Let $A,B\subset\mathbb{R}$ nonempty. Define
\begin{align*}
A+B:&=\big\{x\in\mathbb{R}\,|\,\exists\,(a\in{A},b\in{B})\; {\text{such that}} \, x=a+b\big\}\\
&=\big\{a+b\,|\,a\in{A},b\in{B}\big\}\,.
\end{align*}
- Let $A=[-1,2]=B$. What is $A+B$?
- Let $A=B=\mathbb{N}$. What is $A+B$?
- Let $A=B=\mathbb{Z}$. What is $A+B$?
- Let $A=B=\mathbb{Q}$. What is $A+B$?
- Let $A=B=\mathbb{R}\setminus\mathbb{Q}$. What is $A+B$?