Let $S$ be a subset of $\mathbb{R}$ such that $S \not = \emptyset$ and $S \not = \mathbb{R}$. Define $S^c =\{x\in \mathbb{R}:x \not \in S\}$. Suppose that $S$ is closed under addition AND closed under the additive inverse. Prove $S^c$ is not closed under addition. Find an example where $S$ and $S^c$ are both closed under multiplication.
Can someone help me with this? I do not know what to do for this, I do not even know where to start. I was given a warm-up that is supposed to help that is to prove the result for $S = \mathbb{Q}$ (so that $S^c$ is the set of irrational numbers), but I do not even know how to do this. Can someone offer me some guidance, please?
$\Bbb{Q} \ni 2 = \sqrt{2} + (2 - \sqrt{2}) \in (\Bbb{R} \smallsetminus \Bbb{Q}) + (\Bbb{R} \smallsetminus \Bbb{Q})$. In fact any $s \in S$ can be written as $c + (s - c)$ for any $c \in S^c$, for which $s - c \in S^c$ (since $S$ is closed under addition).