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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
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
@GrijeshChauhan read the first paragraph on the $\textbf{The contemporary standard}$ section:… – Git Gud Jan 26 '13 at 11:53
up vote 5 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.

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.

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

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