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A surjective function is a function whose image is equal to its codomain. Equivalently, ... For example, in the first illustration in the gallery, ...
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.
One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval [0, π]. The image of this restriction is the interval [−1, 1], and thus the restriction has an inverse function from [−1, 1] to [0, π], which is called arccosine and is denoted arccos.
For example, as a function from the integers to the integers, the doubling function () = is not surjective because only the even integers are part of the image. However, a new function f ~ ( n ) = 2 n {\displaystyle {\tilde {f}}(n)=2n} whose domain is the integers and whose codomain is the even integers is surjective.
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.
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 .
In category theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g 1, g 2: Y → Z, = =. Epimorphisms are categorical analogues of onto or surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the ...