A variable for the first 8 integers?

I wish to use algebra to (is the term truncate?) the set of positive integers to the first 8 and call it for example 'n'.

In order to define $r_n = 2n$ or similar.

This means:

$$r_0 = 0$$ $$r_1 = 2$$ $$\ldots$$ $$r_7 = 14$$

However there would not be an $r_8$.

edit: Changed "undefined" to "would not be", sorry about this.

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Please format using LaTeX code when possible. – Emily Aug 21 '12 at 13:56
You're going to have to give some more explanation. I don't see how $r_n=2n$ implies that $r_7=7$ and so on. What do you mean by a "variable for the first 8 integers"? – axblount Aug 21 '12 at 13:57
I would write something like $$r_i = \begin{cases} i,& \text{ if }0\leq i\leq 7, \\ undefined,& \text{otherwise}. \end{cases}$$ Or maybe $r = \mathrm{id}_{\mathbb N_0}|_{\overline{0,7}}$ since Russians use the notation $\overline{m,n}:= \{m,m+1,\dots,n-1,n\}$. – Ilya Aug 21 '12 at 13:57
Sorry axblount, fixed. – alan2here Aug 21 '12 at 14:08

3 Answers

You could simply write

"Let $r_n = 2n$ for all integers $n$ from $0$ to $7$."

or perhaps

"Let $r_n = 2n$ for $n \in \{0,1,2,\dotsc,7\}$."

or, using the compact notation suggested by Ilya,

"Let $r_n = 2n$ for $n \in \overline{0,7}$."

or even, if it's clear from context that $n$ is an integer,

"Let $r_n = 2n$ for $0 \le n \le 7$."

However, if going with Ilya's notation, be aware that many English-speaking readers may not be familiar with it, so you should define it yourself, e.g. "Let $\overline{n,m} = \{n,\,\dotsc,\,m\}$ denote the set of integers between $n$ and $m$ inclusive."

(Ps. What's wrong with MathJax's rendering of \overline, anyway? The lines in my examples above seem to curve upwards at the ends like this, while the first — but not the second — one in Ilya's comment curves the other way.)

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I cannot see a curve (neither up not down) on any of the lines. So maybe the problem is in the rendering of your browser (or maybe it's your monitor; do you happen to still have a CRT?) – celtschk Aug 26 '12 at 17:12
@celtschk: No, I have a flat screen display, and I see the ends clearly shifted by one pixel from the middle of the line. Here's a screenshot. I suppose it might be a browser bug, or something with the OS font rendering. (I'm currently using Chrome on WinXP, which is not my usual browser or OS.) – Ilmari Karonen Aug 26 '12 at 17:23
@alan2here: The notation $x \in y$ has a well defined meaning: "$x$ is an element of the set $y$". When used as in "for all $x \in y$", one would typically read it simply as "for all $x$ in $y$". – Ilmari Karonen Aug 26 '12 at 17:53
I supose even a related statement could be "n∈{0,2,4,…,14}". Mind if I edit "rn=2n if 0≤n≤7" into your answer so that we can unify this into one answer? – alan2here Aug 26 '12 at 17:58
The overline is straight here, on Windows 7 with Firefox 7. Thanks Ilmari Karonen, that makes a lot of sence reguarding ∈. I've removed the comment to which it referred because it was long, wrong and now just in the way. – alan2here Aug 26 '12 at 17:58

$r_n = 2n$ if $0 ≤ n ≤ 7$

Thanks Ilmari Karonen for the correction.

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I've never seen this particular notation, and I'd avoid it because it looks confusingly similar to standard set-builder notation. However, you could simply dispense with the curly braces and write "$r_n = 2n$ if $0 \le n \le 7$", which is perfectly understandable. – Ilmari Karonen Aug 26 '12 at 16:32

You've tagged this and but it's not entirely clear what you mean.

However, by these tags, perhaps you are referring to $\left(\Bbb Z_8, +\right)$?

In such a case, you have $r_1+r_7 = 1+_8 7 = 8 \mod 8 = 0$. So there is no $r_8$ per se; however, the re-definition of the symbols $r_0, r_1, \ldots$ is superfluous.

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I'm not sure about the tags, there wasn't one for algebra. – alan2here Aug 21 '12 at 14:09
It got re-tagged -- there is an algebra-precalculus tag that someone put there for you :) – Emily Aug 21 '12 at 15:16