Why does the definition for the multiplication of dedkind cuts explicitly include the negative rationals?

Question

If A and B are both dedekind cuts. Then $A \times B=\{ab \mid a \in A, b \in B, a \geq 0, b \geq 0 \} \cup \{q \in \mathbb{Q} \mid q <0 \}$. Can someone explain why this definition doesn't work: $A \times B=\{ab \mid a \in A, b \in B\}$.

Answer 1

$\{ab|a\in A, b\in B\}$ is not always a Dedekind cut. For example, if $A=B=\{q\in \mathbb Q: q < -2\}$, then $\{ab|a\in A, b\in B\} = \{q\in Q: q > 4\}$