Question 42
Let $\mathbb Z$ and $\mathbb Q$ denote the sets of integers and rational numbers. For subsets $X,Y$ of $\mathbb Q$, define
$$X+Y=\{x+y\mid x\in X,\ y\in Y\}.$$
(a) Can $\mathbb Z$ be partitioned into three nonempty sets $A,B,C$ such that $A+B,B+C,C+A$ are pairwise disjoint?
(b) Can $\mathbb Q$ be partitioned into three nonempty sets $A,B,C$ with this property?
$$X+Y=\{x+y\mid x\in X,\ y\in Y\}.$$
(a) Can $\mathbb Z$ be partitioned into three nonempty sets $A,B,C$ such that $A+B,B+C,C+A$ are pairwise disjoint?
(b) Can $\mathbb Q$ be partitioned into three nonempty sets $A,B,C$ with this property?