MathDB

Let

\mathbb{Z}

and

\mathbb{Q}

be the sets of integers and rationals respectively. a) Does there exist a partition of

\mathbb{Z}

into three non-empty subsets

A,B,C

such that the sets

A+B, B+C, C+A

are disjoint? b) Does there exist a partition of

\mathbb{Q}

into three non-empty subsets

A,B,C

such that the sets

A+B, B+C, C+A

are disjoint?
Here

X+Y

denotes the set

\{ x+y : x \in X, y \in Y \}

, for

X,Y \subseteq \mathbb{Z}

and for

X,Y \subseteq \mathbb{Q}

Problem Statement