In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.
A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph . The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. 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 y in its codomain for which the equation f(x) = y does not have a solution.
A codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.
For a function
the codomain of f is , but f does not map to any negative number. Thus the image of f is the set ; i.e., the interval [0, ∞).
An alternative function g is defined thus:
While f and g map a given x to the same number, they are not, in this view, the same function because they have different codomains. A third function h can be defined to demonstrate why:
The domain of h cannot be but can be defined to be :
The compositions are denoted
On inspection, h ∘ f is not useful. It is true, unless defined otherwise, that the image of f is not known; it is only known that it is a subset of . For this reason, it is possible that h, when composed with f, might receive an argument for which no output is defined – negative numbers are not elements of the domain of h, which is the square root function.
Function composition therefore is a useful notion only when the codomain of the function on the right side of a composition (not its image, which is a consequence of the function and could be unknown at the level of the composition) is a subset of the domain of the function on the left side.
The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. In the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection.
A second example of the difference between codomain and image is demonstrated by the linear transformations between two vector spaces – in particular, all the linear transformations from to itself, which can be represented by the 2×2 matrices with real coefficients. Each matrix represents a map with the domain and codomain . However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or 0). Take for example the matrix T given by
which represents a linear transformation that maps the point (x, y) to (x, x). The point (2, 3) is not in the image of T, but is still in the codomain since linear transformations from to are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that T does not have full rank since its image is smaller than the whole codomain.
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
In mathematics and computer science, currying is the technique of converting a function that takes multiple arguments into a sequence of functions that each takes a single argument. For example, currying a function that takes three arguments creates three functions:
In mathematics, an inner product space or a Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets. Inner products allow the rigorous introduction of intuitive geometrical notions, such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
In mathematics, a linear map is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.
In mathematics, a function f from a set X to a set Y is surjective, if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
In Euclidean geometry, an affine transformation, or an affinity, is a geometric transformation that preserves lines and parallelism.
In mathematics, an injective 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.
In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as . Since a (total) 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 mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In mathematics, the range of a function may refer to either of two closely related concepts:
In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity into functions of the conjugate quantity. In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.
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 ℝ, or a subset of ℝ 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.
In mathematics, the restriction of a function is a new function, denoted or , obtained by choosing a smaller domain A for the original function .
In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other.
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions, where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions.
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : V → W that is: