isomorphism

A function f is bijective if and only if its inverse relation f-1 is a function. In that case, f-1 is a bijection. The composition (mathematics) gf of two bijections f XY and g YZ is a bijection. The inverse of gf is (gf)-1 = (f-1)(g-1). On the other hand, if the composition g o f of two functions is bijective, we can only say ...

See also:

Bijection, Bijection - Composition and inverses, Bijection - Bijections and cardinality, Bijection - Examples and counterexamples, Bijection - Properties, Bijection - Bijections and category theory, Bijection - Properties, Bijection - Category theory

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This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. Cartan connection applications - Vierbeins et cetera. The vierbein or tetrad theory is the special case of a four-dimensional manifold. It applies to metrics of any signature. In any dimension, for a pseudo Riemannian geometry (with metric signature (p,q)), this Cartan connection theory is an alternative method in differential geometry. In different contexts it ha ...

Including:

• Cartan connection applications - Vierbeins et cetera
• Cartan connection applications - The basic ingredients
• Cartan connection applications - Constructions
• Cartan connection applications - The Palatini action

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In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. Categories appear in virtually every branch of modern mathematics and are a central unifying notion. The study of categories in their own right is known as category theory. For more extensive motivational background and historical notes, see category theory and the list of category theory topics. Category mathematics - Definition. A category C consists of Including:

• Category mathematics - Definition
• Category mathematics - Examples
• Category mathematics - Types of morphisms
• Category mathematics - Types of categories

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In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form: anxn + an−1xn−1 + ··· + a1x + a0 = 0 wh ...

Including:

• Algebraic number - The field of algebraic numbers
• Algebraic number - Numbers defined by radicals
• Algebraic number - Algebraic integers
• Algebraic number - Special classes of algebraic number

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In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. (Some authors use the term "algebra" synonymously with "associative algebra", but Wikipedia does not. Note also the other uses of the word listed in the algebra article.) Algebra over a field - Definitions. To be precise, let K ...

Including:

• Algebra over a field - Definitions
• Algebra over a field - Properties
• Algebra over a field - Kinds of algebras and examples
• Algebra over a field - Index-free notation
• Algebra over a field - K-algebra morphism

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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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Relativism is the view that the meaning and value of human beliefs and behaviors have no absolute reference. Relativists claim that humans understand and evaluate beliefs and behaviors only in terms of, for example, their historical and cultural context. Philosophers identify many different kinds of relativism depending upon what allegedly depends on something and what something depends on. The term is often used for truth relativism - the doctrine th ...

Including:

• Relativism - Advocates of relativism
• Relativism - Arguments against relativism
• Relativism - Counter-arguments
• Relativism - The Catholic Church and relativism
• Relativism - John Paul II
• Relativism - Benedict XVI
• Relativism - See Also

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Analogy is either the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), or a linguistic expression corresponding to such a process. In a narrower sense, analogy is an inference or an argument from a particular to another particular, as opposed to deduction, induction and abduction, where at least one of the premises or the conclusion is general. The word analogy can also refer to the relation between the source and the target themselves, which is often, though not necessarily, a simil ...

Including:

• Analogy - Models and theories of analogy
• Analogy - Identity of relation
• Analogy - Shared abstraction
• Analogy - Special case of induction
• Analogy - Hidden deduction
• Analogy - Shared structure
• Analogy - High-level perception
• Analogy - Applications and types of analogy
• Analogy - Linguistics
• Analogy - Mathematics
• Analogy - Artificial intelligence
• Analogy - Anatomy
• Analogy - Law
• Analogy - Engineering

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In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. For example, 3 + 2i is a complex number, where 3 is called the real part and 2 the imaginary part. Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one corresponde ...

Including:

• Complex number - Definition
• Complex number - The complex number field
• Complex number - The complex plane
• Complex number - Absolute value conjugation and distance
• Complex number - Complex number division
• Complex number - Matrix representation of complex numbers
• Complex number - Geometric interpretation of the operations on complex numbers
• Complex number - Some properties
• Complex number - Real vector space
• Complex number - Solutions of polynomial equations
• Complex number - Algebraic characterization
• Complex number - Characterization as a topological field
• Complex number - Complex analysis
• Complex number - Applications
• Complex number - Control theory
• Complex number - Signal analysis
• Complex number - Improper integrals
• Complex number - Quantum mechanics
• Complex number - Relativity
• Complex number - Applied mathematics
• Complex number - Fluid dynamics
• Complex number - Fractals
• Complex number - History

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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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In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of a (or na when the notation is additive). That is, we say G is cyclic if G = { an for any integer n }. Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group G that contains a is ...

Including:

• Cyclic group - Properties
• Cyclic group - Examples
• Cyclic group - Representation
• Cyclic group - Subgroups
• Cyclic group - Endomorphisms

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In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence. The conceptual difference between uniform and topological structures is that in a uniform space, you can formalize the idea that "x is as close to a as y is to b", while in a topological space you can only formalize "x ...

Including:

• Uniform space - History
• Uniform space - Definition
• Uniform space - Entourage definition
• Uniform space - Uniform cover definition
• Uniform space - Pseudometrics definition
• Uniform space - Intuition
• Uniform space - Examples
• Uniform space - Uniformly continuous functions
• Uniform space - Topology of uniform spaces
• Uniform space - Completeness

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In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices and column vectors), which we know how to do explicitly. Coordinate vector - Definition. Let V be a vector space of dimension n over a field F an ...

Including:

• Coordinate vector - Definition
• Coordinate vector - The standard representation
• Coordinate vector - Examples
• Coordinate vector - Example 1
• Coordinate vector - Example 2
• Coordinate vector - Basis transformation matrix

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In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. Functor - Definition. Let C and D be ca ...

Including:

• Functor - Definition
• Functor - Covariance and contravariance
• Functor - Examples
• Functor - Properties
• Functor - Relation to other categorical concepts

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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. In other words, it is a unital semigroup. Monoid - Definition. A monoid is a magma (M,*), i.e. a set M with binary operation * : M × M → M, obeying the following axioms: Associativity: for all a, b, c in M, (a*b)*c = a*(b*c) Identity ...

Including:

• Monoid - Definition
• Monoid - Examples
• Monoid - Properties
• Monoid - Monoid homomorphisms
• Monoid - Relation to category theory

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In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms. Axiom - Etymology. The word axiomIncluding:

• Axiom - Etymology
• Axiom - Mathematics
• Axiom - Logical axioms
• Axiom - Non-logical axioms
• Axiom - Role in mathematical logic
• Axiom - Further discussion

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A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. If one considers geometrical vectors, and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and combinations of these operations, and so on, we arrive at a description of a m ...

Including:

• Vector space - Formal definition
• Vector space - Elementary properties
• Vector space - Examples
• Vector space - Subspaces and bases
• Vector space - Linear transformations
• Vector space - Generalizations and additional structures

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In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e. one which is considered equivalent to it. In group theory, for example, we may have a group G acting on a set X, in which case we say that two elements of X are equivalent "up to the group action" if they lie in the same orbit. Up to - Exam ...

Including:

• Up to - Examples

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Canonical is an adjective derived from canon. It essentially means "standard", "generally accepted" or "part of the back-story." basic, canonic, canonical: reduced to the simplest and most significant form possible without loss of generality, e.g. "a basic story line"; "a canonical syllable pattern" Canonical - Religion. This word is used by theologians and canon lawyers to refer to the canons of the Eastern Orthodox and Roman Catholic churches, adopted by ecumenical councils.

• Canonical - Religion
• Canonical - Literature and art
• Canonical - Mathematics
• Canonical - Computer science
• Canonical - Physics

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In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion. It should be noted that "complete" here is just a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field, compactification, or Gödel's incompleteness theorem. A metric spac

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