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I have an ordered set $S=\{23,45,32,56\}$ I would like to make an ordered subset $M=\{23,45,32\}$ which contains the elements of $S$ except the last one. In this case. How do I represent this formally using mathematical notation?

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    $\begingroup$ Order is arbitrary in sets so there really is no last element. $\endgroup$ – A.Riesen May 25 '16 at 21:33
  • $\begingroup$ Unless it's an ordered set, there is no last element. $\endgroup$ – pisoir May 25 '16 at 21:41
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If you are going to talk about a "last" element, you have to put an order on the set. So instead of thinking of $S$ as a set, you want to think of it as a tuple $\langle 23,45,32,56\rangle$, which is really a function $\sigma:3\to \mathbb R$ by $\sigma(0)=23$, $\sigma(1)=45$, $\sigma(2)=32$, and $\sigma(3)=56$. If you want everything but the last element, this is just a restriction of sigma; $\sigma\restriction 2=\langle 23,45,32\rangle$.

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Actually, all of these are the subsets of $S=\{23,45,32,56\}$:
$\{23\},\{45\},\{32\}$,
$\{23,45\},\{23,32\}$,
$\{23,45,32\}$

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    $\begingroup$ "which contains all of the elements except the last" $\endgroup$ – JMoravitz May 25 '16 at 21:38

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