogeneous for f. # κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ. # Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
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Next, if , then divide the subsets of into two subclasses depending on whether they contain or not some fixed element in . There is an obvious bijection between these two subclasses, pairing those subsets that have the same complement relative to the subset . Also, one of these two subclasses consists of all the subsets of the set , and therefore, by the induction hypothesis, has an equal number of odd- and even- cardinality subsets. These subsets in turn correspond bijectively to the even- and odd-cardinality -containing subsets of .
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EXPRESS-G is a standard graphical notation for information models.4 EXPRESS-G Language Overview . Accessed 9 Nov 2008. It is a companion to the EXPRESS language for displaying entity and type definitions, relationships and cardinality.
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The notion of saturated model is dual to the notion of prime model in the following way: let T be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and let P be a prime model of T. Then P admits an elementary embedding into any other model of T. The equivalent notion for saturated models is that any "reasonably small" model of T is elementarily embedded in a saturated model, where "reasonably small" means cardinality no larger than that of the model in which it is to be embedded. Any saturated model is also homogeneous. However, while for countable theories there is a unique prime model, saturated models are necessarily specific to a particular cardinality. Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories.
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Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality.
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While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).
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124 (1996) 2555-2560. Zbl. 0876.54016 of a small (cardinality continuum) Dowker space. He also solved Nagami's problem (normal + screenable does not imply paracompact),Z. Balogh, A Normal Screenable Nonparacompact Space in ZFC, Proc. Amer. Math. Soc.
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Given a field extension , a subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coefficients in K exists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendence degree of L/K. It is always possible to find a set S, algebraically independent over K, such that L/K(S) is algebraic. Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension.
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Given any model M of ZFC, the collection of hereditarily finite sets in M will satisfy the GST axioms. Therefore, GST cannot prove the existence of even a countable infinite set, that is, of a set whose cardinality is ℵ0. Even if GST did afford a countably infinite set, GST could not prove the existence of a set whose cardinality is \aleph_1, because GST lacks the axiom of power set. Hence GST cannot ground analysis and geometry, and is too weak to serve as a foundation for mathematics.
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The fact that lambda calculus terms act as functions on other lambda calculus terms, and even on themselves, led to questions about the semantics of the lambda calculus. Could a sensible meaning be assigned to lambda calculus terms? The natural semantics was to find a set D isomorphic to the function space D → D, of functions on itself. However, no nontrivial such D can exist, by cardinality constraints because the set of all functions from D to D has greater cardinality than D, unless D is a singleton set.
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In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.
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This is similar to the case of a circle, whose dimension shrinks from two to zero as it degenerates into a point. As another example, the solution set of a system of equations that depends on parameters generally has a fixed cardinality and dimension, but cardinality and/or dimension may be different for some exceptional values, called degenerate cases. In such a degenerate case, the solution set is said to be degenerate. For some classes of composite objects, the degenerate cases depend on the properties that are specifically studied.
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We have to prove that this leads to a contradiction. By Zorn's lemma, every linearly independent set is contained in a maximal linearly independent set . This maximality implies that spans and is therefore a basis (the maximality implies that every element of is linearly dependent from the elements of , and therefore is a linear combination of elements of ). As the cardinality of is greater than or equal to the cardinality of , one may replace with , that is, one may suppose, without loss of generality, that is a basis.
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First, if we take the powerset of any infinite set x, then that powerset will contain elements which are subsets of x of every finite cardinality (among other subsets of x). Proving the existence of those finite subsets may require either the axiom of separation or the axioms of pairing and union. Then we can apply the axiom of replacement to replace each element of that powerset of x by the initial ordinal number of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals.
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The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms.
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However, if is a net in then it is _not_ in general true that is equal to since, for instance, the domain of a net in (i.e. the directed set ) may have any cardinality (so the class of nets in isn't even a set) whereas the cardinality of the set of prefilters on , which is a subset of , is bounded above. ;Ultranets and ultra prefilters A net in is an ultranet if and only if is an ultra prefilter. A prefilter on is an ultra prefilter if and only if is an ultranet in .
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Other important representability results are related to the cardinality of the semilattice. The following result was prepared for publication by Dobbertin after Huhn's passing away in 1985. The two corresponding papers were published in 1989. Theorem (Huhn 1985).
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"Submodular maximization with cardinality constraints." Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, 2014. are imposed on the output, though often slight variations on the greedy algorithm are required.
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There are uncountably many of these sets and also some recursively enumerable but noncomputable sets of this type. Later, Degtev established a hierarchy of recursively enumerable sets that are (1, n + 1)-recursive but not (1, n)-recursive. After a long phase of research by Russian scientists, this subject became repopularized in the west by Beigel's thesis on bounded queries, which linked frequency computation to the above-mentioned bounded reducibilities and other related notions. One of the major results was Kummer's Cardinality Theory which states that a set A is computable if and only if there is an n such that some algorithm enumerates for each tuple of n different numbers up to n many possible choices of the cardinality of this set of n numbers intersected with A; these choices must contain the true cardinality but leave out at least one false one.
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While the concept of cardinality has fallen out of favor in neoclassical economics, the differences between cardinal utility and ordinal utility are minor for most applications. Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to modern mathematical economics.
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Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types. In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined.
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Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph. First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity.
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In DSDs, attributes are specified inside the entity boxes rather than outside of them, while relationships are drawn as boxes composed of attributes which specify the constraints that bind entities together. DSDs differ from the E–R model in that the E–R model focuses on the relationships between different entities, whereas DSDs focus on the relationships of the elements within an entity. There are several styles for representing data structure diagrams, with the notable difference in the manner of defining cardinality. The choices are between arrow heads, inverted arrow heads (crow's feet), or numerical representation of the cardinality.
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Even in this case, however, it is still possible to list all the elements, because the set is finite. Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...}, has infinitely many elements, and we cannot use any normal number to give its size. Nonetheless, it turns out that infinite sets do have a well- defined notion of size (or more properly, cardinality, the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.
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Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2λ = λℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. gave a simple proof for the general case. The existence of Jónsson functions shows that for any cardinal there is an algebra with an infinitary operation that has no proper subalgebras of the same cardinality. In particular if infinitary operations are allowed then an analogue of Jónsson algebras exists in any cardinality, so there are no infinitary analogues of Jónsson cardinals.
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The rank of a finitely generated group G can be equivalently defined as the smallest cardinality of a set X such that there exists an onto homomorphism F(X) → G, where F(X) is the free group with free basis X. There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups,John R. Stallings. Problems about free quotients of groups. Geometric group theory (Columbus, OH, 1992), pp.
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Clearly the number of distinct subsets that can be constructed this way is as . Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum).
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There are several types of secret sharing schemes. The most basic types are the so- called threshold schemes, where only the cardinality of the set of shares matters. In other words, given a secret S, and n shares, any set of t shares is a set with the smallest cardinality from which the secret can be recovered, in the sense that any set of t-1 shares is not enough to give S. This is known as a threshold access structure. We call such schemes (t,n) threshold secret sharing schemes, or t-out-of-n scheme.
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Suppose X is a Tychonoff space, also called a T3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose M is a maximal ideal in C(X). Then the factor algebra A = C(X)/M is a totally ordered field F containing the reals. If F strictly contains R then M is called a hyperreal ideal (terminology due to Hewitt (1948)) and F a hyperreal field. Note that no assumption is being made that the cardinality of F is greater than R; it can in fact have the same cardinality.
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A similar idea occurs in mathematical logic: a theory is called categorical if all its models of the same cardinality are mutually isomorphic. According to Bourbaki, the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.
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But when n > 2, the omitted set can contain more than two points, and its cardinality can be estimated from above in terms of n and K. In fact, any finite set can be omitted, as shown by David Drasin and Pekka Pankka.
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He used this inconsistent multiplicity to prove the aleph theorem.Hallett 1986, pp. 166–169. In 1932, Zermelo criticized the construction in Cantor's proof.Cantor's proof, which is a proof by contradiction, starts by assuming there is a set S whose cardinality is not an aleph.
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In DSDs, attributes are specified inside the entity boxes rather than outside of them, while relationships are drawn as boxes composed of attributes which specify the constraints that bind entities together. DSDs differ from the ER model in that the ER model focuses on the relationships between different entities, whereas DSDs focus on the relationships of the elements within an entity and enable users to fully see the links and relationships between each entity. There are several styles for representing data structure diagrams, with the notable difference in the manner of defining cardinality. The choices are between arrow heads, inverted arrow heads (crow's feet), or numerical representation of the cardinality.
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Arbitrary-precision arithmetic in most computer software is implemented by calling an external library that provides data types and subroutines to store numbers with the requested precision and to perform computations. Different libraries have different ways of representing arbitrary-precision numbers, some libraries work only with integer numbers, others store floating point numbers in a variety of bases (decimal or binary powers). Rather than representing a number as single value, some store numbers as a numerator/denominator pair (rationals) and some can fully represent computable numbers, though only up to some storage limit. Fundamentally, Turing machines cannot represent all real numbers, as the cardinality of exceeds the cardinality of .
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The Löwenheim number of a logic L is the smallest cardinal κ such that if an arbitrary sentence of L has any model, the sentence has a model of cardinality no larger than κ. Löwenheim proved the existence of this cardinal for any logic in which the collection of sentences forms a set, using the following argument. Given such a logic, for each sentence φ, let κφ be the smallest cardinality of a model of φ, if φ has any model, and let κφ be 0 otherwise. Then the set of cardinals :{ κφ : φ is a sentence in L } exists by the axiom of replacement.
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A maximal antichain is an antichain that is not a proper subset of any other antichain. A maximum antichain is an antichain that has cardinality at least as large as every other antichain. The width of a partially ordered set is the cardinality of a maximum antichain. Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into k chains then the width of the order must be at most k (if the antichain has more than k elements, by the pigeonhole principle, there would be 2 of its elements belonging to the same chain, contradiction).
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The entity–relationship model proposes a technique that produces entity–relationship diagrams (ERDs), which can be employed to capture information about data model entity types, relationships and cardinality. A Crow's foot shows a one-to-many relationship. Alternatively a single line represents a one-to-one relationship.
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The seven contiguous trichords in C major. See also: Cardinality equals variety. In music theory, a trichord () is a group of three different pitch classes found within a larger group . A trichord is a contiguous three-note set from a musical scale or a twelve-tone row.
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As the column cardinality increases, each bitmap becomes sparse and it may take more disk space to store the bitmaps than to store the same content as RID-lists. In this case, it switches to use the RID-lists, which makes it a B+tree index.
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The converse of Jensen's covering theorem is also true: if 0# exists then the countable set of all cardinals less than ℵω cannot be covered by a constructible set of cardinality less than ℵω. In his book Proper Forcing, Shelah proved a strong form of Jensen's covering lemma.
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If H is a maximal subgroup of a monoid M (a semigroup with identity), then H is an H-class, and it is naturally isomorphic to its own Schutzenberger group. In general, one has that the cardinality of H and its Schutzenberger group coincide for any H-class H.
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Every countable subset of the real numbers that (i.e. finite or countably infinite) is null. For example, the set of natural numbers is countable, having cardinality \aleph_0 (aleph-zero or aleph-null), is null. Another example is the set of rational numbers, which is also countable, and hence null.
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The Löwenheim–Skolem theorem shows that if a first-order theory of cardinality λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in model theory, it implies that it is not possible to characterize countability or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ(x) such that an arbitrary structure M satisfies φ if and only if the domain of discourse of M is countable (or, in the second case, uncountable). The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic.
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Scott's trick assigns representatives differently, using the fact that for every set A there is a least rank γA in the cumulative hierarchy when some set of the same cardinality as A appears. Thus one may define the representative of the cardinal number of A to be the set of all sets of rank γA that have the same cardinality as A. This definition assigns a representative to every cardinal number even when not every set can be well-ordered (an assumption equivalent to the axiom of choice). It can be carried out in Zermelo-Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.
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In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(μ+), B) → (H(κ+), A) (where H(κ+) is the set of all sets of cardinality hereditarily less than κ+) with critical point μ and j(μ) = κ. Analogously, κ is a quasicompact cardinal if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(κ+), A) → (H(μ+), B) with critical point κ and j(κ) = μ. H(λ) consists of all sets whose transitive closure has cardinality less than λ.
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In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property which Ramsey cardinals generalize to the uncountable case. Let [κ]<ω denote the set of all finite subsets of κ. An uncountable cardinal number κ is called Ramsey if, for every function :f: [κ]<ω → {0, 1} there is a set A of cardinality κ that is homogeneous for f. That is, for every n, f is constant on the subsets of cardinality n from A. A cardinal κ is called ineffably Ramsey if A can be chosen to be stationary subset of κ.
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In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence the theory contains either the sentence or its negation but not both. According to this test, if a satisfiable theory is κ-categorical (there exists an infinite cardinal κ such that it has only one model up to isomorphism of cardinality κ, with κ at least equal to the cardinality of its language) and in addition it has no finite model, then it is complete. This theorem was proved independently by and , after whom it is named.
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The complement of A forms a vertex cover in G with the same cardinality as this matching. This connection to bipartite matching allows the width of any partial order to be computed in polynomial time. More precisely, n-element partial orders of width k can be recognized in time O(kn2) .
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In mathematics, the bondage number of a nonempty graph is the cardinality of the smallest set E of edges such that the domination number of the graph with the edges E removed is strictly greater than the domination number of the original graph. The concept was introduced by Fink et. al.
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The field C has the following three properties: first, it has characteristic 0. This means that for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Third, it is algebraically closed (see above).
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The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied for its own sake as part of set theory.
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For example, the set of 3-tuples of elements from a 2-element set has cardinality . In cardinal arithmetic, κ0 is always 1 (even if κ is an infinite cardinal or zero). Exponentiation of cardinal numbers is distinct from exponentiation of ordinal numbers, which is defined by a limit process involving transfinite induction.
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First, one extends the signature by adding a new constant symbol for every element of M. The complete theory of M for the extended signature σ' is called the elementary diagram of M. In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for any two distinct new constant symbols c and c'. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly κ. It contains an isomorphic copy of M as an elementary substructure.
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Formally, assuming the axiom of choice, the cardinality of a set X is the least ordinal α such that there is a bijection between X and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed we need to do something different. The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the set of all sets that are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems.
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With Theodore Slaman, Groszek showed that (if they exist at all) non-constructible real numbers must be widespread, in the sense that every perfect set contains one of them, and they asked analogous questions of the non-computable real numbers. With Slaman, she has also shown that the existence of a maximally independent set of Turing degrees, of cardinality less than the cardinality of the continuum, is independent of ZFC. In the theory of ordinal definable sets, an unordered pair of sets is said to be a Groszek–Laver pair if the pair is ordinal definable but neither of its two elements is; this concept is named for Groszek and Richard Laver, who observed the existence of such pairs in certain models of set theory.
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It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non- triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ. If this were the case, then we could identify κ with some collection of 0-1 sequences of length λ. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1\.
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Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces and are both Hilbert spaces. In fact, by choosing a Hilbert basis (i.e., a maximal orthonormal subset of or any Hilbert space), one sees that all Hilbert spaces are isometric to , where is a set with an appropriate cardinality.
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K Nearest Neighbor (k-NN) query takes the cardinality of the input set as an input parameter. For a given query object Q ∈ D and an integer k ≥ 1, the k-NN query NN(Q, k) selects the k indexed objects which have the shortest distance from Q, according to the distance function d.
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The index is 4. In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint equal-size pieces called cosets. There are two types of cosets: left cosets and right cosets. Cosets (of either type) have the same number of elements (cardinality) as does .
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The surrogate key has no intrinsic (inherent) meaning, but rather is useful through its ability to uniquely identify a tuple. Another common occurrence, especially in regard to N:M cardinality is the composite key. A composite key is a key made up of two or more attributes within a table that (together) uniquely identify a record.
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A permutation set with positive entries is equivalent to a perfect matching in the positivity graph. A perfect matching in a bipartite graph can be found in polynomial time, e.g. using any algorithm for maximum cardinality matching. Kőnig's theorem is equivalent to the following: > The positivity graph of any bistochastic matrix admits a perfect matching.
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Overall, the index is organized as a B+tree. When the column cardinality is low, each leaf node of the B-tree would contain long list of RIDs. In this case, it requires less space to represent the RID-lists as bitmaps. Since each bitmap represents one distinct value, this is the basic bitmap index.
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Girths of real linear matroids also arise in compressed sensing, where the same concept is referred to as the spark of a matrix.. The girth of a binary matroid gives the cardinality of a minimum even set, a subcollection of a family of sets that includes an even number of copies of each set element.
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Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph.Moore 1982, p. 51.
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Because of this impracticality, two variant notions of min-wise independence have been introduced: restricted min- wise independent permutations families, and approximate min-wise independent families. Restricted min-wise independence is the min-wise independence property restricted to certain sets of cardinality at most .. Approximate min- wise independence has at most a fixed probability of varying from full independence..
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The rank of a set X\subset T in a gammoid defined from a graph G and vertex subsets S and T is, by definition, the maximum number of vertex-disjoint paths from S to X. By Menger's theorem, it also equals the minimum cardinality of a set Y that intersects every path from S to X.
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This is equivalent to the statement that the number of subsets (the cardinality of the power set) of an n-element set is 2^n, as can be seen by observing that the number of subsets is the sum of the number of combinations of each of the possible lengths, which range from zero through to n.
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Vastupurushamandala is a yantra. The design lays out a Hindu temple in a symmetrical, self-repeating structure derived from central beliefs, myths, cardinality and mathematical principles. The four cardinal directions help create the axis of a Hindu temple, around which is formed a perfect square in the space available. The circle of mandala circumscribes the square.
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The continuum hypothesis postulates that the cardinality of the continuum is equal to \aleph_1, which is regular. Without the axiom of choice, there would be cardinal numbers that were not well- orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore, only the aleph numbers can meaningfully be called regular or singular cardinals.
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In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers . In model theory, the notion of a categorical theory is refined with respect to cardinality.
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In fact, for any infinite cardinal κ, every κ-Suslin tree is a κ-Aronszajn tree (the converse does not hold). The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of height ω1 has an antichain of cardinality ω1 or a branch of length ω1.
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Then Nakayama's lemma says that M has a minimal generating set whose cardinality is \dim_k M / mM = \dim_k M \otimes_R k. If M is flat, then this minimal generating set is linearly independent (so M is free). See also: minimal resolution. A more refined information is obtained if one considers the relations between the generators; cf.
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Theorem. Let X be a Polish space, that is, a topological space such that there is a metric d on X that defines the topology of X and that makes X a complete separable metric space. Then X as a Borel space is Borel isomorphic to one of (1) R, (2) Z or (3) a finite space. (This result is reminiscent of Maharam's theorem.) It follows that a standard Borel space is characterized up to isomorphism by its cardinality, and that any uncountable standard Borel space has the cardinality of the continuum. Borel isomorphisms on standard Borel spaces are analogous to homeomorphisms on topological spaces: both are bijective and closed under composition, and a homeomorphism and its inverse are both continuous, instead of both being only Borel measurable.
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This follows immediately from the definition of a strongly compact cardinal as being a cardinal such that every collection of formulae of infinitary logic each of length smaller than , that is satisfiable for every subcollection of fewer than formulae, is globally satisfiable. See e.g. . The original De Bruijn–Erdős theorem is the case of this generalization, since a set is finite if and only if its cardinality is less than . However, some assumption such as the one of being a strongly compact cardinal is necessary: if the generalized continuum hypothesis is true, then for every infinite cardinal , there exists a graph of cardinality such that the chromatic number of is greater than , but such that every subgraph of whose vertex set has smaller power than has chromatic number at most .
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In a relational model, a relation is the cohesion of attributes that are fully and not of every key in that relation. The coupling between the relations is based on accordant attributes. For every relation, a rectangle has to be drawn and every coupling is illustrated by a line that connects the relations. On the edge of each line, arrows indicate the cardinality.
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In that year he published the Peano axioms, a formal foundation for the collection of natural numbers. The next year, the University of Turin also granted him his full professorship. The Peano curve was published in 1890 as the first example of a space-filling curve which demonstrated that the unit interval and the unit square have the same cardinality.
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That is, a complete type specifies the subgraph that a particular set of vertex variables induces. A saturated model is a model that realizes all of the types that have a number of variables at most equal to the cardinality of the model. The Rado graph has induced subgraphs of all finite or countably infinite types, so it is saturated.
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Finally if G is a divisible group and R is a real closed field, then R((G)) is a real closed field, and if R is algebraically closed, then so is R((G)). This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.
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Multiple studies by Irene Pepperberg and her colleagues suggested that Grey parrots (Psittacus erithacus) have some capacity for recognizing numbers or number- like concepts, appearing to understand ordinality and cardinality of numerals. Recent experiments also indicated that, given some language training and capacity for referencing recognized objects, they also have some ability to make inferences about probabilities and hidden object type ratios.
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This group consists of ‘Relation’, and ‘Relation Type’ for expressing declarative knowledge, and ‘Function’ and ‘Function Type’ for expressing procedural knowledge. This group is to express qualitative and quantitative relations among the various instances stored in the knowledge base. While instantiating the predicates can be characterized by their logical properties of relations, quantifiers and cardinality as monadic predicates of these predicate objects.
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For this, one usually chooses an order on . Then, sorting a subset of is equivalent to convert it into an increasing sequence. The lexicographic order on the resulting sequences induces thus an order on the subsets, which is also called the lexicographical order. In this context, one generally prefer to sort first the subsets by cardinality, such as in the shortlex order.
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In set theory, the Hebrew aleph glyph is used as the symbol to denote the aleph numbers, which represent the cardinality of infinite sets. This notation was introduced by mathematician Georg Cantor. In older mathematics books, the letter aleph is often printed upside down by accident, partly because a Monotype matrix for aleph was mistakenly constructed the wrong way up.
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Furthermore, G can be taken of any given cardinality greater than or equal to ℵ2. It follows from the previously mentioned works of Schmidt, Huhn, Dobbertin, Goodearl, and Handelman that the ℵ2 bound is optimal in all three negative results above. As the ℵ2 bound suggests, infinite combinatorics are involved. The principle used is Kuratowski's Free Set Theorem, first published in 1951.
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Every categorical theory is complete. However, the converse does not hold. Any theory T categorical in some infinite cardinal is very close to being complete. More precisely, the Łoś–Vaught test states that if a satisfiable theory has no finite models and is categorical in some infinite cardinal at least equal to the cardinality of its language, then the theory is complete.
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Consider a matchmaking agency with a pool of men and women. Given the preferences of the candidates, the agency constructs a bipartite graph where there is an edge between a man and a woman iff they are compatible. The ultimate goal of the agency is to create as many compatible couples as possible, i.e., find a maximum-cardinality matching in this graph.
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For uncountable regular cardinals \kappa (and some other cardinals) this can be strengthened to \kappa\rightarrow(\kappa,\omega+1)^2; however, it is consistent that this strengthening does not hold for the cardinality of the continuum. The Erdős–Dushnik–Miller theorem has been called the first "unbalanced" generalization of Ramsey's theorem, and Paul Erdős's first significant result in set theory.
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Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite ones, and reducing the Hadwiger–Nelson problem on the chromatic number of the plane to a problem about finite graphs. It may be generalized from finite numbers of colors to sets of colors whose cardinality is a strongly compact cardinal.
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Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875,The Cantor Set Before Cantor Mathematical Association of America is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers. Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem.
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In his paper on the problem, Erdős credited an anonymous mathematician with the observation that, when each x is mapped to a finite set of values, F is necessarily finite. However, as Erdős showed, the situation for countable sets is more complicated: the answer to Wetzel's question is yes if and only if the continuum hypothesis is false.. That is, the existence of an uncountable set of functions that maps each argument x to a countable set of values is equivalent to the nonexistence of an uncountable set of real numbers whose cardinality is less than the cardinality of the set of all real numbers. One direction of this equivalence was also proven independently, but not published, by another UIUC mathematician, Robert Dan Dixon. It follows from the independence of the continuum hypothesis, proved in 1963 by Paul Cohen,.
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For every x in X, the fiber over x is a discrete subset of C. On every connected component of X, the fibers are homeomorphic. If X is connected, there is a discrete space F such that for every x in X the fiber over x is homeomorphic to F and, moreover, for every x in X there is a neighborhood U of x such that its full pre-image p−1(U) is homeomorphic to . In particular, the cardinality of the fiber over x is equal to the cardinality of F and it is called the degree of the cover . Thus, if every fiber has n elements, we speak of an n-fold covering (for the case , the covering is trivial; when , the covering is a double cover; when , the covering is a triple cover and so on).
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In other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, . In which case, the cardinality of the set is denoted \aleph_0 (aleph-null)—the first in the series of aleph numbers. This terminology is not universal. Some authors use countable to mean what is here called countably infinite, and do not include finite sets.
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He dealt with the problems of cardinality in set theory. During his visiting professorship in Halle, East Germany he contributed to the discovery of the mathematical achievements of Georg Cantor, too. He was the important scholar of the Debrecen algebraic school founded by Tibor Szele. At the Martin Luther University of Halle- Wittenberg he had a great role in the establishment of the Modern algebraic school.
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A binary matroid is graphic if and only if its minors do not include the dual of the graphic matroid of K_5 nor of K_{3,3}., Theorem 10.5.1, p. 176. If every circuit of a binary matroid has odd cardinality, then its circuits must all be disjoint from each other; in this case, it may be represented as the graphic matroid of a cactus graph.
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For example, if a contact record in a payroll database is classified as "former employee" then it must not have any associated salary payments after the separation date (cardinality = 0). ;Check digits :Used for numerical data. To support error detection, an extra digit is added to a number which is calculated from the other digits. ;Consistency checks :Checks fields to ensure data in these fields correspond, e.g.
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As is standard in set theory, we denote by \omega the least infinite ordinal, which has cardinality \aleph_0; it may be identified with the set of all natural numbers. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.
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In graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete.
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The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. As a topological space, the real numbers are separable. This is because the set of rationals, which is countable, is dense in the real numbers. The irrational numbers are also dense in the real numbers, however they are uncountable and have the same cardinality as the reals.
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He also proved that any element in an at most countable conical refinement monoid is measured by a unique (up to isomorphism) V-measure on a unique at most countable Boolean algebra. He raised there the problem whether any conical refinement monoid is measurable. This was answered in the negative by Friedrich Wehrung in 1998. The counterexamples can have any cardinality greater than or equal to ℵ2.
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There is a unique connected 0-dimensional manifold, namely the point, and disconnected 0-dimensional manifolds are just discrete sets, classified by cardinality. They have no geometry, and their study is combinatorics. A connected 1-dimensional manifold without boundary is either the circle (if compact) or the real line (if not). However, maps of 1-dimensional manifolds are a non-trivial area; see below.
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However, the computable numbers are rarely used in practice. One reason is that there is no algorithm for testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides in every case if this number is equal to zero or not. The set of computable numbers has the same cardinality as the natural numbers.
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The round-robin protocol guarantees EF1 when the items are goods (- valued positively by all agents) and when they are chores (- valued negatively by all agents). However, when there are both goods and chores, it does not guarantee EF1. An adaptation of round-robin called double round-robin guarantees EF1 even with a mixture of goods and chores. When agents have cardinality constraints (i.e.
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Namely, the worst case behavior of this algorithm might be very far from the optimal solution. The approximation algorithm is extended by the following way. First, define a modified greedy algorithm, that selects the set S_i that has the best ratio of weighted uncovered elements to cost. Second, among covers of cardinality 1, 2, ..., k-1, find the best cover that does not violate the budget.
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Towards this goal, the agency first chooses an edge in the graph, and suggests to the man and woman on both ends of the edge to meet. Now, the agency must take care to only choose a maximally-matchable edge. This is because, if it chooses a non-maximally- matchable edge, it may get stuck with an edge that cannot be completed to a maximum-cardinality matching.
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In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X may be written uniquely as the disjoint union of a perfect set and a countable set.
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Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, any field of characteristic zero and cardinality at most continuum is (ring-)isomorphic to a subfield of complex numbers).. Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field as an algebraic extension of a transcendental extension of its prime field, from which the result follows immediately. The p-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic pk, as k → ∞. For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero.
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The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object. Every space filling curve hits some points multiple times, and does not have a continuous inverse.
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Basic invariants of a field include the characteristic and the transcendence degree of over its prime field. The latter is defined as the maximal number of elements in that are algebraically independent over the prime field. Two algebraically closed fields and are isomorphic precisely if these two data agree. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic.
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For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : File:Maximum-matching-labels.svg In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cover.
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One can find several finite matrix fields of characteristic p for any given prime number p. In general, corresponding to each finite field there is a matrix field. Since any two finite fields of equal cardinality are isomorphic, the elements of a finite field can be represented by matrices. Contrary to the general case for matrix multiplication, multiplication is commutative in a matrix field (if the usual operations are used).
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In set theory, Jensen's covering theorem states that if 0# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universe is close to the universe of all sets. The first proof appeared in . Silver later gave a fine structure free proof using his machines and finally gave an even simpler proof.
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It is a variation of the "crows foot" style of data modelling that was favoured by many over the original Chen style of ERD modelling because of its readability and efficient use of drawing space. The notation has features that represent the properties of relationships including cardinality and optionality (the crows foot and dashing of lines), exclusion (the exclusion arc), recursion (looping structures) and use of abstraction (nested boxes).
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When G is bipartite, there are no odd cycles in G. In that case, blossoms will never be found and one can simply remove lines B20 - B24 of the algorithm. The algorithm thus reduces to the standard algorithm to construct maximum cardinality matchings in bipartite graphs where we repeatedly search for an augmenting path by a simple graph traversal: this is for instance the case of the Ford–Fulkerson algorithm.
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The fundamental elements of WebML structure model are entities, which are containers of data elements, and relationships, which enable the semantic connection of entities. Entities have named attributes, with an associated type; properties with multiple occurrences can be organized by means of multi-valued components, which corresponds to the classical part-of relationship. Entities can be organized in generalization hierarchies. Relationships may be given cardinality constraints and role names.
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Using the axiom of choice, a lot of small cardinals (the \aleph_n, for instance) can be proved to be not Jónsson. Results like this need the axiom of choice, however: The axiom of determinacy does imply that for every positive natural number n, the cardinal \aleph_n is Jónsson. A Jónsson algebra is an algebra with no proper subalgebras of the same cardinality. (They are unrelated to Jónsson–Tarski algebras).
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A second difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power set is a set of sets, it would necessarily be a subset of the set of all sets, provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.
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An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above. An infinite set may have the same cardinality as a proper subset of itself, as the depicted bijection f(x)=2x from the natural to the even numbers demonstrates. Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows.
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The empty set is the set containing no elements. In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.
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The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension. For the non-free case, this generalizes to the notion of the length of a module.
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In order to state the paradox it is necessary to understand that the cardinal numbers admit an ordering, so that one can speak about one being greater or less than another. Then Cantor's paradox is: :Theorem: There is no greatest cardinal number. This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set. :Proof: Assume the contrary, and let C be the largest cardinal number.
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Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n2 < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).
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For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum. Since the real line is infinite, any theory satisfied by the real line is also satisfied by some nonstandard models. When the Löwenheim–Skolem theorem is applied to first-order set theories, the nonintuitive consequences are known as Skolem's paradox.
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2, p. 50. In logic, a theory that has only one model (up to isomorphism) with a given infinite cardinality is called -categorical. The fact that the Rado graph is the unique countable graph with the extension property implies that it is also the unique countable model for its theory. This uniqueness property of the Rado graph can be expressed by saying that the theory of the Rado graph is ω-categorical.
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Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (\aleph_0).
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In case that mappings arise and arrive on finite sets (discrete bounded value signals), this condition is equivalent to saying that mappings are injective (one-to-one). Moreover, if a mapping goes from one set to a set of the same cardinality, it should be bijective. In the Generalized Lifting Scheme the addition/subtraction restriction is avoided by including this step in the mapping. In this way the Classical Lifting Scheme is generalized.
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In object-oriented modeling these kinds of relations are typically modeled as object properties. In this example, the class would have a property called . would be typed to hold a collection of objects, such as instances of , , , etc. Object modeling languages such as UML include capabilities to model various aspects of "part of" and other kinds of relations – data such as the cardinality of the objects, constraints on input and output values, etc.
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In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.
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Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces. Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.
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There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers". Non-algebraic numbers are called transcendental numbers. Specific examples of transcendental numbers include π and Euler's number e.
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A fundamental problem in combinatorial optimization is finding a maximum matching. This problem has various algorithms for different classes of graphs. In an unweighted bipartite graph, the optimization problem is to find a maximum cardinality matching. The problem is solved by the Hopcroft-Karp algorithm in time time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.
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In computer science, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is the assignment problem, in which the input is restricted to be a bipartite graph. Another special case is the problem of finding a maximum cardinality matching on an unweighted graph: this corresponds to the case where all edge weights are the same.
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Let G = (V,E) be a graph, where V are the vertices and E are the edges. A matching in G is a subset M of E, such that each vertex in V is adjacent to at most a single edge in M. A maximum matching is a matching of maximum cardinality. An edge e in E is called maximally-matchable (or allowed) if there exists a maximum matching M that contains e.
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Hemitonia is also quantified by the number of semitones present. Unhemitonic scales have only one semitone; dihemitonic scales have 2 semitones; trihemitonic scales have 3 semitones, etc. In the same way that an anhemitonic scale is less dissonant than a hemitonic scale, an anhemitonic scale is less dissonant than a dihemitonic scale. The qualification of cohemitonia versus ancohemitonia combines with the cardinality of semitones, giving terms like: dicohemitonic, triancohemitonic, and so forth.
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In that case, Jensen's covering lemma holds: :For every uncountable set x of ordinals there is a constructible y such that x ⊂ y and y has the same cardinality as x. This deep result is due to Ronald Jensen. Using forcing it is easy to see that the condition that x is uncountable cannot be removed. For example, consider Namba forcing, that preserves \omega_1 and collapses \omega_2 to an ordinal of cofinality \omega.
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A subset S of M is said to be a generator of M if M is the smallest set containing S that is closed under the monoid operation, or equivalently M is the result of applying the finitary closure operator to S. If there is a generator of M that has finite cardinality, then M is said to be finitely generated. Not every set S will generate a monoid, as the generated structure may lack an identity element.
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This is an eponymous album as he used one of his stage names, Aleph-1. The concept of the album and its name, Aleph-1, derive from the theories of German mathematician Georg Cantor, who was a teacher in Halle, Saxony-Anhalt, Germany, a city, to which Alva Noto is deeply connected with through his family. In mathematical terms, \aleph_1 is the cardinality of the set of all countable ordinal numbers or a number of elements in endless successions.
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John von Neumann In 1929, von Neumann published an article containing the axioms that would lead to NBG. This article was motivated by his concern about the consistency of the axiom of limitation of size. He stated that this axiom "does a lot, actually too much." Besides implying the axioms of separation and replacement, and the well- ordering theorem, it also implies that any class whose cardinality is less than that of V is a set.
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If he finds such a , then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships: (a) , and (b) = . Any two triplets obeying these relationships must be related by a linear transformation; they represent utility indices differing only by scale and origin. In this case, "cardinality" means nothing more being able to give consistent answers to these particular questions. Note that this experiment does not require measurability of utility.
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The norm of an algebraic integer is again an integer, because it is equal (up to sign) to the constant term of the characteristic polynomial. In algebraic number theory one defines also norms for ideals. This is done in such a way that if I is a nonzero ideal of OK, the ring of integers of the number field K, N(I) is the number of residue classes in O_K / I – i.e. the cardinality of this finite ring.
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The links can have a cardinality of 1:1, 1:many, many:1 or many:many and use the OIDs to speed up the navigation of networks of objects. The OIDs are also used in support of scalable collections (tree, list, set etc.), indices and hash tables. Eliminating the relational Join operations inherent in a relational database gives Objectivity/DB a performance advantage. Objectivity/DB is also different from RDBMSs in the way in which it handles queries.
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Zermelo's paper is notable for what may be the first mention of Cantor's theorem explicitly and by name. This appeals strictly to set theoretical notions, and is thus not exactly the same as Cantor's diagonal argument. Cantor's theorem: "If M is an arbitrary set, then always M < P(M) [the power set of M]. Every set is of lower cardinality than the set of its subsets". Zermelo proves this by considering a function φ: M → P(M).
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Here an algebra means a model for a language with a countable number of function symbols, in other words a set with a countable number of functions from finite products of the set to itself. A cardinal is a Jónsson cardinal if and only if there are no Jónsson algebras of that cardinality. The existence of Jónsson functions shows that if algebras are allowed to have infinitary operations, then there are no analogues of Jónsson cardinals.
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In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the capacity (complexity, expressive power, richness, or flexibility) of a set of functions that can be learned by a statistical binary classification algorithm. It is defined as the cardinality of the largest set of points that the algorithm can shatter. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be.
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In the case of triangles, one has a degenerate triangle if at least one side length or angle is zero (equiv., it becomes a "line segment"). Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one.
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The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension.
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5, Princeton University Press, Princeton 1941, 122 pp. The relative simplicity of Post's lattice is in stark contrast to the lattice of clones on a three-element (or larger) set, which has the cardinality of the continuum, and a complicated inner structure. A modern exposition of Post's result can be found in Lau (2006).D. Lau, Function algebras on finite sets: Basic course on many-valued logic and clone theory, Springer, New York, 2006, 668 pp.
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The following result, proved by Ploščica, Tůma, and Wehrung in 1998, is more striking, because it shows examples of representable semilattices that do not satisfy Schmidt's Condition. We denote by FV(Ω) the free lattice on Ω in V, for any variety V of lattices. Theorem (Ploščica, Tůma, and Wehrung 1998). The semilattice Conc FV(Ω) does not satisfy WURP, for any set Ω of cardinality at least ℵ2 and any non- distributive variety V of lattices.
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Additionally, they appear to understand that adding or > subtracting one from a set will change the quantity of that set, though the > generality of this knowledge is difficult to assess without the ability to > label sets of arbitrary cardinality using number words. (emphasis > added)Michael C. Frank, Daniel L. Everett, Evelina Fedorenko and Edward > Gibson (2008), Number as a cognitive technology: Evidence from Pirahã > language and cognition. Cognition, Volume 108, Issue 3, September 2008, pp. > 819–824.
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In this case, we start with bases uα and wβ of L/K and M/L respectively, where α is taken from an indexing set A, and β from an indexing set B. Using an entirely similar argument as the one above, we find that the products uαwβ form a basis for M/K. These are indexed by the cartesian product A × B, which by definition has cardinality equal to the product of the cardinalities of A and B.
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In 1973 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let each datum be a finite binary string and a model be a finite set of binary strings. Consider model classes consisting of models of given maximal Kolmogorov complexity. The Kolmogorov structure function of an individual data string expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data.
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Dehne, Fellows and Rosamond presented an algorithm that solves it in time O(f(k) n^c) for some function f and constant c. When each element of F is restricted to be of cardinality exactly k, the decision variant is called Ek-Set Splitting and the optimization version Max Ek-Set Splitting. For k > 2 the former remains NP complete, and for k ≥ 2 the latter remains APX complete. For k ≥ 4, Ek-Set Splitting is approximation resistant.
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In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality.
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The case leading to the Iwahori–Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel subgroup. Iwahori showed that the Hecke ring :H(G//B) is obtained from the generic Hecke algebra Hq of the Weyl group W of G by specializing the indeterminate q of the latter algebra to pk, the cardinality of the finite field. George Lusztig remarked in 1984 (Characters of reductive groups over a finite field, xi, footnote): Iwahori and Matsumoto (1965) considered the case when G is a group of points of a reductive algebraic group over a non- archimedean local field F, such as Qp, and K is what is now called an Iwahori subgroup of G. The resulting Hecke ring is isomorphic to the Hecke algebra of the affine Weyl group of G, or the affine Hecke algebra, where the indeterminate q has been specialized to the cardinality of the residue field of F.
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These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel, are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. In 1910, the first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory, which Russell and Whitehead developed in an effort to avoid the paradoxes.
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First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse. Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality.
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Large cardinals are cardinal numbers with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an inaccessible cardinal, already implies the consistency of ZFC. Despite the fact that large cardinals have extremely high cardinality, their existence has many ramifications for the structure of the real line. Determinacy refers to the possible existence of winning strategies for certain two-player games (the games are said to be determined).
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The hitting set problem is W[2]-complete for the parameter OPT, that is, it is unlikely that there is an algorithm that runs in time f(OPT)nO(1) where OPT is the cardinality of the smallest hitting set. The hitting set problem is fixed-parameter tractable for the parameter OPT + d, where d is the size of the largest edge of the hypergraph. More specifically, there is an algorithm for hitting set that runs in time dOPTnO(1).
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A problem with the Flajolet–Martin algorithm in the above form is that the results vary significantly. A common solution has been to run the algorithm multiple times with k different hash functions and combine the results from the different runs. One idea is to take the mean of the k results together from each hash function, obtaining a single estimate of the cardinality. The problem with this is that averaging is very susceptible to outliers (which are likely here).
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In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.. The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete..
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The word problem for a finitely generated group is the decision problem whether two words in the generators of the group represent the same element. The word problem for a given finitely generated group is solvable if and only if the group can be embedded in every algebraically closed group. The rank of a group is often defined to be the smallest cardinality of a generating set for the group. By definition, the rank of a finitely generated group is finite.
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The set of all limit ordinals \alpha<\kappa is closed unbounded in \kappa . In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous). More generally, if X is a nonempty set and \lambda is a cardinal, then C\subseteq[X]^\lambda is club if every union of a subset of C is in C and every subset of X of cardinality less than \lambda is contained in some element of C (see stationary set).
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Felix stays there after Kathy leaves him; the hotel is full, but Felix has the desk clerk move everybody one room up, leaving an empty room for him. He falls in with a loquacious beetle named "Franx", reminiscent of Franz Kafka's The Metamorphosis, which is mentioned in Rucker's Afterword. The two decide to climb "Mount On", which itself is infinite (not aleph-null infinite, but perhaps instead cardinality of the continuum or greater). After many adventures, Franx and Felix find Kathy.
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The differential field Q(t) fails to have a solution to the differential equation : \partial(u) = u but expands to a larger differential field including the function et which does have a solution to this equation. A differential field with solutions to all systems of differential equations is called a differentially closed field. Such fields exist, although they do not appear as natural algebraic or geometric objects. All differential fields (of bounded cardinality) embed into a large differentially closed field.
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In mathematics, aleph numbers denote the cardinality (or size) of infinite sets, as originally described by Georg Cantor in his first set theory article in 1874. This relates to the theme of infinity present in Borges' story. The aleph recalls the monad as conceptualized by Gottfried Wilhelm Leibniz, the 17th-century philosopher and mathematician. Just as Borges' aleph registers the traces of everything else in the universe, so Leibniz' monad is a mirror onto every other object of the world.
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Extract of page 616 For example, if a line is viewed as the set of all of its points, their infinite number (i.e. the cardinality of the line) is larger than the number of integers. In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets.
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When two compact Hausdorff spaces and are homeomorphic, the Banach spaces and are isometric. Conversely, when is not homeomorphic to , the (multiplicative) Banach–Mazur distance between and must be greater than or equal to , see above the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor.
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In a nontrivial bipartite graph, the optimal number of colors is (by definition) two, and (since bipartite graphs are triangle-free) the maximum clique size is also two. Also, any induced subgraph of a bipartite graph remains bipartite. Therefore, bipartite graphs are perfect. In n-vertex bipartite graphs, a minimum clique cover takes the form of a maximum matching together with an additional clique for every unmatched vertex, with size n − M, where M is the cardinality of the matching.
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A minimal Hall violator is a Hall violator such that each of its subsets is not a Hall violator. The above algorithm, in fact, finds a minimal Hall violator. This is because, if any vertex is removed from W, then the remaining vertices can be perfectly matched to the vertices of NG(W) (either by edges of M, or by edges of the M-alternating path from x0). Note: the above algorithm does not necessarily find a minimum-cardinality Hall violator.
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The length of a branch is the ordinal that is order isomorphic to the branch. For each ordinal α, the α-th level of T is the set of all elements of T of height α. A tree is a κ-tree, for an ordinal number κ, if and only if it has height κ and every level has size less than the cardinality of κ. The width of a tree is the supremum of the cardinalities of its levels.
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For example, Lω1ω permits countable conjunctions and disjunctions. The set of free variables in a formula of Lκω can have any cardinality strictly less than κ, yet only finitely many of them can be in the scope of any quantifier when a formula appears as a subformula of another.Some authors only admit formulas with finitely many free variables in Lκω, and more generally only formulas with < λ free variables in Lκλ. In other infinitary logics, a subformula may be in the scope of infinitely many quantifiers.
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Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.
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In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number. Some authors use countable set to mean countably infinite alone.
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So, this w does not lie in the span of the dual set. The dual of an infinite-dimensional space has greater dimensionality (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space. Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist.
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The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number.
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In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, α) for the number of models of T (up to isomorphism) of cardinality α. The spectrum problem is to describe the possible behaviors of I(T, α) as a function of α. It has been almost completely solved for the case of a countable theory T.
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The confusion between cardinality and measurability was not to be solved until the works of Armen Alchian, William Baumol, and John Chipman. The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature at the time. It is helpful to consider the same problem as it appears in the construction of scales of measurement in the natural sciences. In the case of temperature there are two degrees of freedom for its measurement - the choice of unit and the zero.
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A finite group G of cardinality n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of G on the group ring K[G] by left (or right) multiplication is linear and faithful. The finite groups of Lie type (classical groups over finite fields) are an important family of finite simple group, as they take up most of the slots in the classification of finite simple groups.
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At the top, an allegorical female figure carrying a horn of plenty represents Canada. Below, the children symbolize the seven provinces that made up Canada at the time. The bronze is housed under a stone baldachin replete with copper bas reliefs of industrial and agricultural trades practised in the Dominion he first commanded. While the plaza is arranged along the skewed cardinality characteristic of Montreal, Macdonald looks west-northwest, under a canopy created by trades, at the vast expanse awaiting the command coming from Montreal.
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A junk dimension is a convenient grouping of typically low-cardinality flags and indicators. By creating an abstract dimension, these flags and indicators are removed from the fact table while placing them into a useful dimensional framework.Ralph Kimball, Margy Ross, The Data Warehouse Toolkit: The Complete Guide to Dimensional Modeling, Second Edition, Wiley Computer Publishing, 2002. , Pages 202, 405 A Junk Dimension is a dimension table consisting of attributes that do not belong in the fact table or in any of the existing dimension tables.
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There are more extreme examples showing that second-order logic with standard semantics is more expressive than first-order logic. There is a finite second-order theory whose only model is the real numbers if the continuum hypothesis holds and which has no model if the continuum hypothesis does not hold (cf. Shapiro 2000, p. 105). This theory consists of a finite theory characterizing the real numbers as a complete Archimedean ordered field plus an axiom saying that the domain is of the first uncountable cardinality.
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In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable and quasi-finite, and for every subfield E, every absolutely entire E-algebra (regular field extension of E) of smaller cardinality than F can be embedded in F. They were introduced by . Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.
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A well-known application of the inclusion–exclusion principle is to the combinatorial problem of counting all derangements of a finite set. A derangement of a set A is a bijection from A into itself that has no fixed points. Via the inclusion–exclusion principle one can show that if the cardinality of A is n, then the number of derangements is [n! / e] where [x] denotes the nearest integer to x; a detailed proof is available here and also see the examples section above.
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An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0\. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable. Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.
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Since each Uai intersects {Ga} for only finitely many values of a, the union of all such Uai intersects the collection {Ga} for only finitely many values of a. It follows that X (the whole space!) intersects the collection {Ga} at only finitely many values of a, contradicting the infinite cardinality of the collection {Ga}. A topological space in which every open cover admits a locally finite open refinement is called paracompact. Every locally finite collection of subsets of a topological space X is also point-finite.
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The set A is a set of generators of the numerical semigroup ⟨ A ⟩. A set of generators of a numerical semigroup is a minimal system of generators if none of its proper subsets generates the numerical semigroup. It is known that every numerical semigroup S has a unique minimal system of generators and also that this minimal system of generators is finite. The cardinality of the minimal set of generators is called the embedding dimension of the numerical semigroup S and is denoted by e(S).
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It is V_\omega \\!, the class of hereditarily finite sets, with the inherited membership relation. Note that if the axiom of the empty set is not taken as a part of this system (since it can be derived from ZF + Infinity), then the empty domain also satisfies ZFC – Infinity + ¬Infinity, as all of its axioms are universally quantified, and thus trivially satisfied if no set exists. The cardinality of the set of natural numbers, aleph null (\aleph_0), has many of the properties of a large cardinal.
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Unless the module is finitely-generated, there may exist no minimal generating set. The cardinality of a minimal generating set need not be an invariant of the module; Z is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set }. What is uniquely determined by a module is the infimum of the numbers of the generators of the module. Let R be a local ring with maximal ideal m and residue field k and M finitely generated module.
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The contact graph of an arbitrary finite packing of unit balls is the graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. The cardinality of the edge set of the contact graph gives the number of touching pairs, the number of 3-cycles in the contact graph gives the number of touching triplets, and the number of tetrahedrons in the contact graph gives the number of touching quadruples (in general for a contact graph associated with a sphere packing in n dimensions that the cardinality of the set of n-simplices in the contact graph gives the number of touching (n + 1)-tuples in the sphere packing). In the case of 3-dimensional Euclidean space, non-trivial upper bounds on the number of touching pairs, triplets, and quadruples were proved by Karoly Bezdek and Samuel Reid at the University of Calgary. The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the "sticky-sphere problem".
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In mathematics, an unfoldable cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j(κ) ≥ λ. A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ. A cardinal number κ is strongly λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus- power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model "N" with the critical point of j being κ, j(κ) ≥ λ, and V(λ) is a subset of N. Without loss of generality, we can demand also that N contains all its sequences of length λ.
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If we order the integers in the interval [1, 2n] by divisibility, the subinterval [n + 1, 2n] forms an antichain with cardinality n. A partition of this partial order into n chains is easy to achieve: for each odd integer m in [1,2n], form a chain of the numbers of the form m2i. Therefore, by Dilworth's theorem, the width of this partial order is n. The Erdős–Szekeres theorem on monotone subsequences can be interpreted as an application of Dilworth's theorem to partial orders of order dimension two .
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If a(F) and b(F) are identically distributed for all search algorithms a and b, then F has an NFL distribution. This condition holds if and only if F and F o j are identically distributed for all j in J. In other words, there is no free lunch for search algorithms if and only if the distribution of objective functions is invariant under permutation of the solution space.The "only if" part was first published by Set-theoretic NFL theorems have recently been generalized to arbitrary cardinality X and Y.
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In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G = (V, E), a perfect matching in G is a subset M of E, such that every vertex in V is adjacent to exactly one edge in M. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true.
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Since the solid tori are chosen to become arbitrarily small as the iteration number increases, the connected components of A must be single points. It is then easy to verify that A is closed, dense-in-itself, and totally disconnected, having the cardinality of the continuum. This is sufficient to conclude that as an abstract metric space A is homeomorphic to the Cantor set. However, as a subset of Euclidean space A is not ambiently homeomorphic to the standard Cantor set C, embedded in R3 on a line segment.
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Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as is a singleton as it contains a single element (which itself is a set, however, not a singleton). A set is a singleton if and only if its cardinality is .
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In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of GrushkoI. A. Grushko, On the bases of a free product of groups, Matematicheskii Sbornik, vol 8 (1940), pp. 169-182. and then, independently, in a 1943 article of Neumann.
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Find the cardinality (denoted by Ak(m)) of the largest subset of {1, 2, ..., m} which contains no progression of k distinct terms. The elements of the forbidden progressions are not required to be consecutive. For example, A4(10) = 8, because {1, 2, 3, 5, 6, 8, 9, 10} has no arithmetic progressions of length 4, while all 9-element subsets of {1, 2, ..., 10} have one. Paul Erdős set a $1000 prize for a question related to this number, collected by Endre Szemerédi for what has become known as Szemerédi's theorem.
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These sets are then taken to "be" cardinal numbers, by definition. In Zermelo-Fraenkel set theory with the axiom of choice, one way of assigning representatives to cardinal numbers is to associate each cardinal number with the least ordinal number of the same cardinality. These special ordinals are the ℵ numbers. But if the axiom of choice is not assumed, for some cardinal numbers it may not be possible to find such an ordinal number, and thus the cardinal numbers of those sets have no ordinal number as representatives.
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The first one to theorize about the marginal value of money was Daniel Bernoulli in 1738. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns. Since Bernoulli tacitly assumed that an interpersonal measure for the utility reaction of different persons can be discovered, he was then inadvertedly using an early conception of cardinality. Bernoulli's imaginary logarithmic utility function and Gabriel Cramer's function were conceived at the time not for a theory of demand but to solve the St. Petersburg's game.
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In mathematics, a ω1-tree is a tree of power ω1 and height ω1. A Jech–Kunen tree is a ω1-tree in which the number of branches is greater than ω1 and less than 2ω1. It is named after who found the first model in which this tree exists, and who showed that, assuming the continuum hypothesis and 2ω1 > ω2, the existence of a Jech–Kunen tree is equivalent to the existence of a compact Hausdorff space with weight ω1 and cardinality strictly between ω1 and 2ω1.
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By 1889, Windsor Station would take up its prominent position at the corner of Peel and De la Gauchetière, with the plaza providing a path directly to the corner opposite. In 1895, the Macdonald Monument would be constructed under a stone baldachin in the centre of the northern third of the plaza. His baldachin is replete with copper bas reliefs of the various industrial and agricultural trades practised in the Dominion he first commanded. While the plaza is arranged along the skewed cardinality characteristic of Montreal, John A. Macdonald looks more west-northwest than north.
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These two papers by Hausmann and Korte both concerned the problem of finding a maximum cardinality independent set, which is easy for matroids but (as they showed) harder to approximate or compute exactly for more general independence systems represented by an independence oracle. This work kicked off a flurry of papers in the late 1970s and early 1980s showing similar hardness results for problems on matroids; ; ; ; . and comparing the power of different kinds of matroid oracles. Since that time, the independence oracle has become standard for most research on matroid algorithms.E.g.
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Set covering is equivalent to the hitting set problem. That is seen by observing that an instance of set covering can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, the universe represented by vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right- vertices. In the Hitting set problem, the objective is to cover the left- vertices using a minimum subset of the right vertices.
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Three iterations of a Peano curve construction, whose limit is a space-filling curve. In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890.. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve..
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A binary classification model f with some parameter vector \theta is said to shatter a set of data points (x_1,x_2,\ldots,x_n) if, for all assignments of labels to those points, there exists a \theta such that the model f makes no errors when evaluating that set of data points. The VC dimension of a model f is the maximum number of points that can be arranged so that f shatters them. More formally, it is the maximum cardinal D such that some data point set of cardinality D can be shattered by f.
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The width of a decomposition method is a measure of the size of problem it produced. Originally, the width was defined as the maximal cardinality of the sets of original variables; one method, the hypertree decomposition, uses a different measure. Either way, the width of a decomposition is defined so that decompositions of size bounded by a constant do not produce excessively large problems. Instances having a decomposition of fixed width can be translated by decomposition into instances of size bounded by a polynomial in the size of the original instance.
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Let C be a cyclic group generated by an element c of order n. Suppose C acts on a set X. Let X(q) be a polynomial with integer coefficients. Then the triple (X, X(q), C) is said to exhibit the cyclic sieving phenomenon (CSP) if for all integers d, the value X(e2id/n) is the number of elements fixed by cd. In particular X(1) is the cardinality of the set X, and for that reason X(q) is regarded as a generating function for X.
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In his work on set theory, Georg Cantor denoted the collection of all cardinal numbers by the last letter of the Hebrew alphabet, ' (transliterated as Tav', Taw, or Sav.) As Cantor realized, this collection could not itself have a cardinality, as this would lead to a paradox of the Burali-Forti type. Cantor instead said that it was an "inconsistent" collection which was absolutely infinite.The Correspondence between Georg Cantor and Philip Jourdain, I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 73 (1971/72), pp. 111-130, at pp. 116-117.
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More generally, in musical set theory partitioning is the division of the domain of pitch class sets into types, such as transpositional type, see equivalence class and cardinality. Partition is also an old name for types of compositions in several parts; there is no fixed meaning, and in several cases the term was reportedly interchanged with various other terms. A cross-partition is, "a two-dimensional configuration of pitch classes whose columns are realized as chords, and whose rows are differentiated from one another by registral, timbral, or other means."Alegant (2001), p.1.
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In graph theory, a maximally-matchable edge in a graph is an edge that is included in at least one maximum-cardinality matching in the graph. An alternative term is allowed edge. A fundamental problem in matching theory is: given a graph G, find the set of all maximally-matchable edges in G. This is equivalent to finding the union of all maximum matchings in G (this is different than the simpler problem of finding a single maximum matching in G). Several algorithms for this problem are known.
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Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice. The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. Dimension theorem for vector spaces). It is called the dimension of the vector space, denoted by dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.
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In 1890, Peano discovered a continuous curve, now called the Peano curve, that passes through every point of the unit square (). His purpose was to construct a continuous mapping from the unit interval onto the unit square. Peano was motivated by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. The problem Peano solved was whether such a mapping could be continuous; i.e.
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Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width. Similarly, one can define the height of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned.
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Additionally, every well-covered graph is a critical graph for vertex covering in the sense that, for every vertex , deleting from the graph produces a graph with a smaller minimum vertex cover. The independence complex of a graph is the simplicial complex having a simplex for each independent set in . A simplicial complex is said to be pure if all its maximal simplices have the same cardinality, so a well-covered graph is equivalently a graph with a pure independence complex.For examples of papers using this terminology, see and .
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While the parameter k in the examples above is chosen as the size of the desired solution, this is not necessary. It is also possible to choose a structural complexity measure of the input as the parameter value, leading to so-called structural parameterizations. This approach is fruitful for instances whose solution size is large, but for which some other complexity measure is bounded. For example, the feedback vertex number of an undirected graph G is defined as the minimum cardinality of a set of vertices whose removal makes G acyclic.
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1\. The torus is not homeomorphic to R2 because their fundamental groups are not isomorphic (their fundamental groups don’t have the same cardinality). More generally, a simply connected space cannot be homeomorphic to a non-simply connected space; one has a trivial fundamental group and the other does not. 2\. The fundamental group of the unit circle is isomorphic to the group of integers. Therefore, the one-point compactification of R has a fundamental group isomorphic to the group of integers (since the one-point compactification of R is homeomorphic to the unit circle).
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Qvist's theorem: to the proof in case of n odd Qvist's theorem: to the proof in case of n even ;Proof: (a) Let be the tangent to at point and let be the remaining points of this line. For each , the lines through partition into sets of cardinality 2 or 1 or 0. Since the number is even, for any point , there must exist at least one more tangent through that point. The total number of tangents is , hence, there are exactly two tangents through each , and one other.
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The inversion number is the cardinality of inversion set. It is a common measure of the sortedness of a permutation or sequence. It is the number of crossings in the arrow diagram of the permutation, its Kendall tau distance from the identity permutation, and the sum of each of the inversion related vectors defined below. It does not matter if the place-based or the element-based definition of inversion is used to define the inversion number, because a permutation and its inverse have the same number of inversions.
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Special types of filters called ultrafilters have many useful properties that can significantly help in proving results. One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space . In fact, the class of nets in a given set is too large to even be a set (it is a proper class); this is because nets in can have domains of any cardinality. In contrast, filters, like topologies, are in a sense more "intrinsic" to the set .
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The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski (1948) established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. (He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic.) A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem, proved by Michael D. Morley (1965), states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e.
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Rather, they show entity sets (all entities of the same entity type) and relationship sets (all relationships of the same relationship type). Examples: a particular song is an entity; the collection of all songs in a database is an entity set; the eaten relationship between a child and his lunch is a single relationship; the set of all such child-lunch relationships in a database is a relationship set. In other words, a relationship set corresponds to a relation in mathematics, while a relationship corresponds to a member of the relation. Certain cardinality constraints on relationship sets may be indicated as well.
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The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers.
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The situation for complete lattices with complete homomorphisms obviously is more intricate. In fact, free complete lattices do generally not exist. Of course, one can formulate a word problem similar to the one for the case of lattices, but the collection of all possible words (or "terms") in this case would be a proper class, because arbitrary meets and joins comprise operations for argument-sets of every cardinality. This property in itself is not a problem: as the case of free complete semilattices above shows, it can well be that the solution of the word problem leaves only a set of equivalence classes.
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Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
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Reducing Minimum weight bipartite matching to minimum cost max flow problem Given a bipartite graph G = (A ∪ B, E), the goal is to find the maximum cardinality matching in G that has minimum cost. Let w: E → R be a weight function on the edges of E. The minimum weight bipartite matching problem or assignment problem is to find a perfect matching M ⊆ E whose total weight is minimized. The idea is to reduce this problem to a network flow problem. Let G′ = (V′ = A ∪ B, E′ = E). Assign the capacity of all the edges in E′ to 1.
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The continuum hypothesis has been proven independent of the ZF axioms of set theory, so within that system, the proposition can neither be proven true nor proven false. A formalist would therefore say that the continuum hypothesis is neither true nor false, unless you further refine the context of the question. A platonist, however, would assert that there either does or does not exist a transfinite set with a cardinality less than the continuum but greater than any countable set. So, regardless of whether it has been proven unprovable, the platonist would argue that an answer nonetheless does exist.
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Deciding whether a graph admits a perfect matching can be done in polynomial time, using any algorithm for finding a maximum cardinality matching. However, counting the number of perfect matchings, even in bipartite graphs, is #P-complete. This is because computing the permanent of an arbitrary 0–1 matrix (another #P-complete problem) is the same as computing the number of perfect matchings in the bipartite graph having the given matrix as its biadjacency matrix. A remarkable theorem of Kasteleyn states that the number of perfect matchings in a planar graph can be computed exactly in polynomial time via the FKT algorithm.
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Applying the back-and-forth construction to any two isomorphic finite subgraphs of the Rado graph extends their isomorphism to an automorphism of the entire Rado graph. The fact that every isomorphism of finite subgraphs extends to an automorphism of the whole graph is expressed by saying that the Rado graph is ultrahomogeneous. In particular, there is an automorphism taking any ordered pair of adjacent vertices to any other such ordered pair, so the Rado graph is a symmetric graph. The automorphism group of the Rado graph is a simple group, whose number of elements is the cardinality of the continuum.
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The metric on the plane in which the distance between any two points is their Euclidean distance when the two points belong to a ray though the origin, and is otherwise the sum of the distances of the two points from the origin, is sometimes called the Paris metric because navigation in this metric resembles that in the radial street plan of Paris: for almost all pairs of points, the shortest path passes through the center. The Paris metric, restricted to the unit disk, is a hedgehog space where K is the cardinality of the continuum.
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In a pseudorandom number generator (PRNG), a full cycle or full period is the behavior of a PRNG over its set of valid states. In particular, a PRNG is said to have a full cycle if, for any valid seed state, the PRNG traverses every valid state before returning to the seed state, i.e., the period is equal to the cardinality of the state space. The restrictions on the parameters of a PRNG for it to possess a full cycle are known only for certain types of PRNGs, such as linear congruential generators and linear feedback shift registers.
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Frege takes this observation to be the fundamental thought of Grundlagen. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept horse in the barn. Frege attempts to explain our grasp of numbers through a contextual definition of the cardinality operation ('the number of...', or Nx: Fx ). He attempts to construct the content of a judgment involving numerical identity by relying on Hume's principle (which states that the number of Fs equals the number of Gs if and only if F and G are equinumerous, i.e.
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One book can have many pages. One page can only be in one book. In systems analysis, a one-to-many relationship is a type of cardinality that refers to the relationship between two entities (see also entity–relationship model) A and B in which an element of A may be linked to many elements of B, but a member of B is linked to only one element of A. For instance, think of A as books, and B as pages. A book can have many pages, but a page can only be in one book.
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Therefore, in the following, we will consider only orders on subsets of fixed cardinal. For example, using the natural order of the integers, the lexicographical ordering on the subsets of three elements of } is : ::. For ordering finite subsets of a given cardinality of the natural numbers, the colexicographical order (see below) is often more convenient, because all initial segments are finite, and thus the colexicographical order defines an order isomorphism between the natural numbers and the set of sets of natural numbers. This is not the case for the lexicographical order, as, with the lexicographical order, we have, for example, for every .
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Generalizing finite and (ordinary) infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers to transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called uncountable.
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The decision must be made on other grounds. One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.
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Similarly with the higher axioms of infinity. Now \aleph_1 is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set C [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach C. Thus C is greater than \aleph_n, \aleph_\omega, \aleph_a, where a = \aleph_\omega, etc.
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For example, the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra H must be just the nondegenerate quotients of H, namely all smaller linearly ordered nondegenerate Heyting algebras. The conditions cannot be dropped in general: for example, the variety of all Heyting algebras is generated by the set of its finite subdirectly irreducible algebras, but there exist subdirectly irreducible Heyting algebras of arbitrary (infinite) cardinality. There also exists a single finite algebra generating a (non- congruence-distributive) variety with arbitrarily large subdirect irreducibles.R. McKenzie, The residual bounds of finite algebras, Int.
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Suppose that one wants to define what it means for two sets to "have the same number of elements". One way to do this is to say that two sets "have the same number of elements", if and only if all the elements of one set can be paired with the elements of the other, in such a way that each element is paired with exactly one element. Accordingly, one can define two sets to "have the same number of elements"—if there is a bijection between them. In which case, the two sets are said to have the same cardinality.
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In contrast to a basis of a vector space in linear algebra, a base need not be maximal; indeed, the only maximal base is the topology itself. In fact, any open set generated by a base may be safely added to the base without changing the topology. The smallest possible cardinality of a base is called the weight of the topological space. An example of a collection of open sets which is not a base is the set S of all semi-infinite intervals of the forms (−∞, a) and (a, ∞), where a is a real number.
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This result is extended to semilattices of cardinality at most ℵ1 in 2000 by Wehrung, by keeping only the regularity of R (the ring constructed by the proof is not locally matricial). The question whether R could be taken locally matricial in the ℵ1 case remained open for a while, until it was disproved by Wehrung in 2004. Translating back to the lattice world by using the theorem above and using a lattice-theoretical analogue of the V(R) construction, called the dimension monoid, introduced by Wehrung in 1998, yields the following result. Theorem (Wehrung 2004).
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Syntactic pattern recognition or structural pattern recognition is a form of pattern recognition, in which each object can be represented by a variable- cardinality set of symbolic, nominal features. This allows for representing pattern structures, taking into account more complex interrelationships between attributes than is possible in the case of flat, numerical feature vectors of fixed dimensionality, that are used in statistical classification. Syntactic pattern recognition can be used instead of statistical pattern recognition if there is clear structure in the patterns. One way to present such structure is by means of a strings of symbols from a formal language.
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For high- cardinality columns, it is useful to bin the values, where each bin covers multiple values and build the bitmaps to represent the values in each bin. This approach reduces the number of bitmaps used regardless of encoding method. However, binned indexes can only answer some queries without examining the base data. For example, if a bin covers the range from 0.1 to 0.2, then when the user asks for all values less than 0.15, all rows that fall in the bin are possible hits and have to be checked to verify whether they are actually less than 0.15.
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An interval graph with pathwidth two, one less than the cardinality of its four maximum cliques ABC, ACD, CDE, and CDF. The pathwidth of any graph G is equal to one less than the smallest clique number of an interval graph that contains G as a subgraph., Theorem 29, p. 13. That is, for every path decomposition of G one can find an interval supergraph of G, and for every interval supergraph of G one can find a path decomposition of G, such that the width of the decomposition is one less than the clique number of the interval graph.
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Therefore, by the De Bruijn–Erdős theorem, P itself also has a w-colorable incomparability graph, and thus has the desired partition into chains . However, the theorem does not extend so simply to partially ordered sets in which the width, and not just the cardinality of the set, is infinite. In this case the size of the largest antichain and the minimum number of chains needed to cover the partial order may be very different from each other. In particular, for every infinite cardinal number κ there is an infinite partially ordered set of width ℵ0 whose partition into the fewest chains has κ chains .
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Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts). In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions.
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Finally, consider the signature σ consisting of a single unary relation symbol P. Every σ-structure is partitioned into two subsets: Those elements for which P holds, and the rest. Let K be the class of all σ-structures for which these two subsets have the same cardinality, i.e., there is a bijection between them. This class is not elementary, because a σ-structure in which both the set of realisations of P and its complement are countably infinite satisfies precisely the same first- order sentences as a σ-structure in which one of the sets is countably infinite and the other is uncountable.
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Gödel's completeness theorem, proved by Kurt Gödel in 1929, establishes that there are sound, complete, effective deductive systems for first-order logic, and thus the first-order logical consequence relation is captured by finite provability. Naively, the statement that a formula φ logically implies a formula ψ depends on every model of φ; these models will in general be of arbitrarily large cardinality, and so logical consequence cannot be effectively verified by checking every model. However, it is possible to enumerate all finite derivations and search for a derivation of ψ from φ. If ψ is logically implied by φ, such a derivation will eventually be found.
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Such a linearly independent set that spans a vector space is called a basis of . The importance of bases lies in the fact that there are together minimal generating sets and maximal independent sets. More precisely, if is a linearly independent set, and is a spanning set such that S\subseteq T, then there is a basis such that S\subseteq B\subseteq T. Any two bases of a vector space have the same cardinality, which is called the dimension of ; this is the dimension theorem for vector spaces. Moreover, two vector spaces over the same field are isomorphic if and only if they have the same dimension.
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The hitting set problem is equivalent to the set cover problem: An instance of set cover can be viewed as an arbitrary bipartite graph, with sets represented by vertices on the left, elements of the universe represented by vertices on the right, and edges representing the inclusion of elements in sets. The task is then to find a minimum cardinality subset of left-vertices which covers all of the right- vertices. In the hitting set problem, the objective is to cover the left- vertices using a minimum subset of the right vertices. Converting from one problem to the other is therefore achieved by interchanging the two sets of vertices.
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In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function f:[x]^\omega\to x with the property that, for any subset y of x with the same cardinality as x, the restriction of f to [y]^\omega is surjective on x. Here [x]^\omega denotes the set of strictly increasing sequences of members of x, or equivalently the family of subsets of x with order type \omega, using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson. showed that for every ordinal λ there is an ω-Jónsson function for λ.
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Under his leadership, the Institute became a noted centre for attracting young researchers, both within the Federal Republic but also abroad. With Hermes there, among others, were Wilhelm Ackermann and Gisbert Hasenjaeger. In 1966 he accepted an appointment to the newly established Chair of Mathematical Logic and the Foundations of Mathematics at the University of Freiburg and began to build an eponymous department at the Mathematical Institute, becoming Professor Emeritus there in 1977. In 1954 Hermes produced an informal proof, that the possibilities of programmable eigenvalues include the predictable functions , so the calculating machines have the same cardinality as Turing machines re: Turing completeness.
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For planar graphs, the properties of being Eulerian and bipartite are dual: a planar graph is Eulerian if and only if its dual graph is bipartite. As Welsh showed, this duality extends to binary matroids: a binary matroid is Eulerian if and only if its dual matroid is a bipartite matroid, a matroid in which every circuit has even cardinality.. For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down. For instance, the uniform matroid U{}^2_6 is Eulerian but its dual U{}^4_6 is not bipartite, as its circuits have size five. The self-dual uniform matroid U{}^3_6 is bipartite but not Eulerian.
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A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous.
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We say G is a (\tau, \kappa) -disjoint collection if G is the union of at most \tau subcollections G_\alpha, where for each \alpha, G_\alpha is a disjoint collection of cardinality at most \kappa It was proven by Petr Simon that X is a Boolean space with the generating set G of CO(X) being (\tau, \kappa) -disjoint if and only if X is homeomorphic to a closed subspace of \alpha \kappa ^ \tau. The Ramsey-like property for polyadic spaces as stated by Murray Bell for Boolean spaces is then as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint.
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Given a field extension L/K, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of L over K. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. For every finite set S of elements of L, the algebraically independent subsets of S satisfy the axioms that define the independent sets of a matroid. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set T of elements is the intersection of L with the field K[T].Oxley (1992) p.
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The second Prüfer theorem provides a straightforward extension of the fundamental theorem of finitely generated abelian groups to countable abelian p-groups without elements of infinite height: each such group is isomorphic to a direct sum of cyclic groups whose orders are powers of p. Moreover, the cardinality of the set of summands of order pn is uniquely determined by the group and each sequence of at most countable cardinalities is realized. Helmut Ulm (1933) found an extension of this classification theory to general countable p-groups: their isomorphism class is determined by the isomorphism classes of the Ulm factors and the p-divisible part. : Ulm's theorem.
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On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every real- closed field satisfies the same first-order sentences in the signature \langle +,\cdot,\le\rangle as the real numbers.) In second-order logic, it is possible to write formal sentences which say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective.
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The sum of all -choose binomial coefficients is equal to . Consider the set of all -digit binary integers. Its cardinality is . It is also the sums of the cardinalities of certain subsets: the subset of integers with no 1s (consisting of a single number, written as 0s), the subset with a single 1, the subset with two 1s, and so on up to the subset with 1s (consisting of the number written as 1s). Each of these is in turn equal to the binomial coefficient indexed by and the number of 1s being considered (for example, there are 10-choose-3 binary numbers with ten digits that include exactly three 1s).
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An Author can write several Books, and a Book can be written by several Authors The Author-Book many-to-many relationship as a pair of one-to-many relationships with a junction table In systems analysis, a many-to-many relationship is a type of cardinality that refers to the relationship between two entitiesAlso see entity–relationship model. A and B in which A may contain a parent instance for which there are many children in B and vice versa. For example, think of A as Authors, and B as Books. An Author can write several Books, and a Book can be written by several Authors.
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Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a hypercube. If the hypercube consists of all points in Euclidean space of a given dimension whose Cartesian coordinates are in the interval [-1,+1], then a signed subset of the coordinate axes can be used to specify the points whose coordinates within the subset are -1 or +1 (according to the sign in the signed subset) and whose other coordinates may be anywhere in the interval [-1,+1]. This subset of points forms a face, whose codimension is the cardinality of the signed subset.
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In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra A has an essentially unique completion, which is a complete Boolean algebra containing A such that every element is the supremum of some subset of A. As a partially ordered set, this completion of A is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum.
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In her dissertation and postdoctoral research, Malliaris studied unstable model theory and its connection, via characteristic sequences, to graph theoretic concepts such as the Szemerédi regularity lemma. She is also known for two joint papers with Saharon Shelah connecting topology, set theory, and model theory. In this work, Malliaris and Shelah used Keisler's order, a construction from model theory, to prove the equality between two cardinal characteristics of the continuum, 𝖕 and 𝖙, which are greater than the smallest infinite cardinal and less than or equal to the cardinality of the continuum. This resolved a problem in set theory that had been open for fifty years.
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Consider, for example, the number :3.1400010000000000000000050000.... 3.14(3 zeros)1(17 zeros)5(95 zeros)9(599 zeros)2(4319 zeros)6... where the digits are zero except in positions n! where the digit equals the nth digit following the decimal point in the decimal expansion of . As shown in the section on the existence of Liouville numbers, this number, as well as any other non-terminating decimal with its non-zero digits similarly situated, satisfies the definition of a Liouville number. Since the set of all sequences of non-null digits has the cardinality of the continuum, the same thing occurs with the set of all Liouville numbers.
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In particular if is finitely generated, then all its bases are finite and have the same number of elements. While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma,Howard, P., Rubin, J.: "Consequences of the axiom of choice" - Mathematical Surveys and Monographs, vol 59 (1998) . which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all cardinal numbers are comparable, a statement which is also equivalent to the axiom of choice).
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In group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product of simplexes, that is, sets of the form {e,a,a2,...,as-1} where e is the identity element, then at least one of the factors is a subgroup. The theorem was proved by the Hungarian mathematician György Hajós in 1941 using group rings. Rédei later proved the statement when the factors are only required to contain the identity element and be of prime cardinality. In this lattice tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.
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A matching in a graph is a subset of its edges, no two of which share an endpoint. Polynomial time algorithms are known for many algorithmic problems on matchings, including maximum matching (finding a matching that uses as many edges as possible), maximum weight matching, and stable marriage.. In many cases, matching problems are simpler to solve on bipartite graphs than on non- bipartite graphs,, p. 463: "Nonbipartite matching problems are more difficult to solve because they do not reduce to standard network flow problems." and many matching algorithms such as the Hopcroft–Karp algorithm for maximum cardinality matching. work correctly only on bipartite inputs.
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The term "α-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal κ is called α-inaccessible, for α any ordinal, if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal κ is called α-weakly inaccessible if κ is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than κ is unbounded in κ.
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In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.Page 46 of The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved.
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Magidor obtained several important consistency results on powers of singular cardinals substantially developing the method of forcing. He generalized the Prikry forcing in order to change the cofinality of a large cardinal to a predetermined regular cardinal. He proved that the least strongly compact cardinal can be equal to the least measurable cardinal or to the least supercompact cardinal (but not at the same time). Assuming consistency of huge cardinals he constructed models (1977) of set theory with first examples of nonregular ultrafilters over very small cardinals (related to the famous Guilmann Keisler problem concerning existence of nonregular ultrafilters), even with the example of jumping cardinality of ultrapowers.
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Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.
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With the axiom of choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portion does not have the cardinality of the continuum. We create a counterexample by transfinite induction on the set of strategies under this well ordering: We start with the set A undefined. Let T be the "time" whose axis has length continuum. We need to consider all strategies {s1(T)} of the first player and all strategies {s2(T)} of the second player to make sure that for every strategy there is a strategy of the other player that wins against it.
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Therefore, we have constructed an antichain and a partition into chains with the same cardinality. To prove Kőnig's theorem from Dilworth's theorem, for a bipartite graph G = (U,V,E), form a partial order on the vertices of G in which u < v exactly when u is in U, v is in V, and there exists an edge in E from u to v. By Dilworth's theorem, there exists an antichain A and a partition into chains P both of which have the same size. But the only nontrivial chains in the partial order are pairs of elements corresponding to the edges in the graph, so the nontrivial chains in P form a matching in the graph.
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In a game where chance or a random event is involved, the outcome is not known from only the set of strategies, but is only realized when the random event(s) are realized. A set of payoffs can be considered a set of N-tuples, where N is the number of players in the game, and the cardinality of the set is equal to the total number of possible outcomes when the strategies of the players are varied. The payoff set can thus be partially ordered, where the partial ordering comes from the value of each entry in the N-tuple. How players interact to allocate the payoffs among themselves is a fundamental aspect of economics.
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Every subgroup of this group whose index is less than the cardinality of the continuum can be sandwiched between the pointwise stabilizer and the stabilizer of a finite set of vertices., Section 1.8: The automorphism group. The construction of the Rado graph as an infinite circulant graph shows that its symmetry group includes automorphisms that generate a transitive infinite cyclic group. The difference set of this construction (the set of distances in the integers between adjacent vertices) can be constrained to include the difference 1, without affecting the correctness of this construction, from which it follows that the Rado graph contains an infinite Hamiltonian path whose symmetries are a subgroup of the symmetries of the whole graph.
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Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \mu^\lambda = \kappa. However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy u^\lambda = \kappa. The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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A Forte number, "consists of two numbers separated by a hyphen....The first number is the cardinality of the set form...and the second number refers to the ordinal position..." Major and minor chords on C . In the 12-TET tuning system (or in any other system of tuning that splits the octave into twelve semitones), each pitch class may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers. The prime form of a pitch class set is the most compact (i.e., leftwards packed or smallest in lexicographic order) of either the normal form of a set or of its inversion.
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Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem showed in 1922 that every consistent theory of first-order predicate calculus, such as set theory, has an at most countable model. However, Cantor's theorem proves that there are uncountable sets. The root of this seeming paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in which the cardinality is measured. It is possible for a set to be uncountable in one model of set theory but countable in a larger model (because the bijections that establish countability are in the larger model but not the smaller one).
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Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits a basis, but not orthonormal base Linear Functional Analysis Authors: Rynne, Bryan, Youngson, M.A. page 79; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice).
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One approach to the problem, imitating the way children choose teams for a game, is the greedy algorithm, which iterates through the numbers in descending order, assigning each of them to whichever subset has the smaller sum. This approach has a running time of . This heuristic works well in practice when the numbers in the set are of about the same size as its cardinality or less, but it is not guaranteed to produce the best possible partition. For example, given the set S = {4, 5, 6, 7, 8} as input, this greedy algorithm would partition into subsets {4, 5, 8} and {6, 7}; however, S has an exactly balanced partition into subsets {7, 8} and {4, 5, 6}.
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Being a subgroup of a symmetric group, all that is necessary for a set of permutations to satisfy the group axioms and be a permutation group is that it contain the identity permutation, the inverse permutation of each permutation it contains, and be closed under composition of its permutations. A general property of finite groups implies that a finite nonempty subset of a symmetric group is again a group if and only if it is closed under the group operation. The degree of a group of permutations of a finite set is the number of elements in the set. The order of a group (of any type) is the number of elements (cardinality) in the group.
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More generally, the fundamental theorem of algebra asserts that the complex solutions of a polynomial equation of degree always form a multiset of cardinality . A special case of the above are the eigenvalues of a matrix, whose multiplicity is usually defined as their multiplicity as roots of the characteristic polynomial. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the minimal polynomial, and the geometric multiplicity, which is defined as the dimension of the kernel of (where is an eigenvalue of the matrix ). These three multiplicities define three multisets of eigenvalues, which may be all different: Let be a matrix in Jordan normal form that has a single eigenvalue.
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A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see below. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space H, in the sense that the closure of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a basis of H, its cardinality is known as the Hilbert space dimension.A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra above.
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In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University of Illinois at Urbana–Champaign... Let F be a family of distinct analytic functions on a given domain with the property that, for each x in the domain, the functions in F map x to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that F is necessarily itself countable.. As cited by . Paul Erdős in turn learned about the problem at the University of Michigan, likely via Lee Albert Rubel.
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The optimization version of this problem is called Max Set Splitting and requires finding the partition which maximizes the number of split elements of F. It is an APX-complete problem and hence in NPO. The Set k-Splitting problem is stated as follows: given S, F, and an integer k, does there exist a partition of S which splits at least k subsets of F? The original formulation is the restricted case with k equal to the cardinality of F. The Set k-Splitting is fixed-parameter tractable, i.e., if k taken to be a fixed parameter, rather than a part of the input, then a polynomial algorithm exists for any fixed k.
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Not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be independent from the generally accepted set of Zermelo–Fraenkel axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner (much as Euclid's parallel postulate can be taken either as true or false in an axiomatic system for geometry). In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis (in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i.e.
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A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, and there is only one way to put a finite set into a linear sequence (up to isomorphism). When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.
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So ω can be identified with \aleph_0, except that the notation \aleph_0 is used when writing cardinals, and ω when writing ordinals (this is important since, for example, \aleph_0^2 = \aleph_0 whereas \omega^2 > \omega). Also, \omega_1 is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and \omega_1 is the order type of that set), \omega_2 is the smallest ordinal whose cardinality is greater than \aleph_1, and so on, and \omega_\omega is the limit of the \omega_n for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_n).
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More generally if (G,K) is a Gelfand pair then the resulting algebra turns out to be commutative. Example: If G = SL(2,Q) and K = SL(2,Z) we get the abstract ring behind Hecke operators in the theory of modular forms, which gave the name to Hecke algebras in general. The case leading to the Hecke algebra of a finite Weyl group is when G is the finite Chevalley group over a finite field with pk elements, and B is its Borel subgroup. Iwahori showed that the Hecke ring H(G//B) is obtained from the generic Hecke algebra Hq of the Weyl group W of G by specializing the indeterminate q of the latter algebra to pk, the cardinality of the finite field.
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If M is a binary matroid, then so is its dual, and so is every minor of M. Additionally, the direct sum of binary matroids is binary. define a bipartite matroid to be a matroid in which every circuit has even cardinality, and an Eulerian matroid to be a matroid in which the elements can be partitioned into disjoint circuits. Within the class of graphic matroids, these two properties describe the matroids of bipartite graphs and Eulerian graphs (not-necessarily-connected graphs in which all vertices have even degree), respectively. For planar graphs (and therefore also for the graphic matroids of planar graphs) these two properties are dual: a planar graph or its matroid is bipartite if and only if its dual is Eulerian.
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In addition it is possible to represent axioms in F-logic in the following manner: man(X) <\- person(X) AND NOT woman(X). X:person[hasFather->Y] <\- Y:man[hasSon -> X]. These mean "X is a man if X is a person but not a woman" and "if X is the son of Y then X is a person and Y is the father of X". The Flora-2 system introduced a number of changes to the syntax of F-logic, making it more suitable for a knowledge representation and reasoning system as opposed to just a theoretical logic. In particular, variables became prefixed with a ?-mark, the distinction between functional and multi-valued properties was dropped and replaced by cardinality constraints, plus other important changes.
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Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. The name is something of a misnomer as the concepts involved usually apply much more generally, to any periodically repeating scale. Music theorists working in diatonic set theory include Eytan Agmon, Gerald J. Balzano, Norman Carey, David Clampitt, John Clough, Jay Rahn, and mathematician Jack Douthett. A number of key concepts were first formulated by David Rothenberg, who published in the journal Mathematical Systems Theory, and Erv Wilson, working entirely outside of the academic world.
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To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the compactness theorem and the upward Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic. Certain fragments of second order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic which allow non-linear ordering of quantifier dependencies, like first-order logic extended with Henkin quantifiers, Hintikka and Sandu's independence- friendly logic, and Väänänen's dependence logic.
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In North Indian temples, the tallest towers are built over the sanctum sanctorum in which the deity is installed. The north India Nagara style of temple designs often deploy fractal-theme, where smaller parts of the temple are themselves images or geometric re-arrangement of the large temple, a concept that later inspired French and Russian architecture such as the matryoshka principle. One difference is the scope and cardinality, where Hindu temple structures deploy this principle in every dimension with garbhgriya as the primary locus, and each pada as well as zones serving as additional centers of loci. This makes a Nagara Hindu temple architecture symbolically a perennial expression of movement and time, of centrifugal growth fused with the idea of unity in everything.
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Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption. The Chinese remainder theorem can also be used in secret sharing, which consists of distributing a set of shares among a group of people who, all together (but no one alone), can recover a certain secret from the given set of shares. Each of the shares is represented in a congruence, and the solution of the system of congruences using the Chinese remainder theorem is the secret to be recovered. Secret sharing using the Chinese remainder theorem uses, along with the Chinese remainder theorem, special sequences of integers that guarantee the impossibility of recovering the secret from a set of shares with less than a certain cardinality.
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If attribute set K is a superkey of relation R, then at all times it is the case that the projection of R over K has the same cardinality as R itself. A superkey is a set of attributes within a table whose values can be used to uniquely identify a tuple. A candidate key is a minimal set of attributes necessary to identify a tuple; this is also called a minimal superkey. Given an employee schema consisting of the attributes employeeID, name, job, and departmentID, where no value in the employeeID attribute is ever repeated, we could use the employeeID in combination with any or all other attributes of this table to uniquely identify a tuple in the table.
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Girard (1987) to model linear logic), while Chu spaces over K realize any category of vector spaces over a field whose cardinality is at most that of K. This was extended by Vaughan Pratt (1995) to the realization of k-ary relational structures by Chu spaces over 2k. For example, the category Grp of groups and their homomorphisms is realized by Chu(Set, 8) since the group multiplication can be organized as a ternary relation. Chu(Set, 2) realizes a wide range of ``logical`` structures such as semilattices, distributive lattices, complete and completely distributive lattices, Boolean algebras, complete atomic Boolean algebras, etc. Further information on this and other aspects of Chu spaces, including their application to the modeling of concurrent behavior, may be found at Chu Spaces.
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A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others. If is the set of tiles in a tessellation, a set of shapes is called a set of prototiles if no two shapes in are congruent to each other, and every tile in is congruent to one of the shapes in .. It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality, so the number of prototiles is well defined.
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This has been found to help prevent premature convergence at so- called Hamming walls, in which too many simultaneous mutations (or crossover events) must occur in order to change the chromosome to a better solution. Other approaches involve using arrays of real-valued numbers instead of bit strings to represent chromosomes. Results from the theory of schemata suggest that in general the smaller the alphabet, the better the performance, but it was initially surprising to researchers that good results were obtained from using real-valued chromosomes. This was explained as the set of real values in a finite population of chromosomes as forming a virtual alphabet (when selection and recombination are dominant) with a much lower cardinality than would be expected from a floating point representation.
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In mathematics, and particularly in the theory of formal languages, shortlex is a total ordering for finite sequences of objects that can themselves be totally ordered. In the shortlex ordering, sequences are primarily sorted by cardinality (length) with the shortest sequences first, and sequences of the same length are sorted into lexicographical order. Shortlex ordering is also called radix, length-lexicographic, military, or genealogical ordering.. In the context of strings on a totally ordered alphabet, the shortlex order is identical to the lexicographical order, except that shorter strings precede longer strings. For example, the shortlex order of the set of strings on the English alphabet (in its usual order) is [ε, a, b, c, ..., z, aa, ab, ac, ..., zz, aaa, aab, aac, ..., zzz, ...], where ε denotes the empty string.
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The strong version of the paradox claims: : Any two bounded subsets of 3-dimensional Euclidean space with non- empty interiors are equidecomposable. While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein–Schroeder theorem due to Banach that implies that if is equidecomposable with a subset of and is equidecomposable with a subset of , then and are equidecomposable. The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality.
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The Flajolet–Martin algorithm is an algorithm for approximating the number of distinct elements in a stream with a single pass and space-consumption logarithmic in the maximal number of possible distinct elements in the stream (the count-distinct problem). The algorithm was introduced by Philippe Flajolet and G. Nigel Martin in their 1984 article "Probabilistic Counting Algorithms for Data Base Applications". Later it has been refined in "LogLog counting of large cardinalities" by Marianne Durand and Philippe Flajolet, and "HyperLogLog: The analysis of a near-optimal cardinality estimation algorithm" by Philippe Flajolet et al. In their 2010 article "An optimal algorithm for the distinct elements problem", Daniel M. Kane, Jelani Nelson and David P. Woodruff give an improved algorithm, which uses nearly optimal space and has optimal O(1) update and reporting times.
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An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. The diagram to the right gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size". Cantor defined two kinds of infinite numbers: ordinal numbers and cardinal numbers. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted.
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A conceptual model can be described using various notations, such as UML, ORM or OMT for object modelling, ITE, or IDEF1X for Entity Relationship Modelling. In UML notation, the conceptual model is often described with a class diagram in which classes represent concepts, associations represent relationships between concepts and role types of an association represent role types taken by instances of the modelled concepts in various situations. In ER notation, the conceptual model is described with an ER Diagram in which entities represent concepts, cardinality and optionality represent relationships between concepts. Regardless of the notation used, it is important not to compromise the richness and clarity of the business meaning depicted in the conceptual model by expressing it directly in a form influenced by design or implementation concerns.
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If we operate on any point in Euclidean 2-space by the various elements of H we get what is called the orbit of that point. All the points in the plane can thus be classed into orbits, of which there are an infinite number with the cardinality of the continuum. Using the axiom of choice, we can choose one point from each orbit and call the set of these points M. We exclude the origin, which is a fixed point in H. If we then operate on M by all the elements of H, we generate each point of the plane (except the origin) exactly once. If we operate on M by all the elements of A or of B, we get two disjoint sets whose union is all points but the origin.
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Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know :x ⊕ x = (x ⊕ x)2 = x2 ⊕ x2 ⊕ x2 ⊕ x2 = x ⊕ x ⊕ x ⊕ x and since (R,⊕) is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0. A similar proof shows that every Boolean ring is commutative: :x ⊕ y = (x ⊕ y)2 = x2 ⊕ xy ⊕ yx ⊕ y2 = x ⊕ xy ⊕ yx ⊕ y and this yields xy ⊕ yx = 0, which means xy = yx (using the first property above). The property x ⊕ x = 0 shows that any Boolean ring is an associative algebra over the field F2 with two elements, in precisely one way. In particular, any finite Boolean ring has as cardinality a power of two.
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The idea that the ultimate good exists and is not orderable but is globally measurable is reflected in various ways in economic (classical economics, green economics, welfare economics, gross national happiness) and scientific (positive psychology, the science of morality) well-being measuring theories, all of which focus on various ways of assessing progress towards that goal, a so- called genuine progress indicator. Modern economics thus reflects very ancient philosophy, but a calculation or quantitative or other process based on cardinality and statistics replaces the simple ordering of values. For example, in both economics and in folk wisdom, the value of something seems to rise so long as it is relatively scarce. However, if it becomes too scarce, it leads often to a conflict, and can reduce collective value.
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The .pdi format, generation process, and GUI, were invented by Dr. Reimar Hofmann and Dr. Michael Haft from Siemens AG Artificial Intelligence/Machine Learning. The .pdi footprint is typically 100 to 1000 times smaller than the footprint normally found in structured data files or database systems, and is rendered without any loss of detail. The word portable in the name derives from the idea that the smaller footprint allows a .pdi runs in the main memory of a user's’ computer without disk or network input/output (IO). The .pdi is a digitally rights protected, encrypted data source that can be accessed by any ODBO (OLE DB for OLAP) compliant OLAP tool, including Microsoft Excel and the Panoratio's Explorer GUI. The .pdi presents detailed discrete or binned data without pre-calculation or cardinality reduction.
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While Tone-Clock Theory displays many similarities to Allen Forte's pitch-class set theory, it places greater emphasis on the creation of pitch 'fields' from multiple transpositions and inversions of a single set-class, while also aiming to complete all twelve pitch-classes (the 'chromatic aggregate') with minimal, if any, repetition of pitch-classes. While the emphasis of Tone-Clock Theory is on creating the chromatic aggregate, it is not a serial technique, as the ordering of pitch- classes is not important. Having said that, it bears a certain similarity to the technique of 'serial derivation', which was used by Anton Webern and Milton Babbitt amongst others, in which a row is constructed from only one or two set-classes. It also bears a similarity to Josef Hauer's system of 'tropes', albeit generalised to sets of any cardinality.
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Three note sets from the diatonic scale in the chromatic circle: M2M2=red, M2m2=yellow, and m2M2=blue The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the same distance in pitch class space. In general, for a given scale S, the scalar transpositions of a line L can be grouped into categories, or transpositional set classes, whose members are related by chromatic transposition. In diatonic set theory cardinality equals variety when, for any melodic line L in a particular scale S, the number of these classes is equal to the number of distinct pitch classes in the line L. For example, the melodic line C-D-E has three distinct pitch classes.
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Having two distinct options x and y amounts to more FoC than having only the option x. #Independence. If a situation A has more FoC than B, by adding a new option x to both (not contained in A or B), A will still have more FoC than B. They proved that the cardinality is the only measurement that satisfies these axioms, what they observed to be counter- intuitive and suggestive that one or more axioms should be reformulated. They illustrated this with the example of the option set "to travel by train" or "to travel by car", that should yield more FoC than the option set "to travel by red car" or "to travel by blue car". Some suggestions have been made to solve this problem, by reformulating the axioms, usually including concepts of preferences, or rejecting the third axiom.
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In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas.
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For any particular E, the probability that x is missed while y is larger than its median is very small, and the Sauer–Shelah lemma (applied to x\cup y) shows that only a small number of distinct events E need to be considered, so by the union bound, with nonzero probability, x is an ε-net. In turn, ε-nets and ε-approximations, and the likelihood that a random sample of large enough cardinality has these properties, have important applications in machine learning, in the area of probably approximately correct learning.. In computational geometry, they have been applied to range searching, derandomization,. and approximation algorithms... use generalizations of the Sauer–Shelah lemma to prove results in graph theory such as that the number of strong orientations of a given graph is sandwiched between its numbers of connected and 2-edge-connected subgraphs.
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In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
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Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety. The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").
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One-to-one correspondence between an infinite set and its proper subset A different form of "infinity" are the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege, Richard Dedekind and others—using the idea of collections or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers).
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The time complexity of solving some of the applications discussed above, such as multi- dimensional table problems, multicommodity flow problems, and N-fold integer programming problems, is dominated by the cardinality of the relevant Graver basis, which is a polynomial O\left(n^g\right) with typically large degree g which is a suitable Graver complexity of the system. In Raymond Hemmecke, Shmuel Onn, Lyubov Romanchuk: N-fold integer programming in cubic time, Mathematical Programming, 137:325–341, 2013 a much faster algorithm is presented, which finds at each iteration the best improvement x + qg with q positive integer and g element in G(A) without explicitly constructing the Graver basis, in cubic time O\left(n^3\right) regardless of the system. In the terminology of parameterized complexity, this implies that all these problems suitably parameterized, and in particular l × m × n table problems parametrized by l and m, are fixed parameter tractable.
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Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem. The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics. When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA0 (a second- order variant of Peano arithmetic restricted to induction over Σ01 formulas). Weak König's lemma is provable in ZF, the system of Zermelo–Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF. However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.
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The name we put on things is very important: it sets the norm for how we judge them. He introduces three basic dimensions of value, systemic, extrinsic and intrinsic for sets of properties—perfection is to systemic value what goodness is to extrinsic value and what uniqueness is to intrinsic value—each with their own cardinality: finite, \aleph_0 and \aleph_1. In practice, the terms "good" and "bad" apply to finite sets of properties, since this is the only case where there is a ratio between the total number of desired properties and the number of such properties possessed by some object being valued. (In the case where the number of properties is countably infinite, the extrinsic dimension of value, the exposition as well as the mere definition of a specific concept is taken into consideration.) Hartman quantifies this notion by the principle that each property of the thing is worth as much as each other property, depending on the level of abstraction.
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Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y. Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps f: X → Y such that the preimages of all points y in Y consist only of finitely many points): the cardinality of the fibers f−1(y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map :C → C, z ↦ zn has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0. One says that the map is "ramified" in zero.
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The height of a partially ordered set is defined to be the maximum cardinality of a chain, a totally ordered subset of the given partial order. For instance, in the set of positive integers from 1 to N, ordered by divisibility, one of the largest chains consists of the powers of two that lie within that range, from which it follows that the height of this partial order is 1+\lfloor\log_2 N\rfloor. Mirsky's theorem states that, for every finite partially ordered set, the height also equals the minimum number of antichains (subsets in which no pair of elements are ordered) into which the set may be partitioned. In such a partition, every two elements of the longest chain must go into two different antichains, so the number of antichains is always greater than or equal to the height; another formulation of Mirsky's theorem is that there always exists a partition for which the number of antichains equals the height.
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In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition: :For every a ∈ A, there exists some b ∈ B such that a ≤ b. This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”. They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of A is referred to as the cofinality of A. A subset B of A is said to be coinitial (or dense in the sense of forcing) if it satisfies the following condition: :For every a ∈ A, there exists some b ∈ B such that b ≤ a.
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In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of ordinal number. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well- orderable sets.
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Borges in 1976 Jorge Luis Borges and mathematics concerns several modern mathematical concepts found in certain essays and short stories of Argentinian author Jorge Luis Borges (1899-1986), including concepts such as set theory, recursion, chaos theory, and infinite sequences, although Borges' strongest links to mathematics are through Georg Cantor's theory of infinite sets, outlined in "The Doctrine of Cycles" (La doctrina de los ciclos). Some of Borges' most popular works such as "The Library of Babel" (La Biblioteca de Babel), "The Garden of Forking Paths" (El Jardín de Senderos que se Bifurcan), "The Aleph" (El Aleph), an allusion to Cantor's use of the Hebrew letter aleph (\aleph) to denote cardinality of transfinite sets, and "The Approach to Al- Mu'tasim" (El acercamiento a Almotásim) illustrate his use of mathematics. According to Argentinian mathematician Guillermo Martínez, Borges at least had a knowledge of mathematics at the level of first courses in algebra and analysis at a university – covering logic, paradoxes, infinity, topology and probability theory. He was also aware of the contemporary debates on the foundations of mathematics.
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A country has only one capital city, and a capital city is the capital of only one country In systems analysis, a one-to-one relationship is a type of cardinality that refers to the relationship between two entities (see also entity–relationship model) A and B in which one element of A may only be linked to one element of B, and vice versa. In mathematical terms, there exists a bijective function from A to B. For instance, think of A as the set of all human beings, and B as the set of all their brains. Any person from A can and must have only one brain from B, and any human brain in B can and must belong to only one person that is contained in A. In a relational database, a one-to-one relationship exists when one row in a table may be linked with only one row in another table and vice versa. It is important to note that a one- to-one relationship is not a property of the data, but rather of the relationship itself.
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A connected dominating set of a graph G is a set D of vertices with two properties: #Any node in D can reach any other node in D by a path that stays entirely within D. That is, D induces a connected subgraph of G. #Every vertex in G either belongs to D or is adjacent to a vertex in D. That is, D is a dominating set of G. A minimum connected dominating set of a graph G is a connected dominating set with the smallest possible cardinality among all connected dominating sets of G. The connected domination number of G is the number of vertices in the minimum connected dominating set.. Any spanning tree T of a graph G has at least two leaves, vertices that have only one edge of T incident to them. A maximum leaf spanning tree is a spanning tree that has the largest possible number of leaves among all spanning trees of G. The max leaf number of G is the number of leaves in the maximum leaf spanning tree..
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Only the case n=2 is used here. The semilattice part of the result above is achieved via an infinitary semilattice-theoretical statement URP (Uniform Refinement Property). If we want to disprove Schmidt's problem, the idea is (1) to prove that any generalized Boolean semilattice satisfies URP (which is easy), (2) that URP is preserved under homomorphic image under a weakly distributive homomorphism (which is also easy), and (3) that there exists a distributive (∨,0)-semilattice of cardinality ℵ2 that does not satisfy URP (which is difficult, and uses Kuratowski's Free Set Theorem). Schematically, the construction in the theorem above can be described as follows. For a set Ω, we consider the partially ordered vector space E(Ω) defined by generators 1 and ai,x, for i<2 and x in Ω, and relations a0,x+a1,x=1, a0,x ≥ 0, and a1,x ≥ 0, for any x in Ω. By using a Skolemization of the theory of dimension groups, we can embed E(Ω) functorially into a dimension vector space F(Ω).
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Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ < κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.
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This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B. Alternatively, the free abelian group with basis B may be described by a presentation with the elements of B as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.
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He also constructed one-to-one correspondences to prove that the n-dimensional spaces Rn (where R is the set of real numbers) and the set of irrational numbers have the same cardinality as R.. In 1883, Cantor extended the positive integers with his infinite ordinals. This extension was necessary for his work on the Cantor–Bendixson theorem. Cantor discovered other uses for the ordinals—for example, he used sets of ordinals to produce an infinity of sets having different infinite cardinalities.. His work on infinite sets together with Dedekind's set-theoretical work created set theory.. The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets.. In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions.. Building on the work of Borel and Baire, Henri Lebesgue created his theories of measure and integration, which were published from 1899 to 1901.. Countable models are used in set theory.
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Introduction to commutative algebra. Addison-Wesley publishing Company. pp. 11–12.Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma,Mathoverflow discussion it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K. The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K. The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.
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The "girth" terminology generalizes the use of girth in graph theory, meaning the length of the shortest cycle in a graph: the girth of a graphic matroid is the same as the girth of its underlying graph.. The girth of other classes of matroids also corresponds to important combinatorial problems. For instance, the girth of a co-graphic matroid (or the cogirth of a graphic matroid) equals the edge connectivity of the underlying graph, the number of edges in a minimum cut of the graph. The girth of a transversal matroid gives the cardinality of a minimum Hall set in a bipartite graph: this is a set of vertices on one side of the bipartition that does not form the set of endpoints of a matching in the graph.. Any set of points in Euclidean space gives rise to a real linear matroid by interpreting the Cartesian coordinates of the points as the vectors of a matroid representation. The girth of the resulting matroid equals one plus the dimension of the space when the underlying set of point is in general position, and is smaller otherwise.
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N is an elementary substructure of M if N and M are structures of the same signature σ such that for all first- order σ-formulas φ(x1, …, xn) with free variables x1, …, xn, and all elements a1, …, an of N, φ(a1, …, an) holds in N if and only if it holds in M: :N \models φ(a1, …, an) iff M \models φ(a1, …, an). It follows that N is a substructure of M. If N is a substructure of M, then both N and M can be interpreted as structures in the signature σN consisting of σ together with a new constant symbol for every element of N. Then N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σN-structures. If N is an elementary substructure of M, one writes N \preceq M and says that M is an elementary extension of N: M \succeq N. The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
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