Hello

Here is a problem I am doing. Suppose $\displaystyle |A|=n$ and let

$\displaystyle F=\{f|f\mbox{ is a one to one,onto function from }I_n\mbox{ to }A\} $.

Let $\displaystyle L=\{R|R\mbox{ is a total order on }A\}$

Prove that $\displaystyle F\sim L$ and therefore $\displaystyle |L|=n!$.

Velleman gives some hints at the back of the book. To prove $\displaystyle F\sim L$,

he suggets a function $\displaystyle h:F\to L$ defined as

$\displaystyle h(f)=\{(a,b)\in A\times A|f^{-1}(a)\le f^{-1}(b)\} $.

He suggets some things to prove that h is one to one. I could do that. Now to prove

that h is onto, he says , suppose R is a total order on A. Define

$\displaystyle g:A\to I_n$ by the formula $\displaystyle g(a)=|\{x\in A\;|\;xRa\}| $.

He asks the reader to show that

$\displaystyle \forall a\in A \;\forall b\in A(aRb\leftrightarrow g(a)\le g(b))$

I could show this. Then he says use this fact to show that $\displaystyle g^{-1}\in F$

and $\displaystyle h(g^{-1})=R$. I could show that g is one to one. To prove that

$\displaystyle g^{-1}\in F$, I should also show that g is onto. But I have not been able

to show this. Input is welcome.