
Cardinal Exponentials
Show that if κ ≤ λ then κ^μ ≤ λ^μ for any cardinals μ, κ and λ .
I was thinking setting the sets A, B and C to be sets with cardinals κ, λ and μ respectively. Then I'd just need to prove that A^C ≤ B^C gives an injection. If someone could help out with a sketch of the proof that would be great.
(I think it's also helpful to note that A^B is defined to be the set of all functions from B to A and κ^λ = card(A^B), but I'm not sure how to apply these).

Using your notations, what we assume is that $\displaystyle A\prec B$.
You want to send a map from $\displaystyle C$ to $\displaystyle A$ on a map from $\displaystyle C$ to $\displaystyle B$.
$\displaystyle A\prec B$ means there is a map $\displaystyle f:A\rightarrow B$ such that $\displaystyle f$ is a bijection from $\displaystyle A$ into $\displaystyle f(A)\subseteq B$.
Prove that $\displaystyle A^C\rightarrow B^C:g\rightarrow fg$ is an injection.