I found that set notations can be very tricky. I have many dumb questions. Apparently, they also teach me the most. It made me think.

I am thinking: Since elements of $\displaystyle S$ are negative, where $\displaystyle S \subset \mathbb{Z}^-$, and that $\displaystyle -x\in S$, then $\displaystyle x$ ought to be positive, i.e. $\displaystyle x>0$, so $\displaystyle T=\{x\in Z: x>0\}$.

I contemplated about the question you asked--and of course, the poster immediately before you too helped me think. So, I wrote on a piece of paper

If $\displaystyle S\not= \emptyset$ and $\displaystyle S \subset \mathbb{N}$, then the elements of $\displaystyle S$ cannot be negative, despite $\displaystyle T$ being defined as

$\displaystyle T=\{x\in \mathbb{Z}: -x\in S\}$. Since

$\displaystyle -x >0$, it has to be nothing else but $\displaystyle x<0$,

so $\displaystyle T=\{x\in \mathbb{Z}: x<0\}$ as gmatt said.

I am a little too slow, but I will have another 70 years to learn.

Thanks to both of you.