apostrophe ( ' ) ( ’ ) brackets ( ( ) ) ( [ ] ) ( { } ) ( 〈 〉 ) colon ( : ) comma ( , ) dashes ( ‒ ) ( – ) ( — ) ( ― ) ellipsis ( … ) ( ... ) exclamation mark ( ! ) full stop/period ( . ) hyphen ( - ) ( ‐ ) interrobang ( < ...

Including:

• Bracket - In writing
• Bracket - Types of brackets
• Bracket - In computing
• Bracket - Layout rules
• Bracket - In mathematics
• Bracket - In sports
• Bracket - In mechanics and structures
• Bracket - In sociology
• Bracket - Reference

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U is the twenty-first letter of the modern Latin alphabet. U was originally a positional variant of the letter V, as J was of I, used only in lower-case writing and only medially, and representing both the vowels now written with U and the consonants now written with V. The use of the two forms to distinguish the consonants and vowels which they now represent was not standardised until modern times. U - Alternative representations. Uniform represents the letter U in the NATO phonetic alphabet, ...

Including:

• U - Alternative representations
• U - Computing
• U - Meanings for U

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

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

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Aleph or alef has several meanings: Aleph or Alef, first letter of many Semitic alphabets including Phoenician, Hebrew and Aramaic. ʾalif is the equivalent letter of the Arabic alphabet. Alef, the concurrent programming language used in early editions of Plan 9; Alef, one of the main protagonists in SEGA's Shining Force;

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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 mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

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Information theory is the mathematical theory of data communication and storage, generally considered to have been founded in 1948 by Claude E. Shannon. The central paradigm of classic information theory is the engineering problem of the transmission of information over a noisy channel. The most fundamental results of this theory are Shannon's source coding theorem, which establishes that on average the number of bits needed to communicate the result of an uncertain event is given by the entropy of that uncertainty; and ...

Including:

• Information theory - Mathematical preliminaries
• Information theory - Basic Concepts for Discrete Channels
• Information theory - Self-information
• Information theory - Entropy
• Information theory - Joint Entropy
• Information theory - Conditional Entropy Equivocation
• Information theory - Mutual Information Transinformation
• Information theory - Continuous Channels
• Information theory - Channel Capacity
• Information theory - Source Theory
• Information theory - Rate
• Information theory - Fundamental Theorem
• Information theory - Statement Noisy-Channel Coding Theorem
• Information theory - Channel capacity of particular model channels
• Information theory - Related Concepts
• Information theory - Measure Theory
• Information theory - Kolmogorov Complexity
• Information theory - Applications
• Information theory - Coding Theory
• Information theory - Cryptography Cryptanalysis
• Information theory - Relation with thermodynamic entropy
• Information theory - Quantum Information Theory
• Information theory - Detection and Estimation Theory
• Information theory - Gambling
• Information theory - Intelligence
• Information theory - Music
• Information theory - History

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A definition may be a statement of the essential properties of a certain thing, or a statement of equivalence between one expression and another, usually more complex expression that gives the meaning of the first. These two senses are not mutually exclusive, nor are they equivalent. A thing being defined is called (from Latin) a definiendum; the expression which defines it is called a definiens. Definition - Kinds of definition. A number of different kinds and techniques of definition ...

Including:

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

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The distance between two points is the length of a straight line segment between them. In the case of two locations on Earth, usually the distance along the surface is meant: either "as the crow flies" (along a great circle) or by road, railroad, etc. Distance is sometimes expressed in terms of the time to cover it, for example walking or by car. Sometimes a distance thus indicated is ambiguous because the means ...

Including:

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

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Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

• Infinity - History
• Infinity - Ancient view of infinity
• Infinity - Views from the Renaissance to modern times
• Infinity - Modern philosophical views
• Infinity - Infinity symbol
• Infinity - Mathematical infinity
• Infinity - Infinity in real analysis
• Infinity - Infinity in complex analysis
• Infinity - Arithmetic properties of infinity
• Infinity - Infinity in set theory
• Infinity - Mathematics without infinity
• Infinity - Use of infinity in common speech
• Infinity - Physical infinity
• Infinity - Infinity in cosmology
• Infinity - Three types of infinities
• Infinity - Infinity in science fiction
• Infinity - Note

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In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase. Algebraic structure - In the ...

Including:

• Algebraic structure - In the sense of universal algebra
• Algebraic structure - Allowing axioms other than identities
• Algebraic structure - Allowing additional structure
• Algebraic structure - Categories

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The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way. The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — m ...

Including:

• Areas of mathematics - Foundations / general
• Areas of mathematics - Algebra
• Areas of mathematics - Analysis
• Areas of mathematics - Geometry
• Areas of mathematics - Applied mathematics
• Areas of mathematics - Probability and statistics
• Areas of mathematics - Computational sciences
• Areas of mathematics - Physical sciences
• Areas of mathematics - Non-physical sciences

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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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In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). See names of numbers in English. In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. While for finite sets the size is given by a natural number, the number of elements, cardinal numbers (cardinality ...

Including:

• Cardinal number - History
• Cardinal number - Motivation
• Cardinal number - Formal definition
• Cardinal number - Cardinal arithmetic
• Cardinal number - The continuum hypothesis

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Boolean logic, is a complete system for logical operations. It was named after George Boole, an English mathematician at University College Cork who first defined an algebraic system of logic in the mid 19th century. Boolean logic has many applications in electronics, computer hardware and software. In 1938, Claude Shannon showed how electric circuits with relays were a model for Boolean logic. This fact soon proved enormously conseque ...

Including:

• Boolean logic - Terms
• Boolean logic - Example
• Boolean logic - Chaining operations together
• Boolean logic - Use of parentheses
• Boolean logic - Properties
• Boolean logic - Truth tables
• Boolean logic - Other notation
• Boolean logic - Basic mathematics use of Boolean terms
• Boolean logic - English language use of Boolean terms
• Boolean logic - Applications
• Boolean logic - Digital electronic circuit design
• Boolean logic - Database applications
• Boolean logic - Search engine queries

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In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function is injective (one-to-one) if or, equivalently, if . One could also say that every element of the codomain (sometimes called range) is mapped to by at most one element (argument) of the domain; not every element of t ...

Including:

• Bijection injection and surjection - Injection
• Bijection injection and surjection - Surjection
• Bijection injection and surjection - Bijection
• Bijection injection and surjection - Cardinality
• Bijection injection and surjection - Examples
• Bijection injection and surjection - Injective and surjective bijective
• Bijection injection and surjection - Injective and non-surjective
• Bijection injection and surjection - Non-injective and surjective
• Bijection injection and surjection - Non-injective and non-surjective
• Bijection injection and surjection - Properties
• Bijection injection and surjection - Category theory
• Bijection injection and surjection - History

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In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the operation appears more than once in an expression. Put another way, no parentheses are required for an associative operation. Consider for instance the equation (5+2)+1 = 5+(2+1) Adding 5 and 2 gives 7, and adding 1 gives an end result of 8 for the left hand side. To evaluate the right hand side, we start with adding 2 and 1 giving 3, and then add 5 and 3 to get 8, again. S ...

Including:

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

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Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations. In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, ...

Including:

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

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In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). The second Hebrew letter (beth) is also used. To define the beth numbers, start by letting be the cardinality of countably infinite sets; for concreteness, take the set of natural numbers to be the typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define = the cardinality of the power set of A if is the cardina ...

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Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object — a number, a function, a set, a space of one sort or another — is "well-behaved" or not. While the term has no fixed formal definition, it can have fairly precise meaning within a given context. In pure mathematics, "well-behaved" objects are those that can be proved or analyzed by elegant means to have elegant properties. In both pure and applied mathematics, (optimization, numerical integration, or mathematical ph ...

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Intersection has various meanings in different contexts: In road transport, an intersection is a place where two roads meet. If there is a grade separation, it is an interchange. In mathematics (in particular, set theory), an intersection is the set of elements common to some collection of sets. In aviation, an intersection is an virtual navigational fix In post-classical cinema, Intersection (1994) is Mark Rydell's remake of Claude Sautet's 1970 French film, Les Choses de la vi

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