# Notation for an arbitrary set of N elements

Is there any notation for referring to a general set of $N$ elements? Currently I'm using $\{1, \dots, N\}$, but the fact that the set consists of natural numbers is irrelevant. I'd prefer to just write something like ${\mathfrak{S}}(N)$. :-)

For instance, I know natural numbers are some times encoded as $0 = \{\}$, $1 = \{0\}$, $2 = \{0, 1\}$. The $N$:th element in such encoding could for instance do.

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Strange to see no votes on either the question or the answer! (+1 to both). –  Aryabhata Apr 27 '12 at 21:32

If we decode $0=\varnothing, 1=\{0\}, 2=\{0,1\}$ and generally $n+1=n\cup\{n\}$, then we have that $n$ itself is a set of $n$ elements.

If you don't want to confuse with the natural numbers you can use $[n]$ for example.

Whatever notation you chooses, though, just write it down. For example:

We will use ${\frak S}(n)$ to denote a fixed set of $n$ elements.

Or if you prefer

Let $A_n$ be a set of size $n$.

Then you can continue with $A_n$ instead.

As for a concrete notation for the set of size $n$, well.. there is none which I can recall.

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Absolutely, but that is longer than just {1, ..., 32}. What I intended to ask was if there exists a well known notation already. –  aioobe Mar 28 '12 at 12:19
If you only use the set once, or in a very short and specific way... then there is no sense in introducing notation (even a relatively known one). If you wish to have a very general notation (which is what I thought you aim for) then there is nothing I recall as the standard set of size $n$. –  Asaf Karagila Mar 28 '12 at 12:21
I've seen texts that introduce $[n] = \{1,2,3,...,n\}$ and just use $[n]$ throughout. –  user17753 Dec 12 '12 at 17:33
@user17753: This is a question on what is your primitive mathematical notions. If you only have the empty set and the membership relation, then $n$ is already a set of $n$ elements. If you think of the natural numbers as a concrete and atomic object to your mathematical universe then $[n]$ is probably a good choice (do note that often the natural numbers begin at $0$ - as they should - so $[n]=\{0,\ldots,n-1\}$ in such context). –  Asaf Karagila Dec 12 '12 at 19:22