 # I have a set of nonzero vectors in RR^n, where for each vector, its nonzero elements have the same magnitude. For example, when n=4, (1,0,1,0) and (−3,3,0,3) are in this set, while (1,2,3,2) and (0,0,0,0) are not. The formal notation I came up for it is {x in RR^n∣x != 0 text( and ) x_i in {0,k,−k} text( for some ) k in R} but I'm not sure if the "for some k" part is correct. Should it be "for all k" instead? ghulamu51 2022-09-19 Answered
I have a set of nonzero vectors in ${\mathbb{R}}^{n}$, where for each vector, its nonzero elements have the same magnitude. For example, when $n=4$, $\left(1,0,1,0\right)$ and $\left(-3,3,0,3\right)$ are in this set, while $\left(1,2,3,2\right)$ and $\left(0,0,0,0\right)$ are not.
The formal notation I came up for it is

but I'm not sure if the "for some k" part is correct. Should it be "for all k" instead?
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If you say "${x}_{i}\in \left\{0,k,-k\right\}$ for all $k\in \mathbb{R}$" then you say that ${x}_{i}$ belongs to every $\left\{0,k,-k\right\}$, which is not possible unless ${x}_{i}=0.$. And you do not want that for every i.
What you wrote is OK but could be mis-interpreted.
You could also write $\left\{x\in {\mathbb{R}}^{n}:x\ne 0\wedge \mathrm{\exists }k\in \mathbb{R}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall }i\phantom{\rule{thinmathspace}{0ex}}\left({x}_{i}\in \left\{0,k,-k\right\}\right)\right\}.$
If you want to be annoyingly rigorous but still right, you could write ${\cup }_{k\in \mathbb{R}}\left(\left\{0,k,-k{\right\}}^{n}\right)\setminus \left\{0{\right\}}^{n}.$

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