codomain

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. Formally, given a partially ordered set (P, ≤), an element u of P is an upper bound of a subset S of P, if s ≤ u, for all elements s of S. Using ≥ instead of ≤ leads to ...

Including:

• Upper bound - Bounds of functions

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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, 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 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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List of mathematical functions - Polynomials. Polynomials: can be generated by addition and multiplication alone. Constant function: Zero degree polynomial, fixed value regardless of arguments. Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial.See also:

List of mathematical functions, List of mathematical functions - Classes of functions, List of mathematical functions - Elementary functions, List of mathematical functions - Polynomials, List of mathematical functions - Elementary periodic functions, List of mathematical functions - Elementary transcendental functions, List of mathematical functions - Special functions, List of mathematical functions - Antiderivatives of elementary functions, List of mathematical functions - Gamma and related functions, List of mathematical functions - Elliptic and related functions, List of mathematical functions - Bessel and related functions, List of mathematical functions - Riemann zeta and related functions, List of mathematical functions - Hypergeometric and related functions, List of mathematical functions - Other standard special functions, List of mathematical functions - Number theoretic functions, List of mathematical functions - Miscellaneous

Read more here: » List of mathematical functions: Encyclopedia II - List of mathematical functions - Elementary functions

The modern idea of a mathematical function was introduced by Leibniz, and the associated notation y = f(x) was invented by Leonhard Euler, in the 18th century. But the intuitive idea of a function as any rule or procedure that assigns an output to each given input proved to be naive. Joseph Fourier, for example, claimed that every function had a Fourier series, something no mathematician would claim today. The concept of a function was not put on a rigorous basis u ...

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Function mathematics, Function mathematics - Introduction, Function mathematics - Functions of more than one variable, Function mathematics - History, Function mathematics - Formal definition, Function mathematics - Domains codomains and ranges, Function mathematics - Injective surjective and bijective functions, Function mathematics - Images and preimages, Function mathematics - Graph of a function, Function mathematics - Examples of functions, Function mathematics - Properties of functions, Function mathematics - Ambiguous functions, Function mathematics - n-ary function: function of several variables, Function mathematics - Composing functions, Function mathematics - Inverse function, Function mathematics - Restrictions and extensions, Function mathematics - Pointwise operations, Function mathematics - Computable and non-computable functions, Function mathematics - Functions from the categorical viewpoint

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Introduction

Order of a group. Order of a group (G,*) is the cardinality (i.e. number of elements) of G. A group with finite order is called a finite group. Order of an element of a group. Suppose x∈G and there exists a positive integer m such that xm = e, then the smallest possible m is called the order of x. The order of a finite group is divisible by the order of every element. Subgroup. A subset H of a group (G,*) which remains a group when the operation * is r ...

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Glossary of group theory, Glossary of group theory - Basic definitions, Glossary of group theory - Types of groups

Read more here: » Glossary of group theory: Encyclopedia II - Glossary of group theory - Basic definitions

An input to a function is called argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x under f. The image of x can be written as f(x) or as y. Written mathematics sometimes omits the parentheses around the argument, thus: sin x, but calculators and computers require parentheses around the argument. In some branches of mathematics, such as automata theory, th ...

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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 - The vocabulary of functions, 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 - Set of all functions, Function mathematics - Is a function more than its graph?, Function mathematics - Partial functions and multi-functions, 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

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - The vocabulary of functions

When considering graded vector spaces, the nicest linear maps are those which respect the grading. With this in mind, we define a linear map T between M-graded vector space V and N-graded vector space W to be such that for every m in M, there is some n in N with Then the vector space L(V,W) of graded linear maps is itself an M×N-graded vector space, where M×N is the Cartesian product, since for each choice of homogeneous subspace in the domain, the map may choose a different range homogen ...

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Graded vector space, Graded vector space - Graded vector spaces, Graded vector space - I-graded vector spaces, Graded vector space - Linear maps

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Main article: limit of a function Suppose f(x) is a real function and c is a real number. The expression: means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of f(x), as x approaches c". Note that this statement can be true even if f(c) L. Indeed, the function f(x) need not even b ...

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Limit mathematics, Limit mathematics - Limit of a function, Limit mathematics - Formal definition, Limit mathematics - Limit of a function at infinity, Limit mathematics - Limit of a sequence, Limit mathematics - Topological net, Limit mathematics - Limit in category theory

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Let S be a set with a binary operation *. If e is an identity element of (S,*) and a * b = e, then a is called a left inverse of b and b is called a right inverse of a. If an element x is both a left inverse and a right inverse of y, then x is called a two-sided inverse, or simply an inverse, of y. An element with a two-sided inverse ...

See also:

Inverse element, Inverse element - Introduction, Inverse element - Formal definition, Inverse element - Examples

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Douglas Hofstadter provides an informal definition: The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49) Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, ...

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Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications

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A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the domain and codomain, respectively, of the relation, and G is called its graph. The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation corresponds to viewing R as the ...

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Binary relation, Binary relation - Formal definition, Binary relation - Is a relation more than its graph?, Binary relation - Example, Binary relation - Special types of relations, Binary relation - Total or partial, Binary relation - Functional injective surjective bijective, Binary relation - Relations over a set, Binary relation - Operations on binary relations, Binary relation - Sets versus classes, Binary relation - Examples of common binary relations

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A category C is given by two pieces of data: a class of objects and a class of morphisms. There are two operations defined on every morphism, the domain (or source) and the codomain (or target). Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. The set of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y). (Some authors write MorC(X ...

See also:

Morphism, Morphism - Definition, Morphism - Types of morphisms, Morphism - Examples

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A full proof of the theorem would be too long to reproduce here, but the following paragraph outlines a proof omitting the difficult part. It is hoped that this will at least give some idea why the theorem might be expected to be true. Note that the boundary of D n is S n-1, the (n-1)-sphere. Suppose f : D n → D n is a continuous function that has no fixed point. ...

See also:

Brouwer fixed point theorem, Brouwer fixed point theorem - Proof outline, Brouwer fixed point theorem - Generalizations, Brouwer fixed point theorem - External link

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The most common operation on a sieve is pullback. Pulling back a sieve S on c by an arrow f:c′→c gives a new sieve f*S on c′. This new sieve consists of all the arrows in S which factor through c′. There are several equivalent ways of defining f*S. The simplest is: For any object d of C, f*S(d) = { gSee also:

Sieve category theory, Sieve category theory - Definition, Sieve category theory - Pullback of sieves, Sieve category theory - Properties of sieves

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A precise definition is required for the purposes of mathematics. A function is a binary relation, f, with the property that for an element x there is no more than one element y such that x is related to y. This uniquely determined element y is denoted f(x). Because two definitions of binary relation are in use, there are actually two definitions of function, in ...

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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

Read more here: » Function mathematics: Encyclopedia II - Function mathematics - Mathematical definition of a function

As promised in the introduction, the limit of a pointwise convergent, equicontinuous sequence is continuous. Theorem 1: Let {fn} be an equicontinuous sequence of functions. If fn(x) → f(x) for every x ∈ X, then the function f is continuous. The condition in the above theorem can be slightly weakened. It suffices if the se ...

See also:

Equicontinuity, Equicontinuity - Definitions, Equicontinuity - Properties, Equicontinuity - Generalizations

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A stochastic process is a random function, that is a random variable X defined on a probability space (Ω , Pr) with values in a space of functions F. The space F in turn consists of functions I → D. Thus a stochastic process can also be regarded as an indexed collection of random variables {Xi}, where the index i ranges through an index set I, defined on the probability space (Ω, Pr) and taking values on the same codomain D (often the real numbers R). This view of a stochastic process as an indexed colle ...

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Stochastic process, Stochastic process - Definition, Stochastic process - Examples, Stochastic process - Interesting special cases, Stochastic process - Constructing stochastic processes, Stochastic process - The Kolmogorov extension, Stochastic process - Separability or what the Kolmogorov extension does not provide

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The above diagram represents a partial function that is not a total function since the element 1 in X is not associated with anything. Until the second half of the 20th century, only total functions were considered "well-defined". Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the ...

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Partial function, Partial function - Domain of a partial function, Partial function - Discussion and examples

Read more here: » Partial function: Encyclopedia II - Partial function - Discussion and examples