$A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$? 
If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$?

I think the answer is yes, but I can't construct it. If $A$ is odd number set, $B$ is even number set, then $I$ is integer set $\mathbb{Z}$. But how can the set of irrational numbers and the set of other rational numbers be put into the set $A$? I don't understand.

$A$ can be $ \ldots \cup [-2,-1) \cup [1,2) \cup [4,5) \cup \ldots$ and 
$B$ will then be $ \ldots \cup [-1,1) \cup [2,4) \cup [5,7) \cup \ldots$
