Dividing real number into two sets

Question

I wonder the following question:

Is there a partition of $\mathbb{R}$ into two disjoint subsets $A$ and $B$ such that $B$ satisfies $B=A+A$, namely $B=$ {$ x+y|x,y \in A $}?. Here, "partition into disjoint sets" means $A\cup B=\mathbb{R}, A\cap B=\varnothing$.

Apparently, $\mathbb{Z}$ has this kind of separation (you may take $A=${$2k+1|k\in\mathbb{Z}$},$B=${$2k|k\in\mathbb{Z}$}), and it seems to me, that due to some continuity property of $\mathbb{R}$, such separation does not exist. However, the condition ($B=A+A$) given does not tell much about $A$ and $B$ from a global viewpoint.

Thanks in advance!

Here's a quick guide (if nothing else, read up the part on "avoiding no-clue questions"). — 5xum, May 20 '22 at 11:54
What do you mean "separate $\Bbb{R}$"? Are you just looking for two disjoint subsets $A$ and $B$ such that $B = A + A$, or are they supposed to partition $\Bbb{R}$? — Theo Bendit, May 20 '22 at 12:01
What is the source of this problem? Further, it this problem is from a book or class, what is the exact wording with which this problem was posed? — user2661923, May 20 '22 at 12:04
I've edited the question to make it clearer. Sorry for the previous inconvenience. — Wembley Inter, May 20 '22 at 12:07
"Two disjoint subsets $A$ and $B$ such that $B$ satisfies $B=A+A$" and "$A\cup B=\mathbb{R}, A\cap B=\varnothing$" are far from been the same thing. So your "In other words" makes no sense. — jjagmath, May 20 '22 at 13:40

score 4 · Accepted Answer · answered May 20 '22 at 13:28

4

$$A=\left( \bigcup_{n\in\mathbb{N}}\left(-3n+1,-3n+2\right] \right)\\ \cup \left( \bigcup_{n\in\mathbb{N}}\left[3n-2,3n-1\right) \right) $$

$$B= \left( \bigcup_{n\in\mathbb{N}}\left(-3n-1,-3n+1\right] \right)\ \cup (-1,1) \cup\left( \bigcup_{n\in\mathbb{N}}\left[3n-1,3n+1\right) \right)\ $$

answered May 20 '22 at 13:28

Adam Rubinson

20,052

2

Thanks! It seems to me that $A=\bigcup_{n\in\mathbb{Z}} [3n+1,3n+2)$, $B=\bigcup_{n\in\mathbb{Z}} [3n-1,3n+1)$ still satisfies? – Wembley Inter May 20 '22 at 14:00

Dividing real number into two sets

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