--let $A, B \in \mathbb{R}$, with $0\leq A$ and $0 \leq B$, $A \star B :=\{q\in\Bbb Q\mid q<0\}\cup\{a\cdot b\mid a\in A\wedge b\in B\wedge a\ge 0\wedge b\ge 0\}$"
this definition is correct:
--let $ A, B, C \in \mathbb{R}$, $ C $ is product of $ A $ and $ B $, $ C \triangleq A \cdot B $, if $C=\begin{cases} |A| \star |B|, & \mbox{if } (0 \leq A \wedge 0 \leq B )\mbox{ OR }(A \leq 0 \wedge B \leq 0 )\\ -(|A| \star |B|), & \mbox{if } (A \leq 0 \wedge 0 \leq B )\mbox{ OR } (0 \leq A \wedge B \leq 0 ) \end{cases}$
???
Thanks in advance!!
P.S.= OR, the question is edited!
There's a major problem with your original definition: $A\star B$ will not be a real number as written! Instead, you need $$A\star B:=\{q\in\Bbb Q\mid q<0\}\cup\{a\cdot b\mid a\in A\wedge b\in B\wedge a\ge 0\wedge b\ge 0\}.$$ (Edit: This has been fixed.)
There's also a bit of an issue with your general definition, as it doesn't cover the case $0\cdot 0.$ The XORs rule out exactly this case. Switch them to $\vee$s and that issue will go away.