Is comma notation rigorous way to list solutions?

Question

If I like to say that the equation is has roots $x=1$ and $x=2$ then is it formally correct to say that $x=1,2$ or should one use the notation $x\in\{1,2\}$?

Answer 1

I would say there is no need to make nonessential distinction, unless confusion may arise.

The comma appearing in a phrase such as $x = 1,2$ is usually understood as "or". In this sense to write $x = 1,2$ or to write $x \in \{ 1,2 \}$ are equally clear. (And the latter one may look "spurious"...)

But if instead what you are working with already employs too may commas, then maybe the set notation is preferable.

Answer 2

The question is little bit soft. I find following variants and variations useful. Worth noting, I am not a native speaker.

1. Naming the roots: A number $x^*$ is a root of $x^2 -1$ if (and only if) ...

   • $x^* = -1$ or $x^* = 1$
   • $x^* =-1, 1$
   • $x^*\in \{-1, 1\}$

2. Not naming:

   • The roots of $x^2 - 1$ are (exactly) $-1,1$.
   • The set of roots of $x^2 - 1$ is $\{-1, 1\}$.