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In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2] The term map may be used to distinguish some special types of functions, such as homomorphisms.
A map can have any set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of real or complex numbers. [ 13 ] Alternatively, a map is associated with a special structure (e.g. by explicitly specifying a structured codomain in its definition).
The arrows or morphisms between sets A and B are the functions from A to B, and the composition of morphisms is the composition of functions. Many other categories (such as the category of groups, with group homomorphisms as arrows) add structure to the objects of the category of sets or restrict the arrows to functions of a particular kind (or ...
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set.
The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself. Fundamental group Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points.
A function f from X to Y. The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
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At the same time, the mapping of a function to the value of the function at a point is a functional; here, is a parameter. Provided that f {\displaystyle f} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals .