Surjective in a sentence

Examples A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range ) of f. This function is not surjective, because the image does not fill the whole codomain.

A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice ).

Alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function can also be given.

Any surjective function induces a bijection defined on a quotient of its domain by collapsing all arguments mapping to a given fixed image.

A partial function is said to be injective or surjective when the total function given by the restriction of the partial function to its domain of definition is.

A partial function may be both injective and surjective.

A surjective isometry between the normed vector spaces V and W is called an isometric isomorphism, and V and W are called isometrically isomorphic.

Consequently, f is surjective; however, f is not injective interestingly enough, the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed.

Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them.

Dedekind finite naturally means that every injective self-map is also surjective.

Examples Every morphism in a concrete category whose underlying function is surjective is an epimorphism.

For any set X, the identity function id X on X is surjective.

For example, the division example above is surjective (or onto) because every rational number may be expressed as a quotient of an integer and a natural number.

For example, the inclusion map Q → R, is a non-surjective epimorphism.

For instance, since a function is bijective if and only if it is both injective and surjective, in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism.

However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings.

However, talking of set of sets may be counterintuitive, and quotient spaces are commonly considered as a pair of a set of undetermined objects, often called "points", and a surjective map onto this set.

However there are also many concrete categories of interest where epimorphisms fail to be surjective.

If the codomain (-π/2, π/2) was made larger to include an integer multiple of π/2 then this function would no longer be onto (surjective) since there is no real number which could be paired with the multiple of π/2 by this arctan function.

If X and Y are Banach spaces and A : X → Y is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).