First, a definition. Let $(M;*)$ be a magma, and let $S$ be a subset of $M$. I define $S$ to be anti-closed under $*$ if whenever $x,y \in S$, then $x*y \notin S$. For example, the odd integers are anti-closed under addition. Now, the question itself. Does there exist a partition of the complex numbers $\mathbb{C}$ into two disjoint nonempty subsets $S$ and $T$, such that $S$ is closed under multiplication, and $T$ is anti-closed under multiplication? I am asking this question because in the real numbers, there is such a partition, namely the non-negative numbers and the negative numbers.
Asked
Active
Viewed 640 times
1 Answers
20
Take $z \in \Bbb C$ and consider $w\in \Bbb C$ such that $w^2=z$.
If $w\in S$ then $w^2\in S$. If $w\in T$ then $w^2\not\in T$, so $w^2\in S$.
In any case $z=w^2\in S$.
This means $\Bbb C = S$ and this proves there is no such partition of $\Bbb C$.
jjagmath
- 24,713