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Surjective composition: the first function need not be surjective. A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain. In other words, each element of the codomain has a non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain.
Any surjective function induces a bijection defined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows.
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain).
Variations of the horizontal line test can be used to determine whether a function is surjective or bijective: The function f is surjective (i.e., onto) if and only if its graph intersects any horizontal line at least once. f is bijective if and only if any horizontal line will intersect the graph exactly once.
In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable, and the corresponding element of Y is the value of the function at x, or the image of x under the function. A function f, its domain X, and its codomain Y are often specified by the notation :.
Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. [1] An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse.
T is said to be surjective or an epimorphism if any of the following equivalent conditions are true: T is onto as a map of sets. coker T = {0 W} T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S.
A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D.