A function or morphism f: X → Y whose image equals its codomain: for every y in Y there exists x in X with f(x)=y.

Principle

A surjection guarantees that every target value is attained; in Set this is equivalent to existence of a right-inverse and to the codomain being covered by the image.

Demonstration

The cubic map f: R → R given by f(x)=x^3 is surjective because every real y has a real cube root; the canonical projection Z → Z/nZ is surjective because each residue class has a representative integer.

Misapplication

Treating a map as surjective because its image is dense or large, or confusing 'image equals codomain' with 'image is nonempty' or 'image is dense in a topology'.

Consequence

When a map is surjective (in Set) it admits a right-inverse; surjectivity enables quotient constructions and ensures no element of the codomain is unreachable by the map.

Reversal

A non-surjective map has a codomain containing elements not hit by the map; its image is a proper subset of the codomain.

Boundary

Surjectivity is an elementary notion in concrete categories like Set or Top, but in general category theory the term most closely matches 'epimorphism' only in some categories; whether a morphism is surjective depends on how objects and morphisms are presented.

Semantic Tension

Surjection versus epimorphism: in Set they coincide, but in other categories an epimorphism need not be surjective; also confuse 'onto' with merely 'large image' in topological or measure contexts.

Synthesis

Surjection identifies maps that exhaust their codomain; it is the condition that every target element is produced by some source element, a property that in concrete categories connects to right-inverses and quotient descriptions.