Sets of real numbers which are anti-closed under addition

Question

Let $(M,*)$ be a magma, that is, a set with a binary operation. I define a subset $S$ of $M$ to be anti-closed under $*$ iff for all $x,y$ in $S$, $x*y \notin S$. For example, the set of negative real numbers is anti-closed under multiplication, and the set of odd integers is anti-closed under addition. I am interested in the question of whether is a partition of $\mathbb{R}$ into two disjoint non-empty sets $A$ and $B$, such that $A$ is closed under addition, but $B$ is anti-closed under addition. There certainly is such a partition in the analogous case of multiplication, namely, the negative reals and the nonnegative reals.

Answer 1

That is not possible. Suppose $A$ is the subset which is closed, and $B$ is the subset which is anti-closed. Then for any $x\in B$, you have $x+x\in A$. But also for any $x\in A$, $x+x\in A$. So in the end for any $x\in \mathbb{R}$, you get $2x\in A$.

Then for any $y\in \mathbb{R}$, you must have $y\in A$ since $y=2x$ with $x=y/2$. So $A=\mathbb{R}$.