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It's true that I can write the set of the $n^{th}$ natural number as $[ n] $?

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

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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
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@GrijeshChauhan read the first paragraph on the $\textbf{The contemporary standard}$ section: en.wikipedia.org/wiki/… –  Git Gud Jan 26 '13 at 11:53
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1 Answer

up vote 3 down vote accepted

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. That notation is used more often on Elementary Set Theory and Discrete Mathematics. Unfortunately analysts don't use it much.

EDIT: I've never seen it being used in Abstract Algebra or Linear Algebra either.

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N.B. This definition requires asserting that $\mathbb{N}$ begin at $1$. –  user18921 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
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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. –  user18921 Jan 26 '13 at 13:52
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