Question: This is an exercise in *Naive Set Theory* by P. R. Halmos. 

> 
> If $\text{card }A=a$, what is the cardinal number of the set of all
>  one-to-one mappings of $A$ onto itself? What is the cardinal number of
>  the set of all countably infinite subsets of $A$? 
> 
> 
> 

It is easy to see that, for the first question, the cardinal number is less than or equal to $a^a$, and for any finite set the cardinal number is $a!$; for the second one, the cardinal number is less than or equal to $2^a$, and for any finite set the cardinal number is $0$.  

I guess for infinite sets the equality holds in both questions. Is that correct? Can anyone give some suggestions on what to do next?  

**Edit**: As is suggested by Arthur, the second cardinal number is at most $a^\omega$.

Answer: 
No. Counterexample: $c=1$ and $f(x)=k>0$ for all $x$.
Consider $f(x)=\frac{x^{\frac{1}{c}-1}}{c}$.