Set of Permutation and Identity Permutation

Question

I have a "basic" math question that seems easy but I couldn't figure out.

Assume I have $L$ = {0,1,2,3,4,5,6,7,8,9} and $M$ = set of all permutations on set $L$.

From the description above, I know that the $|L|$ = 10 and $|M|$ = 10!

Now let $J$ = $M$ \ {id} where id is an identity permutation i.e id($x$) = $x$ for all $x$ in $L$.

What is $|J$|? Would it make more sense if I set the value of $x$ first?

Also, X \ Y means set of X "subtract" with set of Y

It is more likely to get an answer when you show that you have put some effort. — mucciolo, Dec 14 '17 at 03:13
@mucciolo Hi, I wish I could do such thing. The problem is I don't even understand how to subtract a set with an identity permutation. — ThomasWest, Dec 14 '17 at 04:56
I might be misinterpreting your question, but if $id(x) =x \forall x \in L$ then $M \cap {id} = \null$ so $J = M$ — Francisco José Letterio, Dec 14 '17 at 05:13

Ben · Accepted Answer · 2017-12-14T05:25:27.920

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I may be misinterpreting your question, but I'm assuming that by $M\{id\}$, you're referring to the quotient space $M/{id}$. If that is the case, we know by Lagrange's theorem that $|M/\{id\}|=|M|/|\{id\}|=10!/1=10!$. Let me know if I misinterpreted the problem.

Edit: It is just $|J|=10!−1$, as $M$ is the set of permutations and you are taking one away.

edited Dec 14 '17 at 05:25

answered Dec 14 '17 at 02:44

Ben

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M \ {id} means that set of M "subtract" by the set of {id}, that's the hint I got. – ThomasWest Dec 14 '17 at 04:55
Oh, then it is just $|J|=10!-1$, as $M$ is the set of permutations and you are taking one away. – Ben Dec 14 '17 at 05:22
I'm sorry but where did you get 1? Is {id} always 1? – ThomasWest Dec 14 '17 at 06:01
Think about it in terms of sizes. $M$ is the set of all permutations and $id \in M$. So when we take out $id$ [which is just a single element of $M$], we are taking away one element of the set. – Ben Dec 14 '17 at 09:45
I get it now, thanks! – ThomasWest Dec 14 '17 at 14:28

Set of Permutation and Identity Permutation

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