Editing Surjective function (section)

===Composition and decomposition===
The [[function composition|composition]] of surjective functions is always surjective: If ''f'' and ''g'' are both surjective, and the codomain of ''g'' is equal to the domain of ''f'', then {{Nowrap|''f'' <small>o</small> ''g''}} is surjective. Conversely, if {{Nowrap|''f'' <small>o</small> ''g''}} is surjective, then ''f'' is surjective (but ''g'', the function applied first, need not be). These properties generalize from surjections in the [[category of sets]] to any [[epimorphism]]s in any [[category (mathematics)|category]].

Any function can be decomposed into a surjection and an [[injective function|injection]]: For any function {{Nowrap|''h'' : ''X'' → ''Z''}} there exist a surjection {{Nowrap|''f'' : ''X'' → ''Y''}} and an injection {{Nowrap|''g'' : ''Y'' → ''Z''}} such that {{Nowrap begin}}''h'' = ''g'' <small>o</small> ''f''{{Nowrap end}}. To see this, define ''Y'' to be the set of [[preimage]]s {{Nowrap|''h''<sup>−1</sup>(''z'')}} where ''z'' is in {{Nowrap|''h''(''X'')}}. These preimages are [[disjoint sets|disjoint]] and [[partition of a set|partition]] ''X''. Then ''f'' carries each ''x'' to the element of ''Y'' which contains it, and ''g'' carries each element of ''Y'' to the point in ''Z'' to which ''h'' sends its points. Then ''f'' is surjective since it is a projection map, and ''g'' is injective by definition.