Editing Surjective function (section)

===Cardinality of the domain of a surjection===
The [[cardinality]] of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If {{Nowrap|''f'' : ''X'' → ''Y''}} is a surjective function, then ''X'' has at least as many elements as ''Y'', in the sense of [[cardinal number]]s.   (The proof appeals to the [[axiom of choice]] to show that a function
{{Nowrap|''g'' : ''Y'' → ''X''}} satisfying ''f''(''g''(''y'')) = ''y'' for all ''y'' in ''Y'' exists. ''g'' is easily seen to be injective, thus the [[Cardinal number#Formal definition|formal definition]] of |''Y''| ≤ |''X''| is satisfied.)

Specifically, if both ''X'' and ''Y'' are [[finite set|finite]] with the same number of elements, then {{Nowrap|''f'' : ''X'' → ''Y''}} is surjective if and only if ''f'' is [[injective]].

Given two sets ''X'' and ''Y'', the notation {{nowrap|''X'' ≤<sup>*</sup> ''Y''}} is used to say that either ''X'' is empty or that there is a surjection from ''Y'' onto ''X''. Using the axiom of choice one can show that {{nowrap|''X'' ≤<sup>*</sup> ''Y''}} and {{nowrap|''Y'' ≤<sup>*</sup> ''X''}} together imply that {{nowrap begin}}|''Y''| = |''X''|,{{nowrap end}} a variant of the [[Schröder–Bernstein theorem]].