Editing Cardinality (section)

=== Inequality ===
[[File:Cantor-Bernstein.png|thumb|256x256px|[[Gyula Kőnig]]'s definition of a bijection {{color|#00c000|''h''}}:''A''&nbsp;→&nbsp;''B'' from the given injections {{color|#c00000|''f''}}:''A''&nbsp;→&nbsp;''B'' and {{color|#0000c0|''g''}}:''B''&nbsp;→&nbsp;''A''. ]]
A set <math>A</math> is not larger than a set <math>B</math> if it can be mapped into <math>B</math> without overlap. That is, the cardinality of <math>A</math> is less than or equal to the cardinality of <math>B</math> if there is an [[injective function]] from <math>A</math> to '''<math>B</math>'''. This is written <math>A \preceq B,</math> <math>A \lesssim B</math> and eventually <math>|A| \leq |B|.</math> If <math>A \preceq B,</math> but there is no injection from <math>B</math> to <math>A,</math> then <math>A</math> is said to be ''strictly'' smaller than <math>B,</math> written without the underline as <math>A \prec B</math> or <math>|A| < |B|.</math> For example, if <math>A</math> has four elements and <math>B</math> has five, then the following are true <math>A \preceq A,</math> <math>A \preceq B,</math> and <math>A \prec B.</math>

The basic properties of an inequality are reflexivity (for any <math>a,</math> <math>a \leq a</math>), transitivity (if <math>a \leq b</math> and <math>b \leq c,</math> then <math>a \leq c</math>) and [[Antisymmetric relation|antisymmetry]] (if <math>a \leq b</math> and <math>b \leq a,</math> then <math>a = b</math>) (See [[Inequality (mathematics)#Formal definitions and generalizations|''Inequality § Formal definitions'']]). Cardinal inequality <math>(\preceq)</math> as defined above is reflexive since the [[identity function]] is injective, and is transitive by function composition. Antisymmetry is established by the [[Schröder–Bernstein theorem]]. The proof roughly goes as follows. 

Given sets <math>A</math> and <math>B</math>, where <math>f:A \mapsto B</math> is the function that proves <math>A \preceq B</math> and <math>g: B \mapsto A</math> proves <math>B \preceq A</math>, then consider the sequence of points given by applying <math>f</math> then <math>g</math> over and over. Then we can define a bijection <math>h: A \mapsto B</math> as follows: If a sequence forms a cycle, begins with an element <math>a \in A</math> not mapped to by <math>g</math>, or extends infinitley in both directions, define <math>h(x) = f(x)</math> for each <math>x \in A</math> in those sequences. The last case, if a sequence begins with an element <math>b \in B</math>, not mapped to by <math>f</math>, define <math>h(x) = g^{-1}(x)</math> for each <math>x \in A</math> in that sequence. Then <math>h</math> is a bijection.

The above shows that cardinal inequality is a [[partial order]]. A [[total order]] has the additional property that, for any <math>a</math> and <math>b</math>, either <math>a \leq b</math> or <math>b \leq a.</math> This can be established by the [[well-ordering theorem]]. Every well-ordered set is uniquely [[order isomorphic]] to a unique [[ordinal number]], called the [[order type]] of the well-ordered set. Then, by comparing their order types, one can show that <math>A \preceq B</math> or <math>B \preceq A</math>. This result is equivalent to the [[axiom of choice]].<ref>{{citation | author=Friedrich M. Hartogs | author-link=Friedrich M. Hartogs | editor=Felix Klein | editor-link=Felix Klein |editor2=Walther von Dyck |editor2-link=Walther von Dyck |editor3=David Hilbert |editor3-link=David Hilbert |editor4=Otto Blumenthal |editor4-link=Otto Blumenthal | title=Über das Problem der Wohlordnung | journal=[[Mathematische Annalen]] | volume=76 | number=4 | publisher=B.&nbsp;G. Teubner | location=Leipzig | year=1915 | pages=438–443 | issn=0025-5831 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0076&DMDID=DMDLOG_0037&L=1 | doi=10.1007/bf01458215| s2cid=121598654 }}</ref><ref>{{citation | author=Felix Hausdorff | author-link=Felix Hausdorff | editor=Egbert Brieskorn | editor-link=Egbert Brieskorn |editor2=Srishti D. Chatterji| title=Grundzüge der Mengenlehre | edition=1. | publisher=Springer | location=Berlin/Heidelberg | year=2002 | pages=587 | isbn=3-540-42224-2| url=https://books.google.com/books?id=3nth_p-6DpcC|display-editors=etal}} - [https://jscholarship.library.jhu.edu/handle/1774.2/34091 Original edition (1914)]</ref>