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== Size of sets ==
In both [[ZFC]] and [[NFU]], we say that two sets A and B are the same size (or are <strong>equinumerous</strong>) if and only if there is a bijection f from A to B. We
can write this <math>|A|=|B|</math> as long as we note that for the moment this expresses
a relation between A and B rather than a relation between objects <math>|A|</math> and
<math>|B|</math> which have not yet been defined.
Similarly, we can define <math>|A| \leq |B|</math> as holding iff 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 <math>i_A</math>; if f witnesses
<math>|A|=|B|</math> then <math>f^{-1}</math> witnesses <math>|B|=|A|</math>; if f witnesses
<math>|A|=|B|</math> and g witnesses <math>|B|=|C|</math>, then <math>g\circ f</math> witnesses <math>|A|=|C|</math>.
We can show that <math>|A| \leq |B|</math> is almost a linear order. Reflexivity is obvious
and transitivity is proven just as for equinumerousness. The [[Cantor-Schröder-Bernstein theorem]], provable in either [[ZFC]] or [[NFU]] in an entirely standard way, establishes that
<math>|A| \leq |B| \wedge |B| \leq |A| \rightarrow |A| = |B|</math> (this establishes that
we have antisymmetry on cardinals (not yet defined), but we are now considering a relation on sets),
and <math>|A| \leq |B| \vee |B| \leq |A|</math> follows in a standard way in either theory from
the Axiom of Choice.
== Finite sets and natural numbers ==
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