How to write a symbol showing that two non-negative numbers cannot be zero at the same time? Can I write:
$a,b \in \mathbb{R}_{\geq 0}, \sim(a=b=0)$
or
$a,b \in \mathbb{R}_{\geq 0}, \setminus(a=b=0)$
or
$A=\{(a,b):a,b\in \mathbb{R}_{\geq 0}\}\setminus(a=b=0)$?
This is intended for applied statisticians who might not be too familiar with analysis.
You might try $$ (a,b) \in (\mathbb R_{\geq 0} \times \mathbb R_{\geq 0})\setminus\{(0,0)\}.$$
But your last example is probably a more accessible style, which could be modified to either
$$ \{(a,b) : a, b \in\mathbb R_{\geq 0} \}\setminus\{(0,0)\}$$
or perhaps even more clearly
$$ \{(a,b) : a, b \in\mathbb R_{\geq 0}, \lnot (a = b = 0)\}.$$
Or perhaps just "$a, b$ two non-negative real numbers, not both zero" if you don't absolutely need it to be a formula.