mathematics - A sum and product from first principles

A question was put before Iron Man and Miss Tress. They were allowed to choose from the set $\mathbb{C}$ of complex numbers and asked to find a subset satisfying the conditions below:

• sum is zero


• product is one

Can you provide two suitably different solutions for Iron Man and Miss Tress?

Answer

My guess is that the answer you are after is from first principles
Iron Man

$\{0+i, 0-i\}$
Iron Man $\rightarrow$ IM $\rightarrow \Bbb{Im}$

...or any extension thereof, such as $\{0+ai,0-ai,0+\frac{i}a,0-\frac{i}a\}\forall a \in A \subseteq \Bbb{R}$

Miss Tress

$\emptyset$
Miss Tress $\rightarrow$ MT $\rightarrow$ "em tee" $\rightarrow$ empty