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 R 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 R n x 0  and  x i { 0 , k , k }  for some  k R } ,
but I'm not sure if the "for some k" part is correct. Should it be "for all k" instead?
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Answers (1)

Elias Keller
Answered 2022-09-20 Author has 8 answers
If you say " x i { 0 , k , k } for all k R " then you say that x i belongs to every { 0 , k , k }, 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 { x R n : x 0 k R i ( x i { 0 , k , k } ) } .
If you want to be annoyingly rigorous but still right, you could write k R ( { 0 , k , k } n ) { 0 } n .

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