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Is it true that I can write the set of the the first $n$ natural numbers as $[ n] $?

For example, $[10]= \{1,2,3,4,5,6,7,8,9,10 \}$. And in which math context this is used?

root
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  • It's the set of the first $n$ natural numbers, not of the $n^{\text{th}}$ natural number. This would be ${n}$. – k.stm Jan 26 '13 at 11:45
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    Actually... the $n^{th}$ natural number can be defined as $[n-1]$, so it's almost true. – Git Gud Jan 26 '13 at 11:48
  • @GitGud Why 0 is natural number ? I am not good in math.. – Grijesh Chauhan Jan 26 '13 at 11:50
  • some time n natural numbers are shown as [1-n] in TOC – Grijesh Chauhan Jan 26 '13 at 11:52
  • Here's a question. What's more useful? Defining ${1,...,10}=[10]$, or defining ${0,...,9}=[10]$? And does it depend on context? – goblin GONE Jan 26 '13 at 11:52
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    @GrijeshChauhan read the first paragraph on the $\textbf{The contemporary standard}$ section: http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers#The_contemporary_standard – Git Gud Jan 26 '13 at 11:53
  • Yes, I wanted to mean the set of the first $n$ natural numbers. The set of the natural numbers, til the $n^{th}$ natural number –  Jan 26 '13 at 11:53
  • @user18921 I don't think any of the defitions is more useful than the other, one just has to be careful with $[0]$, if it is meant to be defined. – Git Gud Jan 26 '13 at 12:02
  • @GitGud: Actually $[0]=\emptyset$ is the one case where it makes no difference what you define $[n]$ to be. However, as a combinatorialist I can witness that it is much more useful to define $[n]={, i\in\mathbf N\mid i<n,} = {0,\ldots,n-1}$ than the more common ${1,\ldots,n}$. Also by the former convention and the "contemporary standard" you linked to, one has $[n]=n$ for all $n\in\mathbf N$, which some might consider cute. – Marc van Leeuwen Jan 26 '13 at 12:33
  • @MarcvanLeeuwen Thanks for your insight. – Git Gud Jan 26 '13 at 12:35
  • @MarcvanLeeuwen When I said that one had to be careful with $[0]$, I meant that $0$ might not be considered a natural number, that's all. – Git Gud Jan 26 '13 at 12:43
  • @MarcvanLeeuwen In my opinion, Von Neumann's approach to constructing the natural numbers is merely a "model", and should not be seen as the unique contemporary understanding of what the natural numbers "are." After all, there are other models. However, I'm interested in this comment: "as a combinatorialist I can witness that it is much more useful to define $[n]={0,...,n-1}$" I presume you speak of modular arithemtic? Or do other examples occur? One argument against beginning at $0$ is that sequences with $n$ terms become $(a_0,...,a_{n-1}),$ which is a tad awkward. – goblin GONE Jan 26 '13 at 13:50
  • @user18921: "Cute" implies it is not really a serious argument. I do not however speak of mudular arithmetic or positional number systems only. There are unimportant reasons to prefer $1$..$n$ (our childood habit to start counting at $1$, and the "..." notation) and numerous important reasons to prefer $0$..$n-1$. This margin is too narrow to be exhausitve, but consider $q$-numbers $[n]q=\sum{i\in[n]}q^i$ or the Vandermonde matrix $(X_i^j)_{i,j\in[n]}$ as examples. – Marc van Leeuwen Jan 26 '13 at 15:08

1 Answers1

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You can define $[n]$ however you want, so it can be true. Despite that, it is a common notation for the set $\{k\in \mathbb{N} : k\leq n\}$, yes.

This notation is used more often on Elementary Set Theory and Discrete Mathematics. Unfortunately analysts don't use it much. I've never seen it being used in Abstract Algebra or Linear Algebra either.

Git Gud
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    N.B. This definition requires asserting that $\mathbb{N}$ begin at $1$. – goblin GONE Jan 26 '13 at 11:53
  • Yes, that is correct. That seems to be the case judging by the OP's question. – Git Gud Jan 26 '13 at 11:54
  • I can certainly attest to analysts not using this notation a lot. I count myself as an analyst, and I have never seen it until now. Why this is unfortunate, I don't know. – Harald Hanche-Olsen Jan 26 '13 at 12:19
  • @HaraldHanche-Olsen Because it's more troublesome to write ${1, \dots , n}$ than $[n]$. – Git Gud Jan 26 '13 at 12:21
  • Yes, but I usually have more interesting things to write about. Brief notation is fine, but only if you use it often enough for it to be worth the bother. – Harald Hanche-Olsen Jan 26 '13 at 13:13
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    @HaraldHanche-Olsen If you talk about the components of vectors or matrices, or anything related to finite sequences, you'll probably find ample uses for this notation. – goblin GONE Jan 26 '13 at 13:52