Cross ratio

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nayel
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Cross ratio

Unread post by nayel » Tue Jan 25, 2011 11:33 pm

Let $S$ be the set of all functions generated by $f(x)=1/x$ and $g(x)=1-x$. So $S$ is the set of the functions mentioned in this post.

The cross ratio of the pairwise distinct numbers $z_1,z_2,z_3,z_4$ is given by
\[[z_1,z_2,z_3,z_4]=\frac{z_1-z_4}{z_1-z_2}\frac{z_3-z_2}{z_3-z_4}.\]
Prove that, for any permutation $\sigma$ of $\{1,2,3,4\}$, there exists $f_\sigma\in S$ such that
\[f_\sigma([z_1,z_2,z_3,z_4])\equiv [z_{\sigma(1)}, z_{\sigma(2)}, z_{\sigma(3)}, z_{\sigma(4)}].\]
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

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