Is this a valid notation in set theory?

Question

I have three sets, $A:=\{a_1,\ldots,a_n\}$, $B:=\{b_1,\ldots,b_n\}$ and $C:=\{0\}$.

Let $D:=A\times B \cup C$. I do not know if this is a valid notation?

For example, Is $(0,b_2)\in D$? Or, is $(a_1,0)\in D$? Or, is $(0,0)\in D$?

I want only $(0,0)$ to be in $D$. How to represent it? Is $C=\{(0,0)\}$ enough?

Answer 1

$C=\{(0,0)\}$ will work.

If it suits your needs to keep $C=\{0\}$, then let $D:= A\times B \cup C\times C$, which you can write $D:= (A\times B) \cup (C\times C)$ if the former looks ambiguous to you.