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In mathematics, a surjective function (also known as surjection, or onto function / ˈ ɒ n. t uː /) is a function f such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y.
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. A surjective function is a surjection. [1] The formal definition is the following.
A state transition function is a surjective function when every state has a predecessor (there can be no Garden of Eden). It is an injective function when no two states have the same successor. A surjunctive group is a group with the property that, when its elements are used as the cells of cellular automata, every injective transition function ...
Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets. Closed function: maps closed sets to closed sets.
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.
For some functions, the image and the codomain coincide; these functions are called surjective or onto. For example, consider the function () =, which inputs a real number and outputs its double. For this function, both the codomain and the image are the set of all real numbers, so the word range is unambiguous.
Examples [ edit ] In the category of sets , every monomorphism ( injective function ) with a non-empty domain is a section, and every epimorphism ( surjective function ) is a retraction; the latter statement is equivalent to the axiom of choice .
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.