Školjka

%V0 a) Show that the set $\mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $A,B,C$ satisfying the following conditions: $$BA = B; B^2 = C; BC = A;$$ where $HK$ stands for the set $\{hk: h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb{Q}^{ + }$ and $H^2$ stands for $HH.$ b) Show that all positive rational cubes are in $A$ for such a partition of $\mathbb{Q}^{ + }.$ c) Find such a partition $\mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $n \leq 34,$ both $n$ and $n + 1$ are in $A,$ that is, $$\text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.$$