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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.

Martin Sleziak
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letsmakemuffinstogether
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  • Some posts calculating $\kappa!$: a) For $\kappa=\aleph_0$ http://math.stackexchange.com/questions/367194/cardinality-of-the-set-of-bijective-functions-on-mathbbn, http://math.stackexchange.com/questions/87902/problems-about-countability-related-to-function-spaces, http://mathoverflow.net/questions/29475/an-easy-proof-of-the-uncountability-of-bijections-on-natural-numbers b) for arbitrary $\kappa$: http://mathoverflow.net/questions/27785/cardinality-of-the-permutations-of-an-infinite-set – Martin Sleziak Aug 20 '14 at 09:26
  • @Martin: I'm fairly sure we have $\kappa!$ questions on this site as well. Including some which talk about the relationship with the axiom of choice when talking about arbitrary sets! :-) – Asaf Karagila Aug 20 '14 at 09:40
  • I think we only have finitely many questions on this site... – Martin Sleziak Aug 20 '14 at 09:47
  • To post a more serious comment: I was able to find this question http://math.stackexchange.com/questions/662455/cardinality-of-the-set-of-permutations-of-a-set-a (Maybe there are more posts about the same thing.) – Martin Sleziak Aug 20 '14 at 09:50

1 Answers1

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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 $K_1$).

Asaf Karagila
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