Question 9
(IND 4)
(a) Show that the set $\mathbb Q^+$ of all positive rational numbers can be partitioned into three disjoint subsets $A,B,C$ satisfying the following conditions:
$$BA=B,\qquad B^2=C,\qquad BC=A,$$
where $HK$ stands for the set $\{hk\mid h\in H,k\in K\}$ 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 $n\le34$ are both $n$ and $n+1$ in $A$; that is,
$$\min\{n\in\mathbb N\mid n\in A,n+1\in A\}\gt 34.$$
(a) Show that the set $\mathbb Q^+$ of all positive rational numbers can be partitioned into three disjoint subsets $A,B,C$ satisfying the following conditions:
$$BA=B,\qquad B^2=C,\qquad BC=A,$$
where $HK$ stands for the set $\{hk\mid h\in H,k\in K\}$ 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 $n\le34$ are both $n$ and $n+1$ in $A$; that is,
$$\min\{n\in\mathbb N\mid n\in A,n+1\in A\}\gt 34.$$