Search results
Results From The WOW.Com Content Network
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.
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.
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.
As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement "f maps X onto Y" differs from "f maps X into B", in that the former implies that f is surjective, while the latter makes no assertion about the nature of f. In a complicated reasoning ...
A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic ; they differ only in the notation of their elements (except of identity element) and are identical for all practical purposes.
is essentially surjective if each object of is isomorphic to an object of the form for some object of . Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of categories.
A surjective map : is called an almost open map if for every there exists some () such that is a point of openness for , which by definition means that for every open neighborhood of , is a neighborhood of () in (note that the neighborhood () is not required to be an open neighborhood). Every surjective open map is an almost open map but in ...
If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R. If f is surjective, P is prime (maximal) ideal in R and ker(f) ⊆ P, then f(P) is prime (maximal) ideal in S. Moreover, The composition of ring homomorphisms S → T and R → S is a ring homomorphism R → T.