## [Surjection](/surjection-0) Definition 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 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 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 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 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 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 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 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 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.