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A '''mathematical proof''' is a way to show that a [[mathematics|math]] [[theorem]] is true. One mu == Proof by Induction == ...

...[http://en.wikipedia.org/wiki/VC_dimension '''VC dimension'''] is defined by the power of a family to shatter a set. ...F})</math>, is the size of the largest <math>R\subseteq X</math> shattered by <math>\mathcal{F}</math>. ...

...[http://en.wikipedia.org/wiki/VC_dimension '''VC dimension'''] is defined by the power of a family to shatter a set. ...F})</math>, is the size of the largest <math>R\subseteq X</math> shattered by <math>\mathcal{F}</math>. ...

...antor|Cantor]] published articles on it in 1877, 1891 and 1899. His first proof of the diagonal argument was published in 1890 in the [[journal]] of the [[ By omitting fractions that can still be simplified, there is a [[Bijective fun ...

* <math>R(4,3)\leq 9</math>. (Hint: Proof by contradiction. Color the edges of <math>K_9</math> in red and blue, and assume that there ...</math> and that all the vertices of <math>T\subseteq B</math> are matched by <math>M'</math>. Prove that <math>G</math> contains a matching that matches ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...and how to efficiently find a stable matching. Both questions are answered by the following proposal algorithm due to Gale and Shapley. ...hus <math>m'</math> becomes single again and considers himself as rejected by <math>w</math>) and is married to <math>m</math>; ...