Midpoint circle algorithm: Difference between revisions

Revision as of 19:18, 26 January 2014

Kuratowski's free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several lattice theory problems, such as the Congruence Lattice Problem.

Denote by $[X]^{< ω}$ the set of all finite subsets of a set $X$ . Likewise, for a positive integer $n$ , denote by $[X]^{n}$ the set of all $n$ -elements subsets of $X$ . For a mapping $Φ : [X]^{n} \to [X]^{< ω}$ , we say that a subset $U$ of $X$ is free (with respect to $Φ$ ), if $u \notin Φ (V)$ , for any $n$ -element subset $V$ of $U$ and any $u \in U ∖ V$ . Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form $ℵ_{n}$ .

The theorem states the following. Let $n$ be a positive integer and let $X$ be a set. Then the cardinality of $X$ is greater than or equal to $ℵ_{n}$ if and only if for every mapping $Φ$ from $[X]^{n}$ to $[X]^{< ω}$ , there exists an $(n + 1)$ -element free subset of $X$ with respect to $Φ$ .

For $n = 1$ , Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.

References

P. Erdős, A. Hajnal, A. Máté, R. Rado: Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland, 1984, pp. 282-285.
C. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14--17.
John C. Simms: Sierpiński's theorem, Simon Stevin, 65 (1991) 69--163.

Template:Settheory-stub

@@ Line 1: / Line 1: @@
-Alyson is the name people use to contact me and I believe it sounds fairly good when you say it. Mississippi is the only place I've been residing in but I will have to move in a yr or two. Invoicing is my occupation. The favorite pastime for him and his children is style and he'll be beginning some thing else along with it.<br><br>My web site; [http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 best psychics]
+'''Kuratowski's free set theorem''', named after [[Kazimierz Kuratowski]], is a result of [[set theory]], an area of [[mathematics]]. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several [[lattice theory]] problems, such as the [[Congruence Lattice Problem]].
+Denote by <math>[X]^{<\omega}</math> the [[Set (mathematics)|set]] of all [[Finite set|finite subsets]] of a set <math>X</math>. Likewise, for a [[positive integer]] <math>n</math>, denote by <math>[X]^n</math> the set of all <math>n</math>-elements subsets of <math>X</math>. For a [[Map (mathematics)|mapping]] <math>\Phi\colon[X]^n\to[X]^{<\omega}</math>, we say that a [[subset]] <math>U</math> of <math>X</math> is ''free'' (with respect to <math>\Phi</math>), if <math>u\notin\Phi(V)</math>, for any <math>n</math>-element subset <math>V</math> of <math>U</math> and any <math>u\in U\setminus V</math>. [[Kuratowski]] published in 1951 the following result, which characterizes the [[Infinity|infinite]] [[Cardinal number|cardinals]] of the form <math>\aleph_n</math>.
+The theorem states the following. Let <math>n</math> be a positive integer and let <math>X</math> be a set. Then the [[cardinality]] of <math>X</math> is greater than or equal to <math>\aleph_n</math> if and only if for every mapping <math>\Phi</math> from <math>[X]^n</math> to <math>[X]^{<\omega}</math>,
+there exists an <math>(n+1)</math>-element free subset of <math>X</math> with respect to <math>\Phi</math>.
+For <math>n=1</math>, Kuratowski's free set theorem is superseded by [[Hajnal's set mapping theorem]].
+== References ==
+* [[Paul Erdős|P. Erdős]], [[András Hajnal|A. Hajnal]], A. Máté, [[Richard Rado|R. Rado]]: ''Combinatorial Set Theory: Partition Relations for Cardinals'', North-Holland, 1984, pp. 282-285.
+* [[Kazimierz Kuratowski|C. Kuratowski]], ''Sur une caract&eacute;risation des alephs'', Fund. Math. '''38''' (1951), 14--17.
+*   John C. Simms: Sierpiński's theorem,  ''Simon Stevin'',  '''65'''  (1991) 69--163.
+{{settheory-stub}}
+[[Category:Set theory]]

Midpoint circle algorithm: Difference between revisions

Revision as of 19:18, 26 January 2014

References

Navigation menu

Search