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A function is bijective if it is both injective and surjective. A bijective function is a bijection (one-one correspondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. This equivalent condition is formally expressed as follows. The function is bijective iff for all , there is a unique such that f(a) = b. A function f : A → B is bijective if and only if it ...

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Bijection, injection and surjection, 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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Alternative Health Dictionary , Hinduism Dictionary , Spiritual Dictionary, Sanskrit Dictionary , Parapsychology Dictionary, Paganism Dictionary,
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It is important to specify the domain and codomain of each function since by changing these, functions which we think of as the same may have different jectivity. Bijection, injection and surjection - Injective and surjective bijective. For every set A the identity function idA and thus specifically . and thus also its inverse . The exponential function and thus also its inverse the natural logarithm Bijection, injection and surjection - Injective and non- ...

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Bijection, injection and surjection, 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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The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, that's big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the s ...

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Infinity, 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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Claude E. Shannon (1916–2001) founded information theory with his classic paper "A Mathematical Theory of Communication", published in the Bell System Technical Journal in July and October of 1948. This work drew on earlier publications by Harry Nyquist and Ralph Hartley. At the beginning of his paper, Shannon asserted that "The fundamental problem of communication is that of reproducing at one point, either exactly or approximately ...

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Information theory, 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 category C is concretizable if there exists a faithful functor from C into Set. The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The statement that hTop is not concretizable says more than this simple observation, however; i ...

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Concrete category, Concrete category - Definition, Concrete category - Not all categories are concrete, Concrete category - Alternate definition

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Information theory - Coding Theory. Coding theory is the most important and direct application of information theory. It can be subdivided into data compression theory and error correction theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data. There are two formulations for the compression problem -- in lossless data compression the data must be reconstructed exactly, whereas lossy data compression examines how many bits are needed to recons ...

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Information theory, 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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Information theory - Measure Theory. Here is an interesting and illuminating connection between information theory and measure theory: If to arbitrary discrete random variables X and Y we associate the existence of sets and , somehow representing the information borne by X and Y, respectively, such that: and are disjoint whenever X and Y are independent, and whenever X and Y are such that either one is completel ...

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Information theory, 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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Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat and as the same, leading to the one-point compact ...

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Infinity, 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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Main articles: universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. Yet, in the definition of a category, objects are considered to be atomic; i.e. we do not know, whether an object A is a set, a topology, or any other a ...

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Category theory, 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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Main articles: category, morphism A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) [or Hom(a, b), or homC(a, b)] to denote the hom-class of all morphisms from < ...

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Category theory, 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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Information theory - Statement Noisy-Channel Coding Theorem. 1. For every discrete memoryless channel, the channel capacity has the following property. For any ε > 0 and R < C, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε. 2. If a probability of bit error pb is acceptable, ra ...

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Information theory, 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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Information theory - Self-information. Shannon defined a measure of information content called the self-information or surprisal of a message m: where p(m) = Pr(M = m) is the probability of the message m taken from an implicit message space M. Note: Strictly speaking, p(m) is the probability density func ...

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Information theory, 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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Information theory is based on and intimately connected with probability theory. In information theory, logarithms are used extensively. The base of the logarithms used here is deliberately left unspecified, following standard practice in information theory. As in many other applications of logarithms, it really does not matter much what base is used as long as one is consistent. In the case of information theory, the choice of logarithmic base merely determines the unit that is used to measure information. Common choices of logarithm ...

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Information theory, 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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In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite en ...

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Infinity, 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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Shannon information is appropriate for measuring uncertainty over a discrete space. Its basic measures have been extended by analogy to continuous spaces. The continuous entropy, joint entropy, and conditional entropy are not quite as meaningful as the corresponding measures in the discrete case. Here they can take on negative as well as positive values, and they are not invariant under linear transformation. The mutual information has the distinction of being the only one of Shannon's basic measures of information that remains fundamentally ...

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Information theory, 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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Any process that generates successive messages can be considered a source of information. Sources can be classified in order of increasing generality as memoryless, ergodic, stationary, and stochastic, (with each class strictly containing the previous one). The term "memoryless" as used here has a slightly different meaning than it normally does in probability theory. Here a memoryless source is defined as one that generates successive messages independently of one another and with a fixed probability distribution. However, the position of t ...

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Information theory, 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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Let us return for the time being to our consideration of the communications process over a discrete channel. At this time it will be helpful to have a simple model of the process: +-------+ | Noise | +-------+ | V +-----------+ Y +-------+ X +--------+ |Transmitter|-------->|Channel|-------->|Receiver ...

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Information theory, 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

Read more here: » Information theory: Encyclopedia II - Channel Capacity

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In theory, as long as there is no external interaction, the loop will continue to run for all time. In practice however, most programming loops co ...

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Infinity, 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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The functions f: X → Y and g: Y → Z can be composed by first applying f to an argument x and then applying g to the result. Thus one obtains a composite function g o f: X → Z defined by (g o f)(x) = g(f(x)) for all x in X. As an example, suppose that an airplane's height at time t is ...

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Function mathematics, Function mathematics - Mathematical definition of a function, Function mathematics - First definition, Function mathematics - Second definition, Function mathematics - History of the concept, Function mathematics - Functions in other fields, Function mathematics - Domain codomain argument image, Function mathematics - Graph of a function, Function mathematics - Specifying a function, Function mathematics - Functions with multiple inputs and outputs, Function mathematics - Functions of two or more variables, Function mathematics - Functions with output in a product set, Function mathematics - Binary operations, Function mathematics - Argument order and lambda notation, Function mathematics - Examples of functions, Function mathematics - Image of a set, Function mathematics - Range of a function, Function mathematics - Preimage of a set, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, Function mathematics - Classes of functions, Function mathematics - Injective surjective bijective, Function mathematics - Other properties, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Lambda calculus, Function mathematics - Functions in category theory

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The fundamental assumption of the relational model is that all data are represented as mathematical relationships, i.e., a subset of the Cartesian product of n sets. In the mathematical model, reasoning about such data is done in two-valued predicate logic (that is, without NULLs), meaning there are two possible evaluations for each proposition: either true or false. Data are operated upon ...

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Relational model, Relational model - The model, Relational model - Competition, Relational model - History, Relational model - Misimplementation, Relational model - Implementation, Relational model - Controversies, Relational model - Design, Relational model - Example database, Relational model - Set Theory Formulation, Relational model - Key constraints and functional dependencies

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