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## Top 10 Wanted Proofs

 Frobenius's Theorem --Matt Westwood 18:38, 8 November 2008 (UTC) Wedderburn's Theorem --Matt Westwood 18:38, 8 November 2008 (UTC) Whitney Immersion Theorem --Matt Westwood 08:09, 18 January 2009 (UTC) Whitney Embedding Theorem --Matt Westwood 08:09, 18 January 2009 (UTC) Topological h-Cobordism Theorem --Matt Westwood 08:09, 18 January 2009 (UTC) Thurston's Geometrization Conjecture --Matt Westwood 08:09, 18 January 2009 (UTC) Tartaglia's Formula --Matt Westwood 06:39, 15 March 2009 (UTC) Burnside's Theorem --Joe (talk) 16:50, 16 March 2009 (UTC) Abel-Ruffini Theorem [1] --Matt Westwood 20:45, 16 March 2009 (UTC) Central Limit Theorem --HrMeyer 20:53, 23 April 2009 (UTC)

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## Proof of the Week

Intersection of Relation with Inverse is Symmetric Relation

## Theorem

Let $\mathcal R$ be a relation on a set $S$.

Then $\mathcal R \cap \mathcal R^{-1}$, the intersection of $\mathcal R$ with its inverse, is symmetric.

## Proof

Let $\left({x, y}\right) \in \mathcal R \cap \mathcal R^{-1}$

By definition of intersection:

$\left({x, y}\right) \in \mathcal R$
$\left({x, y}\right) \in \mathcal R^{-1}$

By definition of inverse relation:

$\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R^{-1}$
$\displaystyle \left({x, y}\right) \in \mathcal R^{-1} \implies \left({y, x}\right) \in \left ({\mathcal R^{-1}} \right )^{-1}$

By Inverse of Inverse Relation the second statement may be rewritten:

$\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R^{-1}$
$\left({x, y}\right) \in \mathcal R^{-1} \implies \left({y, x}\right) \in \mathcal R$

Then by definition of intersection:

$\left({y, x}\right) \in \mathcal R \cap \mathcal R^{-1}$

Hence $\mathcal R \cap \mathcal R^{-1}$ is symmetric.

$\blacksquare$