function, with , if the range of is a subset of the domain of . The inverse of , , is defined as the converse of if this is a function. Given any set , the identity function is the set , and this is a set in both ZFC and NFU for different reasons. Special kinds of function A function is an injective (also called one-to-one) if it has an inverse function. A function from to is a: Injection from to if the images under of distinct members of are distinct members of . Surjection from to if the range of is . Bijection from to if is both an injection and a surjection. Defining functions as ordered pairs or ordered triples has the advantages that we do not have to introduce the terminology of being a function "from to ", and that we can speak of "being surjective" outright as opposed to only being able to speak of "being surjective onto ". Size of sets In both ZFC and NFU, two sets A and B are the same size (or are equinumerous) if and only if there is a bijection f from A to B. This can be written as , but note that (for the moment) this expresses a relation between A and B rather than a relation between yet-undefined objects and . Denote this relation by in contexts such as the actual definition of the cardinals where even the appearance of presupposing abstract cardinals should be avoided. Similarly, define as holding if and only if there is an injection from A to B. It is straightforward to show that the relation of equinumerousness is an equivalence relation: equinumerousness of A with A is witnessed by ; if f witnesses , then witnesses ; and if f witnesses and g witnesses , then witnesses . It can be shown that is a linear order on abstract cardinals, but not on sets. Reflexivity is obvious and transitivity is proven just as for equinumerousness. The Schröder–Bernstein theorem, provable in ZFC and NFU in an entirely standard way, establishes that (this establishes antisymmetry on cardinals), and follows in a standard way in either theory from the axiom of choice. Finite sets and natural numbers Natural numbers can be considered either as finite ordinals or finite cardinals. Here consider them as finite cardinal numbers. This is the first place where a