Editing Cardinality (section)

=== Inequality ===
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> or <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>

For example, the set {{tmath|\N}} of all [[natural numbers]] has cardinality strictly less than its [[power set]] {{tmath|\mathcal P (\N)}}, because <math>g(n) = \{n\}</math> is an injective function from {{tmath|\N}} to {{tmath|\mathcal P (\N)}}, and it can be shown that no function from {{tmath|\N}} to {{tmath|\mathcal P (\N)}} can be bijective (see picture). By a similar argument, {{tmath|\N}} has cardinality strictly less than the cardinality of the set {{tmath|\R}} of all [[real number]]s. For proofs, see [[Cantor's diagonal argument]] or [[Cantor's first uncountability proof]].

If <math>|A| \leq |B|</math> and <math>|B| \leq |A|,</math> then <math>|A| = |B|</math> (a fact known as the [[Schröder–Bernstein theorem]]). The [[axiom of choice]] is equivalent to the statement that  <math>|A| \leq |B|</math> or <math>|B| \leq |A|</math> for every {{tmath|A}} and {{tmath|B}}.<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>