Proper summation notation

If you have the sum
[tex] v_x w_x + v_y w_y + v_z w_z[/tex]
then you want [itex] i \in \{ x,y,z \} [/itex], which says sum over every element of the set [itex] \{x,y,z \}[/itex]. If you wrote
[tex] \sum_{i=\{x,y,z \}} v_i w_i[/tex] what you really just wrote is
[tex] v_{ \{x,y,z \}} w_{ \{x,y,z \}}[/tex]
which is strange because it's not a sum, and because indices are unlikely (but might be) sets of variables

 

While one can interpret that, it would make more sense if associated an index set with your label set if you need to do this. So if instead of {x,y,z} just introduce the bijection {x,y,z} = {1,2,3} where the ith component of one set maps to the ith of the other.

This is just my opinion, but the reason is mostly conventional because its easier for everyone with a simple mathematics background to understand and causes less confusion.