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?

Question

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&#039;m not sure if the &quot;for some k&quot; part is correct. Should it be &quot;for all k&quot; instead?

Elias Keller · Accepted Answer

If you say &quot;      x    i    ∈  {  0  ,  k  ,  −  k  } for all   k  ∈      R  &quot; 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    .

Answered question

Answer & Explanation

New Questions in Precalculus