I am working on the following problem and am having difficulties getting started:
We define a permutation of $K$ to be any one-to-one function from $K$ onto $K$. We can then define the factorial operation on cardinal numbers by the equation $K!$ = card$\{f \ | \ f$ is a permutation of $K\}$, where $K$ is any set of cardinality $\kappa$. We show that $\kappa !$ is well defined, i.e., the value of $\kappa !$ is independent of just which set $K$ is chosen.
The idea I have so far is to try the following. Let $|K_0| = \kappa$ and $|K_1| = \kappa$, where $K_0 \neq K_1$ and are arbitrary sets. We show that $K_0! = K_1! = \kappa !$. To do this, we must demonstrate a bijection between $K_0!$ and $K_1!$. That is, we must construct a bijective function $f: K_0 \rightarrow K_1$.
Any advice would be appreciated.
Thank you in advance.
By the definition of $|K_0|=|K_1|$ there is a bijection between the two sets.
You can use this two construct a bijection between between $K_0!$ and $K_1!$ (by conjugating permutations of $K_1$ by a fixed bijection between $K_0$ and $K_1$).