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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (also known as onto, or a surjection), if for every element y in the codomain Y of f
Injective function
non-surjective function (injection, not a bijection) An injective surjective function (bijection) A non-injective surjective function (surjection, not
Bijection, injection and surjection
\forall x\in X,\exists !y\in Y{\text{ such that }}y=f(x).} The function is surjective, or onto, if each element of the codomain is mapped to by at least
Bijection
one-to-one function (a.k.a. injective function) (see figures). An injective non-surjective function (injection, not a bijection) An injective surjective function
Twelvefold way
set X is equivalent to counting injective functions N → X when n = x, and also to counting surjective functions N → X when n = x. Counting multisets of
Partial function
or surjective respectively. A partial function may be both injective and surjective (and thus bijective). Because a function is trivially surjective when
Inverse function
as: f ( x ) = ( 2 x + 8 ) 3 . {\displaystyle f(x)=(2x+8)^{3}.} A surjective function f from the real numbers to the real numbers possesses an inverse
Domain of a function
and only if f {\displaystyle f} is a surjective function, and otherwise it is smaller. A well-defined function must map every element of its domain to
Codomain
the image of a function is a subset of its codomain. Thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements
Epimorphism
categorical analogues of surjective functions (and in the category of sets the concept corresponds exactly to the surjective functions), but it may not exactly