MathDB

Can the set of positive rationals be split into two nonempty disjoint subsets

\mathbb Q_1

and

\mathbb Q_2

, such that both are closed under addition, i.e.

p+q\in\mathbb Q_k

for every

p,q\in\mathbb Q_k

k=1,2

? Can it be done when addition is exchanged for multiplication, i.e.

p\cdot q\in\mathbb Q_k

for every

p,q\in\mathbb Q_k

k=1,2

Problem Statement