Q:

Proving $|A^A|=|2^A|$ for infinite $A$.

Possible Duplicate:
Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$? 

How can one prove that $|A^A|=|2^A|$ for infinite $A$? (summary of proof or providing link with proof will also suffice.)
Thanks!

A:

We have $|2^A|\leq |A^A|$ and by Cantor-Bernstein theorem we are reduced to show that $|A^A|\leq |2^A|$. To see that, we use the fact that $|A\times A|=|A|$, applying Zorn lemma to $S:=\{(B,f), B\subset A, f\colon B\times B\to B\}$ with the partial order $(B_1,f_1)\leq (B_2,f_2)$ if $B_1\subset B_2$ and $f_{2\mid B_1}=f_1$. We also have $$|2^A|=|2^{A\times A}|=|(2^A)^A|$$
which gives what we wanted. 

