empty set

All completeness properties are described along a similar scheme: one describes a certain class of subsets of a partial order that are required to have a supremum or infimum. Hence every completeness property has its dual, obtained by inverting the order-dependent definitions in the given statement. Some of the notions are usually not dualized while others may be self-dual (i.e. equivalent to their dual statements). ...

See also:

Completeness order theory, Completeness order theory - Types of completeness properties, Completeness order theory - Least and greatest elements, Completeness order theory - Finite completeness, Completeness order theory - Further completeness conditions, Completeness order theory - Relationships between completeness properties, Completeness order theory - Completions of domains, Completeness order theory - Completeness in terms of universal algebra, Completeness order theory - Completeness in terms of adjunctions, Completeness order theory - Notes, Completeness order theory - Reference

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Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for utmost rigour and generality, creating some new terminology and concepts along the way. While Nicolas Bourbaki is an invented personage, the Bourbaki group is officially known as the Association des collaborateurs de Nicolas Bourbaki< ...

Including:

• Nicolas Bourbaki - Books by Bourbaki
• Nicolas Bourbaki - Influence on mathematics in general
• Nicolas Bourbaki - The group
• Nicolas Bourbaki - The Bourbaki perspective and its limitations
• Nicolas Bourbaki - Dieudonné as speaker for Bourbaki
• Nicolas Bourbaki - The Bourbachique influence: education institutions trends

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In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. Specifically, it deals with the set operations of intersection, union, complement; and the logic operations of AND, OR, NOT. For example, the logical assertion that a statement a and its negation ¬a cannot both be true, parallels the set-theory assertion that a subset Including:

• Boolean algebra - Formal definition
• Boolean algebra - Examples
• Boolean algebra - Order theoretic properties
• Boolean algebra - Principle of duality
• Boolean algebra - Other notation
• Boolean algebra - Homomorphisms and isomorphisms
• Boolean algebra - Boolean rings ideals and filters
• Boolean algebra - Representing Boolean algebras
• Boolean algebra - Axiomatics for Boolean algebras

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In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Fraenkel axioms, the axiom reads: or in words: There is a set N, such that the empty set is in N and such that whenever x is a member of N, the set formed by taking the union of x with its singleton {x} is also a member of

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

Including:

• Category theory - Background
• Category theory - Historical notes
• Category theory - Categories objects and morphisms
• Category theory - Some properties of morphisms
• Category theory - Functors
• Category theory - Natural transformations and isomorphisms
• Category theory - Universal constructions limits and colimits
• Category theory - Equivalent categories
• Category theory - Further concepts and results
• Category theory - Higher-dimensional categories

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In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms ...

Including:

• Category of topological spaces - Top is a concrete category
• Category of topological spaces - Limits and colimits
• Category of topological spaces - Other properties
• Category of topological spaces - Relationships to other categories

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The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. Algebra of sets - Introduction. The algebra of sets is the development of the fundamental properties of set operations and set relations. These properti ...

Including:

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

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In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the power set of a countably infinite set is un-countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below. Cantor's theorem - The proof. Including:

• Cantor's theorem - The proof
• Cantor's theorem - A detailed explanation of the proof when X is countably infinite
• Cantor's theorem - History

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Gottfried Wilhelm von Leibniz (also Leibnitz) (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath, deemed a genius in his lifetime and since, and the last true polyhistor. Trained as a lawyer and active as a diplomat and librarian, he wrote on philosophy, science, mathematics, theology, history, and comparative philology, even writing verse. Through his service to two major German noble houses, he played a major role in the European ...

Including:

• Gottfried Leibniz - Life
• Gottfried Leibniz - Early life and education
• Gottfried Leibniz - Career
• Gottfried Leibniz - Writings
• Gottfried Leibniz - Posthumous reputation
• Gottfried Leibniz - Philosopher
• Gottfried Leibniz - Metaphysics
• Gottfried Leibniz - Theodicy and optimism
• Gottfried Leibniz - Symbolic thought
• Gottfried Leibniz - Characteristica Universalis Universal characteristic and Calculus Ratiocinator
• Gottfried Leibniz - Formal logic
• Gottfried Leibniz - Mathematician
• Gottfried Leibniz - Topology
• Gottfried Leibniz - The dispute over who first invented the calculus
• Gottfried Leibniz - Science and technology
• Gottfried Leibniz - The vis viva
• Gottfried Leibniz - Information technology
• Gottfried Leibniz - Philology
• Gottfried Leibniz - The Sinophile
• Gottfried Leibniz - Works
• Gottfried Leibniz - Secondary literature
• Gottfried Leibniz - Quotes

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In combinatorics, a greedoid is a type of set system. It rises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of optimization problems that can be solved by greedy algorithms. Around 1980, Korte and Lovász introduced the greedoid to further generalize this characterization of greedy algorithms; hence the name greedoid. Besides mathematical optimization, greedoids have also been connected to graph theory, language theory ...

Including:

• Greedoid - Basic examples
• Greedoid - Greedy algorithm

Read more here: » Greedoid: Encyclopedia - Greedoid

In graph theory, certain vector spaces over the two-element field Z2 are associated with an undirected graph; this allows one to use the tools of linear algebra to study graphs. Let G be a finite simple undirected graph with edge set E. The power set of E becomes a Z2-vector space if we take the symmetric difference as addition. Every element of this vector space can be thought of as a linear combination of edges with coefficient from Z2. In yet another interpr ...

Including:

• Cycle space - The binary cycle space
• Cycle space - The integral cycle space
• Cycle space - The cycle space over a field or commutative ring

Read more here: » Cycle space: Encyclopedia - Cycle space

In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. This article uses mathematical symbols. Union set theory - Basic definition. If A and B are sets, then the union of A and B is the set that contains all elements of A and all elements of B, but no other elements. The union of A and B is usually written "A ∪ B< ...

Including:

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

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"Ø", "ø" is a vowel and a letter used in the Danish, Faroese and Norwegian alphabets. Amongst the English vowels it sounds the most like the "ir" in "bird" [1] or the "ur" in "hurt" [2], as pronounced in a non-rhotic accent, like Received Pronunciation. The name of the letter is the same as the sound it makes. The origin of the letter is a ligature for the diphthong "oe" (the horizontal line of the "e" being written across the "o") that has become a letter in itself. In modern Danish, Faroese and Norwegian, the ...

Including:

• Ø - Not to be confused with

Read more here: » Ø: Encyclopedia - Ø

In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. Universe mathematics - In a specific context. There are several precise versions of this general idea. Perhaps the simplest is that any set can be a universe, so long as you are studying that particular set. So if ...

Including:

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

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In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path. It is usually easy to think about what is not connected. A simple example would be a space consisting of two rectangles, each of whic ...

Including:

• Connected space - Formal definition
• Connected space - Examples
• Connected space - Path connectedness
• Connected space - Local connectedness
• Connected space - Theorems

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In topology and related branches of mathematics, a closed set is a set whose complement is open. This implies that a closed set contains its own boundary; the most familiar example of a closed set is the closed interval [a,b] of real numbers. Intuitively, if you are outside the set, and you "wiggle" a little bit, you will still be outside the set. Note that this notion depends on the concept of "outside", the surrounding space with respect to which the complement is taken. For instance, the unit interval [0,1] is closed ...

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In mathematics a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f(x) = 4, then f is constant since f maps any value to 4. More formally, a function f : A → B, is a constant function if f(x) = f(y) for all x and y in A. Notice that every empty function, that is, any function whose domain equals the empty set, is included in the above definition vacuously, sin ...

Including:

• Constant function - Properties

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In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. Basically, this means the definition is the same as the product but with all arrows reversed. Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The formal definition is as follows: Let C be a category and let {Xj | j ∈ J} be a indexed family of objects in C. The coproduct of the set {Xj} is an object ...

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0 1 2 3 4 5 6 7 8 9 >> List of numbers -- Integers 0 10 20 30 40 50 60 70 80 90 >> 0 (zero), alternatively called naught, nil, ought, or nought, is both a number and a numeral. It was the last numeral to be created in most numerical systems, as it is not a counting number (which is to say, one begins counting at the number 1) and was in many eras and places represented only by a gap or mark very different ...

Including:

• 0 number - 0 as a number
• 0 number - 0 as a numeral
• 0 number - History
• 0 number - Etymology
• 0 number - Babylonians and Greeks
• 0 number - First use of the number
• 0 number - Zero as a decimal digit
• 0 number - In mathematics
• 0 number - Elementary algebra
• 0 number - Extended use of zero in mathematics
• 0 number - In physics
• 0 number - In computer science
• 0 number - Numbering from 1 or 0?
• 0 number - Null value
• 0 number - Null pointer
• 0 number - Negative zero
• 0 number - Distinguishing zero from O
• 0 number - In other fields

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André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. He is known for his foundational work in number theory and algebraic geometry. He was a founding member, and de facto the early leader, of the influential Bourbaki group. The philosopher Simone Weil was his sister. André Weil - Life. Born in Paris to Alsatian parents who fled the annexation of Alsace-Lorraine to Germany, he studied in Paris, Rome and Göttingen and received his doctorate in 1928. He s ...

Including:

• André Weil - Life
• André Weil - Work
• André Weil - Books
• André Weil - External link
• André Weil - Bibliography

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