How to prove that $a^{x}\leq b^{y}$ for any cardinal number a,b, $x,y$ with a$\leq$b , $x\leq y$?
First, I want to show that $p^r\leq\ q^r$ and $r^p\leq\ r^q$ for any cardinal number $p,q,r$ with $p\leq q$.
If $P,Q,R$ are sets such that $cardP=p, cardQ=q, cardR=r,$ it is easy to show that $p^r\leq\ q^r$ because $P$ is equipotent with some subset of $Q$
But I don't know how to show that $r^p\leq\ r^q$
Could anyone help me?
Remember that $C^D$ is the set of functions from $D$ to $C$.
Without loss of generality, assume $P\subseteq Q$.
If $q=0$ then the statement is trivial (because then $p=0$ as well), therefore assume $q>0$, that is, $Q$ is not empty. Therefore you can choose a single element $a\in Q$ (there are no further requirements on $a$).
Now you can define a map $\phi:R^P\to R^Q$ as follows: $$\phi(f)(x) = \begin{cases} f(x) & x\in P\\ a & \text{otherwise} \end{cases}$$ It is not hard to check that $\phi$ is indeed an injection, and therefore $|R^P|\le |R^Q|$, that is, $r^p\le r^q$.