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Vitali Set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measure, Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. Each Vitali set is Uncountable set, uncountable, and there are uncountably many Vitali sets. The proof of their existence depends on the axiom of choice. Measurable sets Certain sets have a definite 'length' or 'mass'. For instance, the interval (mathematics), interval [0, 1] is deemed to have length 1; more generally, an interval [''a'', ''b''], ''a'' ≤ ''b'', is deemed to have length ''b'' − ''a''. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2. There is a natural question here: if ''E'' is an arbitr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solovay Model
In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal. In this way Solovay showed that in the proof of the existence of a non-measurable set from ZFC (Zermelo–Fraenkel set theory plus the axiom of choice), the axiom of choice is essential, at least granted that the existence of an inaccessible cardinal is consistent with ZFC. Statement ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension ''V'' 'G''such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property. Construction Solov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert M
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' () "fame, glory, honour, praise, renown, godlike" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin.Reaney & Wilson, 1997. ''Dictionary of English Surnames''. Oxford University Press. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe, the name entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inaccessible Cardinal
In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it has meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced by . Strongly inaccessible cardinals were introduced by and ; in the latter they were referred to along with \aleph_0 as ''Grenzzahlen'' ( English "limit numbers"). Every strongly inaccessible cardinal is a weakly inaccessible cardinal. The generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well. The two notions of an inaccessible cardinal \kappa describe a cardinality \kappa which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Measure
In the mathematics, mathematical discipline of measure theory, a Banach measure is a certain way to assign a size (or area) to all subsets of the Euclidean plane, consistent with but extending the commonly used Lebesgue measure. While there are certain Non-measurable set, subsets of the plane which are not Lebesgue measurable, all subsets of the plane have a Banach measure. On the other hand, the Lebesgue measure is countably additive while a Banach measure is only finitely additive (and is therefore known as a "Content (measure theory), content"). Stefan Banach proved the existence of Banach measures in 1923. This established in particular that paradoxical decompositions as provided by the Banach–Tarski paradox, Banach-Tarski paradox in Euclidean space R3 cannot exist in the Euclidean plane R2. Definition A Banach measure on R''n'' is a function \mu: (\R^n)\to [0,\infty] (assigning a non-negative Extended real number line, extended real number to each subset of R''n'') su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Property Of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such that A \bigtriangleup U is meager (where \bigtriangleup denotes the symmetric difference).. Definitions A subset A \subseteq X of a topological space X is called almost open and is said to have the property of Baire or the Baire property if there is an open set U\subseteq X such that A \bigtriangleup U is a meager subset, where \bigtriangleup denotes the symmetric difference. Further, A has the Baire property in the restricted sense if for every subset E of X the intersection A\cap E has the Baire property relative to E. Properties The family of sets with the property of Baire forms a σ-algebra. That is, the complement of an almost open set is almost open, and any countable union or intersection of almost open sets is again almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Argument
Diagonal argument can refer to: * Diagonal argument (proof technique), proof techniques used in mathematics. A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: *Cantor's diagonal argument (the earliest) *Cantor's theorem *Russell's paradox * Diagonal lemma ** Gödel's first incompleteness theorem ** Tarski's undefinability theorem *Halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ... * Kleene's recursion theorem See also * Diagonalization (other) {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representative (mathematics)
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
