a) Show that the set
![\mathbb{Q}^{ + }](/media/m/6/8/1/68175b3a50ef693f217095e3c7186a3f.png)
of all positive rationals can be partitioned into three disjoint subsets.
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
satisfying the following conditions:
![BA = B; B^2 = C; BC = A;](/media/m/3/5/5/355847a3ad78909acd845584b1efce04.png)
where
![HK](/media/m/1/6/c/16c38f8d1bdbcdca5b5573926d5999bf.png)
stands for the set
![\{hk: h \in H, k \in K\}](/media/m/8/d/1/8d1df8ead6964d929ccdd9f85b74eea5.png)
for any two subsets
![H, K](/media/m/4/a/4/4a43e626cbdbdbccd58d33c1d5f84928.png)
of
![\mathbb{Q}^{ + }](/media/m/6/8/1/68175b3a50ef693f217095e3c7186a3f.png)
and
![H^2](/media/m/f/d/e/fde2b7d72887d56a408ea40db9561878.png)
stands for
b) Show that all positive rational cubes are in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
for such a partition of
c) Find such a partition
![\mathbb{Q}^{ + } = A \cup B \cup C](/media/m/0/c/3/0c3413559c3a245f55d8506ee9a0e97b.png)
with the property that for no positive integer
![n \leq 34,](/media/m/e/2/7/e27b0bc416013351c7ab32b24a2fd5d5.png)
both
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and
![n + 1](/media/m/3/6/d/36dc98984132471cc8b030d766fd893a.png)
are in
![A,](/media/m/8/6/5/865743fba196abcc2b01372b2f0205c1.png)
that is,
%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, $$\text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34.$$