112 Sentences With "codomain" | Random Sentence Generator

A codomain is part of a function if is defined as a triple where is called the domain of , its codomain, and its graph. The set of all elements of the form , where ranges over the elements of the domain , is called the image of . The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements in its codomain for which the equation does not have a solution.

Pointwise convergence is the same as convergence in the product topology on the space YX, where X is the domain and Y is the codomain. If the codomain Y is compact, then, by Tychonoff's theorem, the space YX is also compact.

Every permutation of has the codomain equal to its domain and is bijective and invertible. If has more than one element, a constant function on has an image that is a proper subset of its codomain, and thus is not bijective (and hence not invertible). The function associating to each natural number the floor of has its image equal to its codomain and is not invertible. Finite endofunctions are equivalent to directed pseudoforests.

However, it is often assumed to have a structure of -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an -algebra, such as the complex numbers or the quaternions. The structure -vector space of the codomain induces a structure of -vector space on the functions. If the codomain has a structure of -algebra, the same is true for the functions.

In many instances, one can also construct a canonical inclusion into the codomain known as the range of .

The values for which an operation is defined form a set called its domain of definition or active domain. The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its codomain of definition, active codomain, image or range. For example, in the real numbers, the squaring operation only produces non- negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers. Operations can involve dissimilar objects: a vector can be multiplied by a scalar to form another vector (an operation known as scalar multiplication), and the inner product operation on two vectors produces a quantity that is scalar.

However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own doesn't determine the codomain. It is common to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective.

Vanamo 6:1-231 The codomain of this distance function ranges from 1 (identical proportional abundances) to 0 (no taxa shared).

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

The sequence is a consequence of the definition of convergence of a spectral sequence. The second page differential with codomain E21,0 originates from E2−1,1, which is zero by assumption. The differential with domain E21,0 has codomain E23,−1, which is also zero by assumption. Similarly, the incoming and outgoing differentials of Er1,0 are zero for all .

The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve. When the codomain of a function of a real variable is a finite- dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications.

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to- one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures.

The most common case of the pointwise product of two functions is when the codomain is a ring (or field), in which multiplication is well-defined.

A codomain is not part of a function if is defined as just a graph., [ pp. 10-11] For example in set theory it is desirable to permit the domain of a function to be a proper class , in which case there is formally no such thing as a triple . With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form .

These embedding results are useful because Banach spaces have a number of useful properties not shared by all metric spaces: they are vector spaces which allows one to add points and do elementary geometry involving lines and planes etc.; and they are complete. Given a function with codomain X, it is frequently desirable to extend this function to a larger domain, and this often requires simultaneously enlarging the codomain to a Banach space containing X.

The concept can be extended to operators that are homogeneous and subadditive. This requires only that the codomain be, say, an ordered vector space to make sense of the conditions.

This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).

A −1 that appears as a superscript of a function does not mean taking the (pointwise) reciprocal of that function, but rather, the inverse function (or more generally inverse relation) of the function. For example, f−1(x) is the inverse of f(x), or sin−1(x) is a notation of arcsine function. When a subset of the codomain is specified inside the function, it instead denotes the preimage of that subset of the codomain under the function.

Formally, if is a set, the identity function on is defined to be that function with domain and codomain which satisfies : for all elements in . In other words, the function value in (that is, the codomain) is always the same input element of (now considered as the domain). The identity function on is clearly an injective function as well as a surjective function, so it is also bijective. The identity function on is often denoted by .

Similarly, the "subset of" relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set of a specific set A): the resulting set relation can be denoted by ⊆A. Also, the "member of" relation needs to be restricted to have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown that assuming ∈ to be defined over all sets leads to a contradiction in naive set theory. Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment.

Generalizing matrices to linear transformations of vector spaces, the corank of a linear transformation is the dimension of the cokernel of the transformation, which is the quotient of the codomain by the image of the transformation.

The `TypeFamilies` extension in the Glasgow Haskell Compiler supports both type synonym families and data families. Type synonym families are the more flexible (but harder to type- check) form, permitting the types in the codomain of the type function to be any type whatsoever with the appropriate kind. Data families, on the other hand, restrict the codomain by requiring each instance to define a new type constructor for the function's result. This ensures that the function is injective, allowing clients' contexts to deconstruct the type family and obtain the original argument type.

Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification problem instead. To some extent, the different problems (regression, classification, fitness approximation) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.

Related terms such as domain, codomain, injective, and continuous can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.

A function g is the left (resp. right) inverse of a function f (for function composition), if and only if g \circ f (resp. f \circ g) is the identity function on the domain (resp. codomain) of f.

An endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism. Let be an arbitrary set. Among endofunctions on one finds permutations of and constant functions associating to every in the same element in .

The technique of using continuous color to map points from domain to codomain or image plane was used in 1999 by George Abdo and Paul Godfrey and colored grids were used in graphics by Doug Arnold that he dates to 1997.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any set.

The domain anti-restriction (or domain subtraction) of a function or binary relation (with domain and codomain ) by a set may be defined as ; it removes all elements of from the domain . It is sometimes denoted ⩤ .Dunne, S. and Stoddart, Bill Unifying Theories of Programming: First International Symposium, UTP 2006, Walworth Castle, County Durham, UK, February 5-7, 2006, Revised Selected ... Computer Science and General Issues). Springer (2006) Similarly, the range anti-restriction (or range subtraction) of a function or binary relation by a set is defined as ; it removes all elements of from the codomain .

Each axiom defines a relation of immediate domination of a parent over its children. The union of these relations is control. Among other things, the axioms establish the relationships of an object for invocation in time and space, input and output (domain and codomain), input access rights and output access rights (domain access rights and codomain access rights), error detection and recovery, and ordering during its developmental and operational states. Every system can ultimately be defined in terms of three primitive control structures, each of which is derived from the six axioms – resulting in a universal semantics for defining systems.

An n-ary operation ω from to Y is a function . The set is called the domain of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer n (the number of operands) is called the arity of the operation. Thus a unary operation has arity one, and a binary operation has arity two. An operation of arity zero, called a nullary operation, is simply an element of the codomain Y. An n-ary operation can also be viewed as an -ary relation that is total on its n input domains and unique on its output domain.

The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general.

As long as one does not quantify over functions in the latter sense, all such uses are in principle eliminable. Outside of formal set theory, we usually specify a function in terms of its domain and codomain, as in the phrase "Let f: A \to B be a function". The domain of a function is just its domain as a relation, but we have not yet defined the codomain of a function. To do this we introduce the terminology that a function is from A to B if its domain equals A and its range is contained in B. In this way, every function is a function from its domain to its range, and a function f from A to B is also a function from A to C for any set C containing B. Indeed, no matter which set we consider to be the codomain of a function, the function does not change as a set since by definition it is just a set of ordered pairs.

Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. If A is some algebraic structure, the set of all functions X to the carrier set of A can be turned into an algebraic structure of the same type in an analogous way.

Each primitive function (or type) can be realized as a top node of a map on a lower (more concrete) layer of the system. Resident at every node on a map is the same kind of object (for example, a function on every node of an FMap and a type on a TMap). The object at each node plays multiple roles; for example, the object can serve as a parent (in control of its children) or a child (being controlled by its parent). Whereas each function on an FMap has a mapping from its input to output (domain to codomain), each type on a TMap has a relation between its domain and codomain.

In a treatment of predicate logic that allows one to introduce new predicate symbols, one will also want to be able to introduce new function symbols. Given the function symbols F and G, one can introduce a new function symbol F ∘ G, the composition of F and G, satisfying (F ∘ G)(X) = F(G(X)), for all X. Of course, the right side of this equation doesn't make sense in typed logic unless the domain type of F matches the codomain type of G, so this is required for the composition to be defined. One also gets certain function symbols automatically. In untyped logic, there is an identity predicate id that satisfies id(X) = X for all X. In typed logic, given any type T, there is an identity predicate idT with domain and codomain type T; it satisfies idT(X) = X for all X of type T. Similarly, if T is a subtype of U, then there is an inclusion predicate of domain type T and codomain type U that satisfies the same equation; there are additional function symbols associated with other ways of constructing new types out of old ones.

Often, the use of the term operation implies that the domain of the function includes a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain), although this is by no means universal, as in the case of dot product, where vectors are multiplied and result in a scalar. An n-ary operation is called an internal operation. An n-ary operation where is called an external operation by the scalar set or operator set S. In particular for a binary operation, is called a left-external operation by S, and is called a right-external operation by S. An example of an internal operation is vector addition, where two vectors are added and result in a vector.

In category theory, "map" is often used as a synonym for "morphism" or "arrow", and thus is more general than "function". For example, a morphism f:\, X \to Y in a concrete category (i.e. a morphism which can be viewed as functions) carries with it the information of its domain (the source X of the morphism) and its codomain (the target Y). In the widely used definition of a function f:X\to Y, f is a subset of X\times Y consisting of all the pairs (x,f(x)) for x\in X. In this sense, the function does not capture the information of which set Y is used as the codomain; only the range f(X) is determined by the function.

Suppose one has two (or more) functions having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as . Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) a composition monoid. In general, transformation monoids can have remarkably complicated structure.

In fact, to turn an injective function into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range . That is, let such that for all x in X; then g is bijective. Indeed, f can be factored as , where is the inclusion function from J into Y. More generally, injective partial functions are called partial bijections.

Similarly, function defined on domain and having the same codomain is an upper bound of , if for each in . Function is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set. The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.

We outline here a formal group equivalent of the Frobenius element, which is of great importance in class field theory,e.g. Serre (1967). generating the maximal unramified extension as the image of the reciprocity map. For this example we need the notion of an endomorphism of formal groups, which is a formal group homomorphism f where the domain is the codomain.

More generally, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra. If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps.

Kelley did not require the map to be order preserving while the definition of an AA-subnet does away entirely with any map between the two nets' domains and instead focuses entirely on (i.e. the nets' common codomain). Every Willard-subnet is a Kelley-subnet and both are AA-subnets. In particular, if is a Willard-subnet or a Kelley-subnet of then .

The codomain for the models that utilize adaptive collaborative control are queries, information statements, and responses from the agent. Queries and information statements are elements of the dialogue exchange at the finite-state machine level. Queries from the agent are the system's way of soliciting a response from a human operator. This is particularly important when the agent is physically stuck or at a logical impasse.

Uniform convergence implies pointwise convergence and uniform Cauchy convergence. Uniform Cauchy convergence and pointwise convergence of a subsequence imply uniform convergence of the sequence, and if the codomain is complete, then uniform Cauchy convergence implies uniform convergence. If the domain of the functions is a topological space, local uniform convergence (i.e. uniform convergence on a neighborhood of each point) and compact (uniform) convergence (i.e.

In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introduced by in order to reformulate the notion of a Grothendieck topology.

The composition of functions is always associative—a property inherited from the composition of relations. That is, if , , and are composable, then . Since the parentheses do not change the result, they are generally omitted. In a strict sense, the composition is only meaningful if the codomain of equals the domain of ; in a wider sense, it is sufficient that the former be a subset of the latter.

If maps to , then maps back to . Let be a function whose domain is the set , and whose codomain is the set . Then is invertible if there exists a function with domain and image (range) , with the property: : f(x) = y\,\,\Leftrightarrow\,\,g(y) = x. If is invertible, then the function is unique, which means that there is exactly one function satisfying this property.

A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. 1.18 Theorem Let \Lambda be a linear functional on a topological vector space .

The definitions can be generalized to functions and even to sets of functions. Given a function with domain and a preordered set as codomain, an element of is an upper bound of if for each in . The upper bound is called sharp if equality holds for at least one value of . It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.

More generally, the restriction (or domain restriction or left-restriction) of a binary relation between and may be defined as a relation having domain , codomain and graph . Similarly, one can define a right-restriction or range restriction . Indeed, one could define a restriction to -ary relations, as well as to subsets understood as relations, such as ones of for binary relations. These cases do not fit into the scheme of sheaves.

In the mathematical discipline of category theory, a strict initial object is an initial object 0 of a category C with the property that every morphism in C with codomain 0 is an isomorphism. If C is a Cartesian closed category, then any initial object 0 of C is strict. Also, if C is a distributive or extensive category, then the initial object 0 of C is strict.

In mathematics, specifically category theory, an essential monomorphism is a monomorphism f in a category C such that for a morphism g in C, g \circ f is a monomorphism only when g is a monomorphism. Essential monomorphisms in a category of modules are those whose image is an essential submodule of the codomain. An injective hull of an object X is an essential monomorphism from X to an injective object.

Under this convention, all functions are surjective, so bijectivity and injectivity are the same. Authors using this convention may use the phrasing that a function is invertible if and only if it is an injection. The two conventions need not cause confusion, as long as it is remembered that in this alternate convention, the codomain of a function is always taken to be the image of the function.

A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If is such a complex valued function, it may be decomposed as : = + , where and are real- valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions.

The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from \Delta.

Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even, as could a complex-valued function of a vector variable, and so on.

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.

In other words, it is both a polynomial function of degree three, and a real function. In particular, the domain and the codomain are the set of the real numbers. Setting produces a cubic equation of the form :ax^3+bx^2+cx+d=0, whose solutions are called roots of the function. A cubic function has either one or three real roots; all odd-degree polynomials have at least one real root.

We may speak of a total functional relation f\subset A\times C when :\forall (a\in A). \exists! (c\in C). \langle a, c\rangle \in f , which notably involves a existential quantifier. (Variants of the functional predicate definition using apartness relations on setoids have been defined as well.) Using the standard class terminology, one can always make use of functions, given their domain is a set. They will be sets if their codomain is, see also Replacement.

Not all mathematical structures are F-algebras. For example, a poset P may be defined in categorical terms with a morphism s:P × P -> Ω, on a subobject classifier (Ω = {0,1} in the category of sets and s(x,y)=1 precisely when x≤y). The axioms restricting the morphism s to define a poset can be rewritten in terms of morphisms. However, as the codomain of s is Ω and not P, it is not an F-algebra.

A binary operation \circ is a calculation that combines the arguments and to x\circ y In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation on a set is an operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, multiplication.

Suppose that there exists a closed immersion . If the morphism is an isomorphism, then p is a covering morphism for the cdh topology. The cd stands for completely decomposed (in the same sense it is used for the Nisnevich topology). An equivalent definition of a covering morphism is that it is a proper morphism p such that for any point x of the codomain, the fiber p−1(x) contains a point rational over the residue field of x.

The map :x \bmod N \mapsto (x \bmod n_1, \ldots, x\bmod n_k) maps congruence classes modulo to sequences of congruence classes modulo . The proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the same number of elements, the map is also surjective, which proves the existence of the solution. This proof is very simple but does not provide any direct way for computing a solution.

Such functions are called bijections. The inverse of an injection that is not a bijection (that is, not a surjection), is only a partial function on , which means that for some , is undefined. If a function is invertible, then both it and its inverse function are bijections. Another convention is used in the definition of functions, referred to as the "set- theoretic" or "graph" definition using ordered pairs, which makes the codomain and image of the function the same.

In many branches of mathematics, the term map is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc. Some authors, such as Serge Lang, use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of R or C), and reserve the term mapping for more general functions.

In mathematics, the pointwise product of two functions is another function, obtained by multiplying the image of the two functions at each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can be multiplied (for instance, Y could be some set of numbers), then the pointwise product of f and g is another function from X to Y which maps x in X to f(x)g(x) in Y.

Where a unique representation is needed for any point besides the pole, it is usual to limit r to positive numbers () and φ to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2) or (−, ]). Another convention, in reverence to the usual codomain of the arctan-function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to . In all cases a unique azimuth for the pole (r = 0) must be chosen, e.g., φ = 0\.

For example, all objects are cofibrant in the standard model category of simplicial sets and all objects are fibrant for the standard model category structure given above for topological spaces. Left homotopy is defined with respect to cylinder objects and right homotopy is defined with respect to path space objects. These notions coincide when the domain is cofibrant and the codomain is fibrant. In that case, homotopy defines an equivalence relation on the hom sets in the model category giving rise to homotopy classes.

Since a function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain. A domain is part of a function if is defined as a triple , where is called the domain of , its codomain, and its graph. A domain is not part of a function if is defined as just a graph.

Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be the real coordinate space (which yields a real multivariable function), a topological vector space,Different definitions of derivative exist in general, but for finite dimensions they result in equivalent definitions of classes of smooth functions. an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above, and are a subclass of continuous functions.

As in animals, the plant homeobox genes code for the typical 60 amino acid long DNA- binding homeodomain or in case of the TALE (three amino acid loop extension) homeobox genes for an "atypical" homeodomain consisting of 63 amino acids. According to their conserved intron–exon structure and to unique codomain architectures they have been grouped into 14 distinct classes: HD-ZIP I to IV, BEL, KNOX, PLINC, WOX, PHD, DDT, NDX, LD, SAWADEE and PINTOX. Conservation of codomains suggests a common eukaryotic ancestry for TALE and non-TALE homeodomain proteins.

The explicitly foundational role of the category Set in his treatment is noteworthy in view of the folk intuition that enriched categories liberate category theory from the last vestiges of Set as the codomain of the ordinary external hom-functor. In 1967 Kelly was appointed Professor of Pure Mathematics at the University of New South Wales. In 1972 he was elected a Fellow of the Australian Academy of Science. He returned to the University of Sydney in 1973, serving as Professor of Mathematics until his retirement in 1994.

An endomorphism is a homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to the target. The endomorphisms of an algebraic structure, or of an object of a category form a monoid under composition. The endomorphisms of a vector space or of a module form a ring. In the case of a vector space or a free module of finite dimension, the choice of a basis induces a ring isomorphism between the ring of endomorphisms and the ring of square matrices of the same dimension.

A complex-valued function of several real variables may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If is such a complex valued function, it may be decomposed as :f(x_1,\ldots, x_n)=g(x_1,\ldots, x_n)+ih(x_1,\ldots, x_n), where and are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions. This reduction works for the general properties.

Instead of using the areas of rectangles, which put the focus on the domain of the function, Lebesgue looked at the codomain of the function for his fundamental unit of area. Lebesgue's idea was to first define measure, for both sets and functions on those sets. He then proceeded to build the integral for what he called simple functions; measurable functions that take only finitely many values. Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written : and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and .

For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere can be thought of as the complex projective line P1(C), the projective space of all complex lines in C2.

In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another. A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation. Equivariant maps generalize the concept of invariants, functions whose value is unchanged by a symmetry transformation of their argument.

A random function is a type of random element in which a single outcome is selected from some family of functions, where the family consists some class of all maps from the domain to the codomain. For example, the class may be restricted to all continuous functions or to all step functions. The values determined by a random function evaluated at different points from the same realization would not generally be statistically independent but, depending on the model, values determined at the same or different points from different realisations might well be treated as independent.

As a special case, whenever x itself is on the boundary, then the intersection point F(x) must be x. Consequently, F is a special type of continuous function known as a retraction: every point of the codomain (in this case Sn−1) is a fixed point of F. Intuitively it seems unlikely that there could be a retraction of Dn onto Sn−1, and in the case n = 1, the impossibility is more basic, because S0 (i.e., the endpoints of the closed interval D1) is not even connected.

In logic, a pseudoelementary class is a class of structures derived from an elementary class (one definable in first-order logic) by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of (the codomain of) a forgetful functor, and in physics of (hypothesized) hidden variable theories purporting to explain quantum mechanics. Elementary classes are (vacuously) pseudoelementary but the converse is not always true; nevertheless pseudoelementary classes share some of the properties of elementary classes such as being closed under ultraproducts.

The "upward" property of filters is less important for topologicial intuition but it is sometimes useful to have for technical reasons. ;Prefilters vs. filters With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non-surjective map is never a filter on the codomain, although it will be a prefilter.

That function is then called the inverse of , and is usually denoted as , a notation introduced by John Frederick William Herschel in 1813. Stated otherwise, a function, considered as a binary relation, has an inverse if and only if the converse relation is a function on the codomain , in which case the converse relation is the inverse function. Not all functions have an inverse. For a function to have an inverse, each element must correspond to no more than one ; a function with this property is called one-to-one or an injection.

If F is a topological field then a symbol c is weakly continuous if for each y in F∗ the set of x in F∗ such that c(x,y) = 1 is closed in F∗. This makes no reference to a topology on the codomain G. If G is a topological group, then one may speak of a continuous symbol, and when G is Hausdorff then a continuous symbol is weakly continuous.Milnor (1971) p.165 The only weakly continuous symbols on R are the trivial symbol and the Hilbert symbol: the only weakly continuous symbol on C is the trivial symbol.Milnor (1971) p.

This question asks if a group is finite if the group has a definite number of generators and meets the criteria xn=1, for x in the group. Many word problems are undecidable based on the Post correspondence problem. Any two homomorphisms g,h with a common domain and a common codomain form an instance of the Post correspondence problem, which asks whether there exists a word w in the domain such that g(w)=h(w). Post proved that this problem is undecidable; consequently, any word problem that can be reduced to this basic problem is likewise undecidable.

In computer science, a null function (or null operator) is a subroutine that leaves the program state unchanged. When it is part of the instruction set of a processor, it is called a NOP or NOOP (No OPeration). Mathematically, a (computer) function f is null if and only if its execution leaves the program state s unchanged. That is, a null function is an identity function whose domain and codomain are both the state space S of the program, and for which: : f(s)=s for all elements s \in S. Less rigorous definitions may also be encountered.

Sometimes, a set is endowed with more than one structure simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology structure and a group structure, such that these two structures are related in a certain way, then the set becomes a topological group. Mappings between sets which preserve structures (i.e., structures in the source or domain are mapped to equivalent structures in the destination or codomain) are of special interest in many fields of mathematics.

The augmented simplex category, denoted by \Delta_+ is the category of all finite ordinals and order-preserving maps, thus \Delta_+=\Delta\cup [-1], where [-1]=\emptyset. Accordingly, this category might also be denoted FinOrd. The augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category. A contravariant functor defined on \Delta_+ is called an augmented simplicial object and a covariant functor out of \Delta_+ is called an augmented cosimplicial object; when the codomain category is the category of sets, for example, these are called augmented simplicial sets and augmented cosimplicial sets respectively.

Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series. Generating functions are not functions in the formal sense of a mapping from a domain to a codomain. Generating functions are sometimes called generating series,This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", Canadian Journal of Mathematics 3, p. 405–411, but its use is rare before the year 2000; since then it appears to be increasing.

In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space (X, ≤) into a preordered vector space (Y, ≤) is a linear operator f on X into Y such that for all positive elements x of X, that is x ≥ 0, it holds that f(x) ≥ 0. In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain. Every positive linear functional is a type of positive linear operator. The significance of positive linear operators lies in results such as Riesz–Markov–Kakutani representation theorem.

Many treatments of predicate logic don't allow functional predicates, only relational predicates. This is useful, for example, in the context of proving metalogical theorems (such as Gödel's incompleteness theorems), where one doesn't want to allow the introduction of new functional symbols (nor any other new symbols, for that matter). But there is a method of replacing functional symbols with relational symbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result. Specifically, if F has domain type T and codomain type U, then it can be replaced with a predicate P of type (T,U).

It is a simple calculation to verify that g thus defined has the property that f\circ g \circ f = f, which is the proof (when the domain and codomain of f are the same set) that the full transformation semigroup is a regular semigroup. g acts as a (not necessarily unique) quasi-inverse for f; within semigroup theory this is simply called an inverse. Note however that for an arbitrary g with the aforementioned property the "dual" equation g \circ f \circ g= g may not hold. However if we denote by h= g \circ f \circ g, then f is a quasi-inverse of h, i.e.

One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as IQ or 1Q and has domain and codomain both equal to the real numbers. IQ(x) equals 1 if x is a rational number and 0 if x is not rational. More generally, if E is any subset of a topological space X such that both E and the complement of E are dense in X, then the real-valued function which takes the value 1 on E and 0 on the complement of E will be nowhere continuous.

As the green ball travels on the graph of the given function, the length of the path travelled by that ball's projection on the y-axis, shown as a red ball, is the total variation of the function. In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ ℝ, its total variation on the interval of definition is a measure of the one- dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b].

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 natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function.

They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets. There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.

Like Set, FinSet and FinOrd are topoi. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product , the categorical sum is given by the disjoint union , and the exponential object BA is given by the set of all functions with domain A and codomain B. In FinOrd, the categorical product of two objects n and m is given by the ordinal product , the categorical sum is given by the ordinal sum , and the exponential object is given by the ordinal exponentiation nm. The subobject classifier in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.

Hence Ω(X) is not an arbitrary complete lattice but a complete Heyting algebra (also called frame or locale – the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras). Now an obvious question is: To what extent is a topological space characterized by its locale of open sets? As already hinted at above, one can go even further. The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f −1(O) of any open set in the codomain of f is open in the domain of f.

However, there is no order preserving map such that the image of is cofinal in its codomain and . If "subnet" is defined to mean Willard-subnet or Kelley-subnet, then nets and filters are not completely interchangeable since there are relationships that filters (and subfilters) can express that nets and subnets can not. In particular, the problem is that Kelley-subnets and Willard-subnets are _not_ fully interchangeable with subfilters. This issue is not present with AA-subnets since AA-subnets have a defining characterization that immediately shows that they are fully interchangeable with sub-filters (in the sense that the statement above becomes true when "Kelley-subnet" is replaced with "AA-subnet").

A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point f(c) as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood N_1(f(c)) there is a neighborhood N_2(c) in its domain such that f(x)\in N_1(f(c)) whenever x\in N_2(c). This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition.

The Macaulay degree is the integer D=d_1+\cdots+d_n-n+1, which is fundamental in Macaulay's theory. For defining the resultant, one considers the Macaulay matrix, which is the matrix over the monomial basis of the -linear map :(Q_1, \ldots, Q_n)\mapsto Q_1P_1+\cdots+Q_nP_n, in which each Q_i runs over the homogeneous polynomials of degree D-d_i, and the codomain is the -module of the homogeneous polynomials of degree . If , the Macaulay matrix is the Sylvester matrix, and is a square matrix, but this is no longer true for . Thus, instead of considering the determinant, one considers all the maximal minors, that is the determinants of the square submatrices that have as many rows as the Macaulay matrix.

Therefore, if the sections never simultaneously vanish, they determine a form [s0 : ... : sr] which gives a map from X to Pr, and the pullback of the dual of the tautological bundle under this map is L. In this way, projective space acquires a universal property. The universal way to determine a map to projective space is to map to the projectivization of the vector space of all sections of L. In the topological case, there is a non-vanishing section at every point which can be constructed using a bump function which vanishes outside a small neighborhood of the point. Because of this, the resulting map is defined everywhere. However, the codomain is usually far, far too big to be useful.

In mathematics, a rational function is any function which can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero and the codomain is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

In mathematics, more specifically in general topology and related branches, a net or Moore-Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the codomain of this function is usually any topological space. However, in the context of topology, sequences do not fully encode all information about a function between topological spaces. In particular, the following two conditions are not equivalent in general for a map f between topological spaces X and Y: #The map f is continuous in the topological sense; #Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f(x) (continuous in the sequential sense).

In a more quantified version: for natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k+1 objects. For arbitrary n and m this generalizes to k + 1 = \lfloor(n - 1)/m \rfloor + 1 = \lceil n/m\rceil, where \lfloor\cdots\rfloor and \lceil\cdots\rceil denote the floor and ceiling functions, respectively. Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smaller than its domain".

Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, if we try to model the general concept of "equality" as a binary relation =, we must take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory. In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set A, that contains all the objects of interest, and work with the restriction =A instead of =.

In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.). Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points (a time series) of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable.

Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG. Monoids are F-algebras of signature F(M) = 1 + M×M. In the same vein, semigroups are F-algebras of signature F(S) = S×S Rings, domains and fields are also F-algebras with a signature involving two laws +,•: R×R -> R, an additive identity 0: 1 -> R, a multiplicative identity 1: 1 -> R, and an additive inverse for each element -: R -> R. As all these functions share the same codomain R they can be glued into a single signature function 1 + 1 + R + R×R + R×R -> R, with axioms to express associativity, distributivity, and so on.

Additionally, one can define functional predicates after proving an appropriate theorem. (If you're working in a formal system that doesn't allow you to introduce new symbols after proving theorems, then you will have to use relation symbols to get around this, as in the next section.) Specifically, if you can prove that for every X (or every X of a certain type), there exists a unique Y satisfying some condition P, then you can introduce a function symbol F to indicate this. Note that P will itself be a relational predicate involving both X and Y. So if there is such a predicate P and a theorem: : For all X of type T, for some unique Y of type U, P(X,Y), then you can introduce a function symbol F of domain type T and codomain type U that satisfies: : For all X of type T, for all Y of type U, P(X,Y) if and only if Y = F(X).

The reign of Castruccio Castracani, member of the House of Antelminelli, leader of great political and military capacity, represented the highest peak of the Republic's power, whose territories included the Garfagnana to the north, the coast from the city of Carrara to Pisa to the west, the city of Pistoia to the east (under the codomain of Lucca and Florence), and south the Valdarno constantly disputed with the Republic of Florence. Castracani also succeeded in making Lucca the only antagonist to the expansion of Republic of Florence leading to the victory in the Battle of Altopascio, in 1325, where he defeated the powerful Florentine army chasing them up to the walls of Florence. When Castruccio died, the city fell into a period of anarchy which saw it subject to the dominion of the Visconti family and subsequently to the government of Giovanni Dell'Agnello, Lord of the Republic of Pisa. Having regained its freedom in 1370, Lucca gave himself a republican government and with a shrewd foreign policy returned to having a remarkable fame in Europe thanks to its bankers and the silk trade.